An Easton-like Theorem for Zermelo-Fraenkel Set Theory with the Axiom of Dependent Choice
aa r X i v : . [ m a t h . L O ] D ec An Easton-like Theorem for Zermelo-Fraenkel SetTheory with the Axiom of Dependent Choice
Anne Fernengel and Peter Koepke
Abstract
We show that in the theory ZF + DC + “for every cardinal λ , the set [ λ ] ℵ is well-ordered”( AX ), the θ -function measuring the surjective size of thepowersets ℘( κ ) can take almost arbitrary values on any set of uncountablecardinals. This complements our results from [FK16], where we prove that in ZF (without DC ), any possible behavior of the θ -function can be realized; andanswers a question of Shelah in [She16], where he emphasizes that ZF + DC + AX is a “reasonable” theory, where much of set theory and combinatorics ispossible. Introduction.
The
Continuum Function κ ↦ κ , which maps any cardinal κ to the cardinality of its power set ℘( κ ) , has been investigated since the early begin-nings of set theory. In 1878, Georg Cantor advanced the Continuum Hypothesis(CH) ([Can78]), which states that 2 ℵ = ℵ , i.e. there is no set the cardinalityof which is strictly between ℵ and the cardinality of ℘(ℵ ) . The Continuum Hy-pothesis was among the first statements that were shown to be independent of ZF :Firstly, Kurt G¨odel proved in [G¨o40] that CH holds in the constructible universe L . On the other hand, when Paul Cohen invented the method of forcing in [Coh63]and [Coh64], he proved that 2 ℵ could be any cardinal κ of uncountably cofinality.The Generalized Continuum Hypothesis (GCH) was formulated by FelixHausdorff in [Hau14] (with earlier versions in [Hau07] and [Hau08]). Assertingthat 2 κ = κ + for all cardinals κ , it is a global statement about possible behaviorsof the Continuum Function. The GCH is consistent with
ZF C , since it holdstrue in L (see [G¨o40]). In 1970, William Easton proved the following global result:For regular cardinals κ , any reasonable behavior of the 2 κ -function is consistentwith ZF C ([Eas70]). In his forcing construction, he takes “many” Cohen forc-ings and combines them in a way that was henceforth known as the
Easton product .For singular cardinals κ , however, the situation is a lot more involved, since thevalue of 2 κ for singular κ is strongly influenced by the behavior of the ContinuumFunction below. The Singular Cardinals Hypothesis (SCH) implies that forany singular cardinal κ with the property that 2 λ < κ holds for all λ < κ , it alreadyfollows that 2 κ = κ + . 1t turned out that the negation of the SCH is tightly linked with the existenceof large cardinals. Among the first results in this direction was a theorem byMenachem Magidor ([Mag77b] and [Mag77a]), who proved that, assuming a hugecardinal, it is possible that
GCH first fails at a singular strong limit cardinal.On the other hand, Ronald Jensen and Keith Devlin proved in [DJ75] that thenegation of 0 ♯ implies SCH . Motik Gitik determined in [Git89] and [Git91] theconsistency strength of ¬ SCH being the existence of a measurable cardinal λ ofMitchell order σ ( λ ) = λ ++ .There are many more results about possible behaviors of the Continuum Functionstarting from large cardinals. For instance, by a theorem of Carmi Merimovich,the theory ZF C + ∀ κ ( κ = κ + n ) is consistent for each n < ω ([Mer07]).On the other hand, Silver’s Theorem ([Sil75]) states that for any singular cardinal κ of uncountable cofinality such that 2 λ = λ + holds for all λ < κ , it already follows that2 κ = κ + . Hence, the SCH holds if it holds for all singular cardinals of countablecofinality. This result was extended by Fred Galvin and Andr´as Hajnal shortlyafter ([GH75]).Another prominent example concerning upper bounds on the Continuum Functionof singular cardinals is the following theorem by Saharon Shelah ([She94]): If ℵ n < ℵ ω for all n < ω , then ℵ ω < ℵ ω . This makes clear that there are significant constraints on possible behaviors of theContinuum Function in
ZF C . In particular, a result like Easton’s Theorem cannot exist for singular cardinals.All of the aforementioned results essentially involve the Axiom of Choice. Without AC , however, there is a lot more possible. In [AK10], Arthur Apter and PeterKoepke examine the consistency strength of the negation of SCH in ZF + ¬ AC .In this context, one has to distinguish between injective and surjective failures.An injective failure of SCH at κ is a model of ZF + ¬ AC with a singular cardinal κ such that GCH holds below κ , but there is an injective function ι ∶ λ ↪ ℘( κ ) for some λ ≥ κ ++ . A surjective failure of SCH at κ is a model of ZF + ¬ AC with a singular cardinal κ such that GCH holds below κ , but there is a surjectivefunction f ∶ ℘( κ ) → λ for some cardinal λ ≥ κ ++ .On the one hand, Arthur Apter and Peter Koepke construct injective failures of the SCH at ℵ ω , ℵ ω and ℵ ω that would contradict the theorems by Shelah and Silverin the ZF C -context, but have fairly mild consistency strengths in ZF + ¬ AC . Onthe other hand, regarding a surjective failure of the SCH , they prove that for every α ≥ ZF C together with the existence of a measurable cardinal is equiconsistentwith the theory ZF + ¬ AC + “GCH holds below ℵ ω ” + “there exists a surjective function f ∶ [ℵ ω ] ω → ℵ ω + ” . It follows that also without the Axiom of Choice, injective failures of the
SCH are inevitably linked to large cardinals. Regarding surjective failures however, it isnot possible to replace in their argument the surjective function f ∶ [ℵ ω ] ω → ℵ ω + by a surjection f ∶ ℘(ℵ ω ) → ℵ ω + , so the following question remained: Is it pos-sible, for λ ≥ ℵ ω + , to construct a model of ZF + ¬ AC where GCH holds below ℵ ω and there is a surjection f ∶ ℘(ℵ ω ) → λ without any large cardinal assumptions?This question was positively answered by Motik Gitik and Peter Koepke in [GK12],where a ground model V ⊧ ZF C + GCH with a cardinal λ ≥ ℵ ω + is extendedvia symmetric forcing, in a way such that the extension N = V ( G ) preserves all V -cardinals, the GCH holds in N below ℵ ω , and there is a surjective function f ∶ ℘(ℵ ω ) → λ .More generally, in the absence of the Axiom of Choice, where the power set ℘( κ ) of a cardinal κ is generally not well-ordered, the “size” of ℘( κ ) can be measuredsurjectively by the θ -function θ ( κ ) ∶ = sup { α ∈ Ord ∣ ∃ f ∶ ℘( κ ) → α surjective function } , which generalizes the value θ ∶ = θ ( ω ) prominent in descriptive set theory. In the ¬ AC -context, the θ -function provides a surjective substitute for the ContinuumFunction κ ↦ κ .One can show that in the model constructed in [GK12], it follows that indeed, θ (ℵ ω ) = λ . This gives rise to the question whether the behavior of the θ -functionmight be essentially undetermined in ZF .In [FK16], we could prove that indeed, the only constraints on the θ -function in ZF are the obvious ones: weak monotonicity and θ ( κ ) ≥ κ ++ for all κ . In otherwords: In ZF , there is an analogue of Easton’s Theorem for regular and singularcardinals. Theorem. ([FK16]) Let V be a ground model of ZF C + GCH with a function F on the class of infinite cardinals such that the following properties hold: • ∀ κ F ( κ ) ≥ κ ++ • ∀ κ, λ ( κ ≤ λ → F ( κ ) ≤ F ( λ )) .Then there is a cardinal-preserving extension N ⊇ V with N ⊧ ZF such that θ N ( κ ) = F ( κ ) holds for all κ . In our construction, we introduce a forcing notion P whose elements p are functionson trees ( t, ≤ t ) with finitely many maximal points. The trees’ levels are indexedby cardinals, and on any level κ , there are finitely many vertices ( κ, i ) with i < ( κ ) . For successor cardinals κ + , the value p ( κ + , i ) is a partial 0-1-function onthe interval [ κ, κ + ) . Thus, for any condition p and ( κ, i ) ∈ dom p , it follows that ⋃ { p ( ν + , j ) ∣ ( ν + , j ) ≤ t ( κ, i )} is a partial function on κ with values in { , } . Sincewe do not allow splitting at limits for the trees, it follows that this forcing indeedadds F ( κ ) -many new κ -subsets for every cardinal κ .We discussed in [FK16, Chapter 6] whether it might be possible to modify ourconstruction and use trees with countably many maximal points, which could re-sult in a countably closed forcing, giving rise to a symmetric extension N with N ⊧ ZF + DC . However, this modification would have a drastic impact on theforcing notion, destroying a crucial homogeneity property. Seemingly, our con-struction relies on certain finiteness properties, and hence does not opt for a sym-metric extension N with N ⊧ DC .In this paper, we treat the question whether the θ -function is still essentially un-determined, if we consider a model N ⊧ ZF + DC . Starting from a ground model V ⊧ ZF C + GCH , we construct a cardinal-preserving symmetric extension N ⊇ V with N ⊧ ZF + DC , and this time, we generalize the forcing in [GK] to obtain acountably closed forcing notion P .The Axiom of Dependent Choice (DC) , introduced by Paul Bernays in 1942([Ber42]), states that for every nonempty set X with a binary relation R such thatfor all x ∈ X there is y ∈ X with yRx , it follows that there is a sequence ( x n ∣ n < ω ) in X such that x n + Rx n for all n < ω .When dealing with real numbers, surprisingly often DC is sufficient (instead of thefull Axiom of Choice), and the theory ZF + DC provides an interesting frameworkfor real analysis.Concerning combinatorial set theory however, investigations under ZF + DC seemed rather hopeless in the first place. A crucial step in the other directionwas a paper by Saharon Shelah ([She97]) with the main result in ZF + DC thatwhenever µ is a singular cardinal of uncountable cofinality such that ∣ H ( µ )∣ = µ ,then µ + is regular and non-measurable. In the case that the power sets ℘( α ) arewell-orderable for all α < ℵ ω with ∣ ⋃ α <ℵ ω ℘( α )∣ = ℵ ω , it essentially follows thatalso ℘(ℵ ω ) is well-orderable.Subsequently (see [She10] and [She16]), Shelah showed that much of pcf -theory ispossible in ZF + DC , if an additional axiom is adopted: ( AX ) For every cardinal λ, the set [ λ ] ℵ can be well-ordered .Starting from a ground model V ⊧ ZF C , any symmetric extension by countablyclosed forcing yields a model of ZF + DC + AX (see [She10, p.3 and p.15]).In [She16, 0.1], Shelah concludes that ZF + ZF + AX is “ a reasonable theory,for which much of combinatorial set theory can be generalized ”. For example, he4roves a rather strong version of the pcf -theorem, gives a representation of λ κ for λ >> κ , and proves that certain covering numbers exist. Concerning appli-cations to cardinal arithmetic, Shelah emphasizes that we “cannot say much” onpossible cardinalities of ℘( κ ) , and suggests to investigate possible cardinalities of ( κ ℵ ∣ κ ∈ Card ) rather than (℘( κ ) ∣ κ ∈ Card ) ([She10, p. 2]). In [She16, 0.2] heasks, referring to [GK12], if there are any bounds on θ ( κ ) for singular cardinals κ in ZF + DC + AX .In this paper, we give a negative answer to this question. We prove that in ZF + DC + AX again, the only restrictions on the θ -function on a set of un-countable cardinals are the obvious ones:Given a ground model V ⊧ ZF C + GCH with “reasonable” sequences of uncount-able cardinals ( κ η ∣ η < γ ) and ( α η ∣ η < γ ) for some ordinal γ , we construct acardinal-preserving symmetric extension N ⊇ V with N ⊧ ZF + DC + AX , suchthat θ N ( κ η ) = α η holds for all η < γ .More precisely, we prove: Theorem.
Let V be a ground model of ZF C + GCH with γ ∈ Ord and sequencesof uncountable cardinals ( κ η ∣ η < γ ) and ( α η ∣ η < γ ) , such that ( κ η ∣ η < γ ) isstrictly increasing and closed, and the following properties hold: • ∀ η < η ′ < γ α η ≤ α η ′ , i.e. the sequence ( α η ∣ η < γ ) is increasing, • ∀ η < γ α η ≥ κ + + η , • ∀ η < γ cf α η > ω , • ∀ η < γ ( α η = α + → cf α > ω ) .Then there is a cardinal- and cofinality-preserving extension N ⊇ V with N ⊧ ZF + DC + AX such that that θ N ( κ η ) = α η holds for all η < γ . Our paper is structured as follows: In Chapter 1, we briefly review some basicdefinitions and fact about forcing and symmetric extensions. In Chapter 2, we stateour theorem and then argue why it is not possible to drop any of our requirementson the sequences ( κ η ∣ η < γ ) and ( α η ∣ η < γ ) .In Chapter 3, we introduce our forcing notion which blows up the power sets (℘( κ η ) ∣ η < γ ) according to ( α η ∣ η < γ ) ; and in Chapter 4 construct a group A of P -automorphisms and a normal filter F on A , giving rise to our symmetricextension N ∶ = V ( G ) . In Chapter 5, we prove that sets of ordinals located in N canbe captured in fairly “mild” V -generic extensions, which implies that all cardinalsand cofinalities are N - V -absolute. It remains to show that indeed, θ N ( κ η ) = α η holds for all η < γ . The first part, θ N ( κ η ) ≥ α η , follows by our construction(see Chapter 6.1), while for the second part, θ N ( κ η ) ≤ α η , we assume that there5as a surjective function f ∶ ℘( κ η ) → α η in N , and obtain a contradiction bycapturing a restricted version f β in an intermediate generic extension V [ G β ] whichis sufficiently cardinal-preserving (see Chapter 6.2 and 6.3). We treat the values θ ( λ ) for cardinals λ in the “gaps” λ ∈ ( κ η , κ η + ) and λ ≥ κ γ ∶ = sup { κ η ∣ η < γ } inChapter 6.4 and 6.5, respectively, and show that they are the smallest possible.We conclude with several remarks in Chapter 7. In this chapter, we briefly establish some basic notations and terminology aboutforcing and symmetric extensions.We write
Ord and
Card for the class of all ordinals and cardinals, respectively.For our construction, we work with a countable transitive model V of ZF C ,our ground model , with a notion of forcing ( P , ≤ , ) ∈ V . The class of P -names , N ame ( P ) , is defined recursively as follows: Name ( P ) ∶ = ∅ , Name α + ( P ) ∶ = ℘( Name α ( P )× P ) for α ∈ Ord, and Name λ ( P ) ∶ = ⋃ α < λ Name α ( P ) whenever λ is a limit ordinal.Then Name ( P ) ∶ = ⋃ α ∈ Ord
Name α ( P ) . As usual, we denote P -names as ˙ x . For ˙ x ∈ Name ( P ) with ˙ x ∈ Name α + ( P ) ∖ Name α ( P ) , we write rk ( ˙ x ) ∶ = α for the rank of ˙ x .Let G be a V -generic filter on P . The V -generic extension by G is V [ G ] ∶ = { ˙ x G ∣ ˙ x ∈ Name ( P )} , where the interpretation function ( ⋅ ) G is defined recursively on theName α ( P ) -hierarchy as usual. Then V [ G ] is a transitive model of ZF C with V ⊆ V [ G ] .For an element a of the ground model, its canonical name is denoted by ˇ a . When-ever x , y ∈ V [ G ] with x = ˙ x G , y = ˙ y G , there is a canonical P -name for the pair ( x, y ) , which will be abbreviated by OR P ( ˙ x, ˙ y ) .Regarding the construction of symmetric extensions, we follow the presentation in[Dim11], where the standard method for forcing with Boolean values as describedin [Jec06] and [Jec73] is translated to partial orders.For the rest of this chapter, fix a partial order P . Let Aut ( P ) denote the automor-phism group of P . Any π ∈ Aut ( P ) can be extended to an automorphism ̃ π of thename space N ame ( P ) by the following recursive definition: ̃ π ( ˙ x ) ∶ = { (̃ π ( ˙ y ) , πp ) ∣ ( ˙ y, p ) ∈ ˙ x } . We confuse any π ∈ Aut ( P ) with its extension ̃ π (which does not lead to ambigu-ity). For any canonical name ˇ a and π ∈ Aut ( P ) , it follows recursively that π ( ˇ a ) = ˇ a .The forcing relation ⊩ can be defined in an outer model as follows:6f ϕ ( v , . . . , v n − ) is a formula of set theory and ˙ x , . . . , ˙ x n − ∈ Name ( P ) , then p ⊩ ϕ ( ˙ x , . . . , ˙ x n − ) if for every V -generic filter G on P , it follows that V [ G ] ⊧ ϕ ( ˙ x G , . . . , ˙ x Gn − ) .The forcing relation ⊩ can also be defined in the ground model V , and the forcingtheorem holds: Theorem (Forcing Theorem) . If ϕ ( v , . . . , v n − ) is a formula of set theory and G a V -generic filter on P , then for every ˙ x , . . . , ˙ x n − ∈ Name ( P ) , it follows that V [ G ] ⊧ ϕ ( ˙ x G , . . . , ˙ x Gn − ) if and only if there is a condition p ∈ P with p ⊩ ϕ ( ˙ x , . . . , ˙ x n − ) . Let now A ⊆ Aut ( P ) denote a group of P -automorphisms. A normal filter on A isa collection F of subgroups B ⊆ A such that F ≠ ∅ , F is closed under supersetsand finite intersections, and for any B ∈ F and π ∈ A , it follows that the conjugate π − Bπ is contained in F , as well.An important property of P -automorphisms is the symmetry lemma : Lemma (Symmetry Lemma) . For a formula of set theory ϕ ( v , . . . , v n − ) , an auto-morphism π ∈ Aut ( P ) and P -names ˙ x , . . . , x n − , it follows that p ⊩ ϕ ( ˙ x , . . . , ˙ x n − ) if and only if πp ⊩ ϕ ( π ˙ x , . . . , π ˙ x n − ) . The proof is by induction over the complexity of ϕ .Fix a normal filter F on A . A P -name ˙ x is symmetric if the stabilizer group { π ∈ A ∣ π ˙ x = ˙ x } is contained in F . Recursively, a name ˙ x is hereditarily symmet-ric , ˙ x ∈ HS , if ˙ x is symmetric and ˙ y ∈ HS for all ˙ y ∈ dom ˙ x .For a V -generic filter G on P , the symmetric extension by F and G is defined asfollows: N ∶ = V ( G ) ∶ = { ˙ x G ∣ ˙ x ∈ HS } . Then N is a transitive class with V ⊆ N ⊆ V [ G ] and N ⊧ ZF .The symmetric forcing relation ⊩ s can be defined informally as follows:If ϕ ( v , . . . , v n − ) is a formula of set theory, and ˙ x , . . . , ˙ x n − ∈ HS , then p ⊩ s ϕ ( ˙ x , . . . , ˙ x n − ) if for every V -generic filter G on P , it follows that V ( G ) ⊧ ϕ ( ˙ x G , . . . , ˙ x Gn − ) .Note that the symmetric forcing relation ⊩ s can be defined in the ground modelsimilar to the ordinary forcing relation ⊩ , but with the quantifiers and variablesranging over HS . It has most of the basic properties as ⊩ . In particular, theforcing theorem holds for ⊩ s , and the symmetry lemma is true, as well.We will consider the following weak versions of the Axiom of Choice:7he Axiom of Dependent Choice (DC) states that whenever X is a nonemptyset with a binary relation R , such that for all x ∈ R there exists y ∈ R with y R x ,there is a sequence ( x n ∣ n < ω ) in X with x n + R x n for all n < ω .The Axiom of Countable Choice ( AC ω ) states that every countable family ofnonempty sets has a choice function.The Axiom of Choice implies DC , and DC implies AC ω .The following lemma follows from [Kar14, Lemma 1]: Lemma ([Kar14, Lemma 1]) . If P is countably closed with a group of automor-phisms A ⊆ Aut ( P ) and a normal filter F on A , such that F is countably closedas well, then DC holds in the corresponding symmetric extension N = V ( G ) . The theory ZF + DC is sufficient to develop most of real analysis; while combi-natorial set theory in ZF + DC seemed rather hopeless in the first place.In [She10] and [She16], Shelah suggests that when working with ZF + DC , anotheraxiom should be adopted to set the framework for a “reasonable” set theory, wherenow, surprisingly much of combinatorial set theory can be realized: ( AX ) For every cardinal λ , the set [ λ ] ℵ can be well-ordered.Note that ( AX ) holds true in any symmetric extension by countably closed forcing(see [She10, p.3 and p.15]). We start from a ground model V ⊧ ZF C + GCH and a reasonable behavior of the θ -function : There will be sequences of uncountable cardinals ( κ η ∣ < η < γ ) and ( α η ∣ < η < γ ) in V , where γ ∈ Ord, for which we aim to construct a symmetricextension N ⊇ V with N ⊧ ZF + DC + AX , such that V and N have the samecardinals and cofinalities and θ N ( κ η ) = α η holds for all η .(Later on, we will set κ ∶ = ℵ , α ∶ = ℵ for technical reasons – therefore, we talkabout sequences ( κ η ∣ < η < γ ) , ( α η ∣ < η < γ ) here, excluding κ and α . )First, we want to discuss what properties the sequences ( κ η ∣ < η < γ ) and ( α η ∣ < η < γ ) must have to allow for such construction.W.l.o.g. we can assume that ( κ η ∣ < η < γ ) is strictly increasing and closed.The following conditions must be satisfied: • For η < η ′ , it follows from κ η < κ η ′ that α η ≤ α η ′ must hold, i.e. the sequence ( α η ∣ < η < γ ) must be increasing.8 For any cardinal κ , it is possible to construct a surjection s ∶ ℘ ( κ ) → κ + without making use of the Axiom of Choice. Hence, α η ≥ κ ++ η must hold forall η . • Since N ⊧ AC ω , it follows that cf α η > ω for all η : Assume towards acontradiction, there were cardinals κ , α with θ N ( κ ) = α , but cf N ( α ) = ω .Let α = ⋃ i < ω α i . By definition of θ N ( κ ) , it follows that for every i < ω , thereexists in N a surjection from ℘ ( κ ) onto α i . Now, AC ω allows us to pick in N a sequence ( s i ∣ i < ω ) such that each s i ∶ ℘ ( κ ) → α i is a surjection. Thisyields a surjective function s ∶ ℘ ( κ ) × ω → α , where s ( X, i ) ∶ = s i ( X ) for each ( X, i ) ∈ ℘ ( κ ) × ω ; which can be easily turned into a surjection s ∶ ℘ ( κ ) → α .Contradiction, since θ N ( κ ) = α . Hence, it follows that cf α η > ω for all η . • Finally, for every α η a successor cardinal with α η = α + , we will need that cf α > ω .In this setting, it is not possible to drop this requirement: We start from aground model V ⊧ ZF C + GCH with sequences ( κ η ∣ < η < γ ) , ( α η ∣ < η < γ ) , and aim to construct N ⊇ V with N ⊧ ZF + DC such that V and N have the same cardinals and cofinalities, and θ N ( κ η ) = α η holds for all η . Ifthere was some η with θ N ( κ η ) = α + , where cf α = ω , one could construct in N a surjective function s ∶ ℘ ( κ η ) → α + as follows:Take a surjection s ∶ ℘ ( κ η ) → α in N . Firstly, the canonical bijection κ ↔ κ × ω gives a surjection s ∶ κ → ( κ ) ω . Secondly, the surjection s ∶ ℘ ( κ η ) → α yields in N a surjection s ∶ ( κ ) ω → α ω , by setting s ( X i ∣ i < ω ) ∶ = ( s ( X i ) ∣ i < ω ) . Then s is surjective, since for ( α i ∣ i < ω ) ∈ α ω given, one can use AC ω to obtain a sequence ( Y i ∣ i < ω ) with Y i ∈ s − [{ α i }] for all i < ω . Then s ( Y i ∣ i < ω ) = ( α i ∣ i < ω ) . Thirdly, it follows from cf α = ω that there is a surjection s ∶ α ω → α + in V . Then s ∈ N , and since ( α ω ) N ⊇ ( α ω ) V and ( α + ) N = ( α + ) V , we obtain a surjection s ∶ α ω → α + in N .Thus, it follows that s ○ s ○ s ∶ κ → α + is a surjective function in N ;contradicting that θ N ( κ η ) = α + .Hence, in our setting, where we want to extend a ground model V ⊧ ZF C + GCH cardinal-preservingly, it is not possible to have α η = α + with cf α = ω .The following question arises: More generally, without referring to a groundmodel V , could there be N ⊧ ZF + DC + AX with cardinals κ , α , suchthat cf N ( α ) = ω and θ N ( κ ) = α + ? The answer is no: Let s ∶ κ → α denote a surjective function in N . Then with DC , it follows as before thatthere is also a surjective function s ∶ ( κ ) ω → α ω in N ; and we also havea surjective function s ∶ κ → ( κ ) ω . Since α ω is well-ordered by ( AX ) ,9 diagonalization argument as in K¨onig’s Lemma shows that there is also asurjection s ∶ α ω → α + . Hence, s ○ s ○ s ∶ α → α + is a surjective functionin N as desired.We conclude that all the requirements on the sequences ( κ η ∣ < η < γ ) and ( α η ∣ < η < γ ) listed above, are necessary for a model N ⊧ ZF + DC + AX .In addition, one could ask if there exists a model N ⊧ ZF + DC (without AX )with cardinals κ , α , such that θ N ( κ ) = α + and cf N ( α ) = ω . It is not difficult tosee that this is not possible under ¬ ♯ (cf. Chapter 7). Hence, if one wishes toavoid large cardinal assumptions, then for every α η = α + a successor cardinal, onehas to require cf α > ω .Our main theorem states that these are the only restrictions on the θ -function forset-many uncountable cardinals in ZF + DC + AX : Theorem.
Let V be a ground model of ZF C + GCH with γ ∈ Ord and sequences ofuncountable cardinals ( κ η ∣ < η < γ ) and ( α η ∣ < η < γ ) such that ( κ η ∣ < η < γ ) is strictly increasing and closed, and the following properties hold: • ∀ < η < η ′ < γ α η ≤ α η ′ , i.e. the sequence ( α η ∣ < η < γ ) is increasing, • ∀ < η < γ α η ≥ κ + + η , • ∀ < η < γ cf α η > ω , • ∀ < η < γ ( α η = α + → cf α > ω ) .Then there is a cardinal- and cofinality-preserving extension N ⊇ V with N ⊧ ZF + DC + AX such that that θ N ( κ η ) = α η holds for all < η < γ . In our construction, we will make sure that for any cardinal λ in a “gap” ( κ η , κ η + ) ,the value θ N ( λ ) is the smallest possible, i.e. θ N ( λ ) = max { α η , λ ++ } . Also, if weset κ γ ∶ = ⋃ { κ η ∣ < η < γ } , α γ ∶ = ⋃ { α η ∣ < η < γ } , then for every λ ≥ κ γ , wewill again make sure that θ N ( λ ) takes the smallest possible value: We will have θ N ( λ ) = max { α ++ γ , λ ++ } in the case that cf α γ = ω , θ N ( λ ) = max { α + γ , λ ++ } in thecase that α γ = α + for some cardinal α with cf α = ω , and θ N ( λ ) = max { α γ , λ ++ } ,else.This allows us to assume w.l.o.g. that the sequence ( α η ∣ < η < γ ) is strictly increasing: If not, one can start with the original sequences ( κ η ∣ < η < γ ) and ( α η ∣ < η < γ ) , and successively strike out all κ η for which the value α η is not larger than the values α η before. This procedure results in sequences (̃ κ η ∣ < η < ̃ γ ) ∶ = ( κ s ( η ) ∣ < η < ̃ γ ) and (̃ α η ∣ < η < ̃ γ ) ∶ = ( α s ( η ) ∣ < η < ̃ γ ) for some ̃ γ ≤ γ and a strictly increasing function s ∶ ̃ γ → γ , such that ̃ α ̃ γ ∶ = ⋃ {̃ α η ∣ < η < ̃ γ } = ⋃ { α s ( η ) ∣ < η < ̃ γ } = ⋃ { α η ∣ < η < γ } = α γ ,10nd (̃ α η ∣ < η < ̃ γ ) = ( α s ( η ) ∣ < η < ̃ γ ) is strictly increasing. If we then usethe sequences (̃ κ η ∣ < η < ̃ γ ) , (̃ α η ∣ < η < ̃ γ ) for our construction and makesure that not only θ N (̃ κ η ) = ̃ α η holds for all 0 < η < ̃ γ , but additionally, θ N ( λ ) takes the smallest possible value for all cardinals λ within the “gaps” (̃ κ η , ̃ κ η + ) ,and also make sure that θ N ( λ ) takes the smallest possible value for all cardinals λ ≥ ̃ κ ̃ γ ∶ = ⋃ {̃ κ η ∣ < η < ̃ γ } , then it follows, that for all κ η in the original sequence ( κ η ∣ < η < γ ) , the values θ N ( κ η ) = α η are as desired.Hence, from now on, we assume w.l.o.g. that the sequence ( α η ∣ < η < γ ) is strictly increasing . In this chapter, we will define our forcing notion P .We start from a ground model V ⊧ ZF C + GCH with sequences ( κ η ∣ < η < γ ) , ( α η ∣ < η < γ ) that have all the properties mentioned in Chapter 2.We will have to treat limit cardinals and successor cardinals separately. Let Lim ∶ = { < η < γ ∣ κ η is a limit cardinal } , and Succ ∶ = { < η < γ ∣ κ η is a successor cardinal } .For η ∈ Succ, we denote by κ η the cardinal predecessor of κ η ; i.e. κ η = κ η + . Ourforcing will be a product P = P × P , where P deals with the limit cardinals κ η ,and P is in charge of the successor cardinals.The forcing P is a generalized version of the forcing notion in [GK12].Roughly speaking, for every η ∈ Lim we add α η -many κ η -subsets, which will belinked in a certain fashion, in order to make sure that not too many κ -subsets forcardinals κ < κ η make their way into the eventual model N .For technical reasons, let κ ∶ = ℵ , α ∶ = ℵ . For all η with η + ∈ Lim, we take asequence of cardinals ( κ η,j ∣ j < cf κ η + ) cofinal in κ η + , such that κ η, = κ η , thesequence ( κ η,j ∣ j < cf κ η + ) is strictly increasing and closed, and any κ η,j + is asuccessor cardinal with κ η,j + ≥ κ ++ η,j for all j < cf κ η + .These “gaps” between the cardinals κ η,j and κ η,j + will be necessary for furtherfactoring arguments.For all 0 < η < γ for which η + ∈ Succ, i.e. κ η + is a successor cardinal, we set κ η, ∶ = κ η , and cf κ η + ∶ = η ∈ Lim, the forcing P η will be defined like an Easton-supportproduct of Cohen forcings for the intervals [ κ ν,j , κ ν,j + ) ⊆ κ η : Definition 1.
For η ∈ Lim , we let the forcing notion ( P η , ⊇ , ∅ ) consist of all func-tions p ∶ dom p → such that dom p is of the following form: here is a sequence ( δ ν,j ∣ ν < η , j < cf κ ν + ) with δ ν,j ∈ [ κ ν,j , κ ν,j + ) for all ν < η , j < cf κ ν + with dom p = ⋃ ν < ηj < cf κ ν + [ κ ν,j , δ ν,j ) , and for any regular κ ν,j , the domain dom p ∩ κ ν,j is bounded below κ ν,j . For a set S ⊆ κ η , we let P η ↾ S ∶ = { p ∈ P η ∣ dom p ⊆ S } = { p ↾ S ∣ p ∈ P η } . Then forany κ ν,j < κ η , the forcing P η is isomorphic to the product P η ↾ κ ν,j × P η ↾ [ κ ν,j , κ η ) ,where the first factor has cardinality ≤ κ + ν,j , and the second factor is ≤ κ ν,j -closed.This helps to establish: Lemma 2.
For all η ∈ Lim , the forcing P η preserves cardinals and the GCH .Proof.
Let G η denote a V -generic filter on P η . It suffices to show that for allcardinals α in V , ( α ) V [ G η ] ≤ ( α + ) V , which implies that cardinals are V - V [ G η ] -absolute: If not, there would be a V -cardinal α with a surjection s ∶ β → α in V [ G η ] for some V [ G η ] -cardinal β < α .Then there is also a surjection s ∶ β → ( β + ) V in V [ G η ] , which gives a surjection s ∶ β → ( β ) V [ G η ] . Contradiction. • In the case that α ≥ κ + η , it follows that ( α ) V [ G η ] ≤ ∣ ℘ ( α ⋅ ∣ P η ∣)∣ V ≤ ( α ) V = ( α + ) V by the GCH in V . • Now, assume α ∈ ( κ ν,j , κ ν,j + ) for some κ ν,j < κ η . Then the forcing P η canbe factored as P η ↾ κ ν,j × P η ↾ [ κ ν,j , κ η ) , where P η ↾ κ ν,j has cardinality ≤ κ + ν,j ≤ α , and P η ↾ [ κ ν,j , κ η ) is ≤ α -closed. Hence, ( α ) V [ G η ] ≤ ( α ) V [ G η ↾ κ ν,j ] ≤ ∣ ℘ ( α ⋅ ∣ P η ↾ κ ν,j ∣) ∣ V ≤ ( α ) V = ( α + ) V . • If α = κ ν,j for some regular κ ν,j < κ η , then ∣ P η ↾ κ ν,j ∣ = κ ν,j and P η ↾ [ κ ν,j , κ η ) is ≤ κ ν,j -closed; so the same argument applies.If α = κ η is regular, then ( α ) V [ G η ] ≤ ( α + ) V follows from ∣ P η ∣ ≤ κ η .It remains to show that ( κ ν,j ) V [ G η ] = ( κ + ν,j ) V for all singular κ ν,j < κ η , and ( κ η ) V [ G η ] ≤ ( κ + η ) V in the case that κ η itself is singular.We only prove the first part (the argument for κ η is similar). • Assume the contrary and let κ ν,j least with λ ∶ = cf κ ν,j < κ ν,j and ( κ ν,j ) V [ G η ] > ( κ + ν,j ) V . Take ( α i ∣ i < λ ) cofinal in κ ν,j . By assumption and by what wehave shown before, it follows that ( α ) V [ G η ] = ( α + ) V for all α < κ ν,j . Hence,Card V ∩ ( κ ν,j + ) = Card V [ G ] ∩ ( κ ν,j + ) , and ( α i ) V [ G η ] = ( α + i ) V for all i < λ . Thus, 2 κ ν,j ≤ ∏ i < λ α i ≤ κ λν,j ≤ κ κ ν,j ν,j = κ ν,j V and V [ G η ] . Let λ ∈ [ κ µ,m , κ µ,m + ) for some κ µ,m < κ ν,j . If λ > κ µ,m , then ∣ P η ↾ κ µ,m ∣ ≤ ( κ µ,m ) + ≤ λ , and P η ↾ [ κ µ,m , κ η ) is ≤ λ -closed. Inthe case that λ = κ µ,m , it follows by regularity of λ that ∣ P η ↾ κ µ,m ∣ ≤ κ µ,m = λ ,as well. In either case, ( κ ν,j ) V [ G η ] = ( κ λν,j ) V [ G η ] ≤ ( κ λν,j ) V [ G η ↾ κ µ,m ] ≤ ( κ ν,j ) V [ G η ↾ κ µ,m ] ≤≤ ∣ ℘ ( κ ν,j ⋅ ∣ P η ↾ κ µ,m ∣) ∣ V ≤ ∣ ℘ ( κ ν,j ⋅ κ + µ,m ) ∣ V = ( κ + ν,j ) V , which gives the desired contradiction. Corollary 3.
For every η ∈ Lim , the forcing P η preserves cofinalites.Proof. We show that every regular V -cardinal λ is still regular in V [ G η ] . If not,there would be in V [ G η ] a regular cardinal λ < λ with a cofinal function f ∶ λ → λ .Let λ ∈ [ κ ν,j , κ ν,j + ) . The forcing P η is isomorphic to the product P η ↾ κ ν,j × P η ↾ [ κ ν,j , κ η ) , where the second factor is ≤ λ -closed. If λ > κ ν,j , then the first factorhas cardinality ≤ κ + ν,j ≤ λ . In the case that λ = κ ν,j , the first factor has cardinality ≤ κ ν,j = λ by regularity of λ . Hence, f ∈ V [ G ↾ κ ν,j ] . However, since ∣ P η ↾ κ ν,j ∣ < λ ,it follows that λ is still a regular cardinal in the generic extension V [ G η ↾ κ ν,j ] .Contradiction.Thus, it follows that P η preserves cofinalities as desired.Our eventual forcing notion P will contain α σ -many copies of P σ for every σ ∈ Lim.They will be labelled P σi , where i < α σ . All the P σi for σ ∈ Lim, i < α σ , will belinked with a forcing notion P ∗ , which is a two-dimensional version of P γ , adding κ ν,j + -many Cohen subsets to every interval [ κ ν,j , κ ν,j + ) : Definition 4.
We denote by ( P ∗ , ⊇ ∅ ) the forcing notion consisting of all func-tions p ∗ ∶ dom p ∗ → such that dom p ∗ is of the following form:There is a sequence ( δ ν,j ∣ ν < γ , j < cf κ ν + ) with δ ν,j ∈ [ κ ν,j , κ ν,j + ) for all ν < γ , j < cf κ ν + with dom p ∗ = ⋃ ν < γj < cf κ ν + [ κ ν,j , δ ν,j ) , and for any κ ν,j a regular cardinal, it follows that ∣ dom p ∗ ∩ κ ν,j ∣ < κ ν,j , and in thecase that κ γ itself is regular, we require that ∣ dom p ∗ ∣ < κ γ . For p ∗ ∈ P ∗ and ξ < κ γ , let p ∗ ( ξ ) ∶ = { ( ζ, p ∗ ( ξ, ζ )) ∣ ( ξ, ζ ) ∈ dom p ∗ } denote the ξ -thsection of p ∗ . If a ⊆ κ γ is a set that hits every interval [ κ ν,j , κ ν,j + ) in at most onepoint, we let p ∗ ( a ) ∶ = { ( ζ, p ∗ ( ξ, ζ )) ∣ ξ ∈ a, ( ξ, ζ ) ∈ dom p ∗ } . As in Lemma 2, it follows that P ∗ preserves cardinals and the GCH .13ow, we are ready to define our forcing notion P . Every p ∈ P is of the form p = ( p ∗ , ( p σi , a σi ) σ ∈ Lim , i < α σ ) with p ∗ ∈ P ∗ and p σi ∈ P σ for all ( σ, i ) .The linking ordinals a σi will determine how the i -th generic κ σ -subset G σi , givenby the projection of the generic filter G onto P σi , will be eventually linked withthe P ∗ -generic filter G ∗ . Definition 5.
Let P be the collection of all p = ( p ∗ , ( p σi , a σi ) σ ∈ Lim , i < α σ ) such that: • The support of p , supp p , is countable with p σi = a σi = ∅ whenever ( σ, i ) ∉ supp p . • We have p ∗ ∈ P ∗ , and p σi ∈ P σ for all ( σ, i ) ∈ supp p . • The domains of the p σi are coherent in the following sense:If dom p ∗ = ⋃ ν < γ,j < cf κ ν + [ κ ν,j , δ ν,j ) , then for every ( σ, i ) ∈ supp p , it followsthat dom p σi = ⋃ ν < σ,j < cf κ ν + [ κ ν,j , δ ν,j ) .We set dom p ∶ = ⋃ ν,j [ κ ν,j , δ ν,j ) . • For all ( σ, i ) ∈ supp p , we have a σi ⊆ κ σ with ∣ a σi ∩ [ κ ν,j , κ ν,j + )∣ = for allintervals [ κ ν,j , κ ν,j + ) ⊆ κ σ .If ( σ , i ) ≠ ( σ , i ) , then a σ i ∩ a σ i = ∅ . (We call this the independenceproperty ). Concerning the partial ordering ≤ , any linking ordinal { ξ } = a σi ∩ [ κ ν,j , κ ν,j + ) set-tles that whenever q ≤ p , the extension q σi ⊇ p σi within in the interval [ κ ν,j , κ ν,j + ) is determined by q ∗ ( ξ ) : For p = ( p ∗ , ( p σi , a σi ) σ,i ) , q = ( q ∗ , ( q σi , b σi ) σ,i ) ∈ P , let q ≤ p if the follow-ing holds: q ∗ ⊇ p ∗ ; q σi ⊇ p σi , b σi ⊇ a σi for all ( σ, i ) ∈ supp p , and whenever ζ ∈ ( dom q σi ∖ dom p σi ) ∩ [ κ ν,j , κ ν,j + ) with a σi ∩ [ κ ν,j , κ ν,j + ) = { ξ } , then ξ ∈ dom q with q σi ( ζ ) = q ∗ ( ξ, ζ ) (we call this the linking property ).The maximal element of P is ∶ = ( ∅ , ( ∅ , ∅ ) σ < γ,i < α σ ) . Let G denote a V -generic filter on P , and g σi ∶ = ⋃ { a σi ∣ p = ( p ∗ , ( p σi , a σi ) σ,i ) ∈ G } .Note that by our strong independence property , every interval [ κ ν,j , κ ν,j + ) will beblown up to size sup { α σ ∣ σ ∈ Lim } in a P -generic extension.Hence, since we want our eventual symmetric submodel N preserve all V -cardinals,we will have so make sure that N “does not know” the sequence of linking ordinals ( g σi ∣ σ ∈ Lim , i < α σ ) .A major difference between our forcing and the basic construction in [GK12] isthe following: The forcing conditions in [GK12, Definition 2] have finite linking14rdinals a σi ; so the according generics g σi are not contained in the ground model V . With our definition however, it follows for any p ∈ G with ( σ, i ) ∈ supp p that g σi = a σi ∈ V . By countable support, also countable sequences of linkingordinals ( g σ j i j ∣ j < ω ) are contained in V ; but for σ ∈ Lim not the whole sequence ( g σi ∣ i < α σ ) .This modification helps to establish that any generic G σi can be described usingonly G ∗ and sets from the ground model V (see below).Next, we define our forcing notion P , which will be in charge of the successorcardinals. For every σ ∈ Succ with κ σ = κ σ + , it follows that σ = ∶ σ + ( κ σ ∣ < σ < γ ) is closed.We denote by P σ the Cohen forcing P σ ∶ = { p ∶ dom p → ∣ dom p ⊆ [ κ σ , κ σ ) , ∣ dom p ∣ < κ σ } , and let C σ ∶ = { p ∶ dom p → ∣ dom p = dom x p × dom y p ⊆ α σ × [ κ σ , κ σ ) , ∣ dom p ∣ < κ σ } . Then both P σ and C σ are < κ σ -closed, and if 2 < κ σ = κ σ , i.e. 2 κ σ = κ σ , then theysatisfy the κ + σ -chain condition and hence, preserve cardinals.In particular, any forcing P σ or C σ preserves cardinals if we are working in ourground model V with V ⊧ GCH , or any V -generic extension by ≤ κ σ -closed forcing. Definition 6.
The forcing notion ( P , ≤ , ∅ ) consists of all p = ( p σ ) σ ∈ Succ withcountable support supp p ∶ = { σ ∈ Succ ∣ p σ ≠ ∅ } , and p σ ∈ C σ for all σ ∈ Succ . For p = ( p σ ) σ ∈ Succ , q = ( q σ ) σ ∈ Succ ∈ P , we let q ≤ p if q σ ⊇ p σ for all σ ∈ Succ ; and ∶ = ( ∅ ) σ ∈ Succ is the maximal element.
For σ ∈ Succ and i < α σ , we set p σi = {( ζ, p σ ( i, ζ )) ∣ ( i, ζ ) ∈ dom p σ } .Our main forcing will be the product P ∶ = P × P with maximal element ∶ = ( , ) and order relation ≤ . In order to simplify notation, we write conditions p ∈ P in the form p = ( p ∗ , ( p σi , a σi ) σ ∈ Lim ,i < α σ , ( p σ ) σ ∈ Succ ) .It is not difficult to verify: Proposition 7. P is countably closed. This is important to make sure that DC holds in our eventual symmetric extension N .For 0 < η ≤ γ (with η ∈ Lim or η ∈ Succ or η = γ ), we define a forcing P η like P η isdefined in the case that η ∈ Lim : 15 et P η consist of all functions p ∶ dom p → such that there is a sequence ( δ ν,j ∣ ν < η, j < cf κ ν + ) with δ ν,j ∈ [ κ ν,j , κ ν,j + ) for all κ ν,j < κ η , and dom p = ⋃ ν,j [ κ ν,j , δ ν,j ) , such that ∣ p ↾ κ ν, ∣ < κ ν, whenever κ ν, is a regular cardinal, and ∣ p ∣ < κ η in the casethat κ η itself is regular. For any 0 < η < λ with κ λ a limit cardinal, it follows that P η = P λ ↾ κ η .Let now G be a V -generic filter on P . It induces G ∗ ∶ = { q ∗ ∈ P ∗ ∣ ∃ p = ( p ∗ , ( p σi , a σi ) σ,i , ( p σ ) σ ) ∈ G ∶ q ∗ ⊆ p ∗ } , and for λ ∈ Lim , k < α λ : G λk ∶ = { q λk ∈ P λ ∣ ∃ p = ( p ∗ , ( p σi , a σi ) σ,i , ( p σ ) σ ) ∈ G ∶ q λk ⊆ p λk } . As usually, these filters G ∗ , G λk are identified with their union ⋃ G ∗ , ⋃ G λk . Thenany G λk can be regarded a subset of κ λ .Moreover, let g λk ∶ = ⋃ { a λk ∣ p = ( p ∗ , ( p σi , a σi ) σ,i , ( p σ ) σ ) ∈ G } . Then g λk = a λk for any p ∈ G with ( λ, k ) ∈ supp p ; and g λk hits any interval [ κ ν,j , κ ν,j + ) ⊆ κ λ in exactly one point. By the independence property , it followsthat g λ k ∩ g λ k = ∅ whenever ( λ , k ) ≠ ( λ , k ) .For λ ∈ Succ , set G λ ∶ = { p λ ∣ p = ( p ∗ , ( p σi , a σi ) η,i , ( p σ ) σ ) ∈ G } , and G λk ∶ = { p λk ∣ p = ( p ∗ , ( p σi , a σi ) σ,i , ( p σ ) σ ) ∈ G } for any k < α λ .Again, we confuse these filters G λ , G λk with their union ⋃ G λ , ⋃ G λk .Let now ξ ∈ [ κ ν,j , κ ν,j + ) . We denote by G ∗ ( ξ ) ∶ = { q ∶ [ κ ν,j , δ ν,j ) → ∣ δ ν,j ∈ [ κ ν,j , κ ν,j + ) , ∃ p = ( p ∗ , ( p σi , a σi ) σ,i , ( p σ ) σ ) ∈ G ∶∀ ζ ∈ dom q q ( ζ ) = p ∗ ( ξ, ζ ) } the ξ -th section of G ∗ .If a ⊆ κ γ is a set that hits any interval [ κ ν,j , κ ν,j + ) ⊆ κ γ in at most one point, wedenote by G ∗ ( a ) the set of all q ∈ P γ such that there is p ∈ G with q ⊆ p ∗ ( a ) .16s before, we identify any G ∗ ( ξ ) and G ∗ ( a ) with their union ⋃ G ∗ ( ξ ) and ⋃ G ∗ ( a ) ,respectively. Then any G ∗ ( ξ ) with ξ ∈ [ κ νj , κ ν,j + ) can be regarded as a function G ∗ ( ξ ) ∶ [ κ ν,j , κ ν,j + ) →
2, and any G ∗ ( a ) becomes a function G ∗ ( a ) ∶ dom G ∗ ( a ) →
2, where dom G ∗ ( a ) ⊆ κ γ is the union of those intervals [ κ ν,j , κ ν,j + ) with a ∩ [ κ ν,j , κ ν,j + ) ≠ ∅ .Now, the linking property implies that any G λk ↾ [ κ ν,j , κ ν,j + ) with λ ∈ Lim, k < α λ ,is eventually equal to G ∗ ( ξ ) , where { ξ } ∶ = a λk ∩ [ κ ν,j , κ ν,j + ) .Indeed, the symmetric difference G λk ⊕ G ∗ ( g λk ) is always an element of the groundmodel V : Take a condition p ∈ G with ( λ, k ) ∈ supp p , such that for any interval [ κ ν,j , κ ν,j + ) ⊆ κ λ with dom p ∩ [ κ ν,j , κ ν,j + ) ≠ ∅ and { ξ } ∶ = a λk ∩ [ κ ν,j , κ ν,j + ) , itfollows that ξ ∈ dom p . ( This does not interfere with the condition that dom p has to be bounded below all regular κ ν, , since we do not bother the intervals [ κ ν,j , κ ν,j + ) with dom p ∩ [ κ ν,j , κ ν,j + ) = ∅ . ) • Firstly, G λk ( ζ ) ⊕ G ∗ ( g λk )( ζ ) = ζ ∉ dom p : Let ζ ∈ [ κ ν,j , κ ν,j + ) , ζ ∉ dom p with { ξ } ∶ = g λk ∩ [ κ ν,j , κ ν,j + ) = a λk ∩ [ κ ν,j , κ ν,j + ) . Take q ∈ G , q ≤ p with ζ ∈ dom q . Then by the linking property , it follows that ξ ∈ dom q with q λk ( ζ ) = q ∗ ( ξ, ζ ) . Hence, G λk ( ζ ) = q λk ( ζ ) = q ∗ ( ξ, ζ ) = G ∗ ( g λk )( ζ ) , and G λk ( ζ ) ⊕ G ∗ ( g λk )( ζ ) = • If ζ ∈ dom p then G λk ( ζ ) ⊕ G ∗ ( g λk )( ζ ) = p λk ( ζ ) ⊕ p ∗ ( ξ, ζ ) , where again, ζ ∈ [ κ ν,j , κ ν,j + ) and { ξ } ∶ = g λk ∩ [ κ ν,j , κ ν,j + ) = a λk ∩ [ κ ν,j , κ ν,j + ) . Here we usethat for any interval [ κ ν,j , κ ν,j + ) with dom p ∩ [ κ ν,j , κ ν,j + ) ≠ ∅ , it followsthat a λk ∩ [ κ ν,j , κ ν,j + ) ⊆ dom p .Hence, G λk ⊕ G ∗ ( g λk ) can be calculated in V .This will be employed to keep control over the surjective size of ℘ ( κ λ ) in the even-tual symmetric extension N .Now, we consider countable products ∏ m < ω P σ m and ∏ m < ω P σ m : Definition 8.
Let (( σ m , i m ) ∣ m < ω ) be a sequence of pairwise distinct pairswith < σ m < γ , i < α σ m for all m < ω . We denote by ∏ m < ω P σ m the set of all ( p ( m ) ∣ m < ω ) with p ( m ) ∈ P σ m for all m < ω (with full support), and similarly, ∏ m < ω P σ m ∶ = { ( p ( m ) ∣ m < ω ) ∣ ∀ m < ω p ( m ) ∈ P σ m } . For any interval [ κ ν,j , κ ν,j + ) ⊆ κ γ , it follows that ∏ m < ω P σ m ↾ κ ν,j has cardinality ≤ κ ν,j in the case that κ ν,j is regular, and cardinality ≤ κ + ν,j , else. Moreover, ∏ m < ω P σ m ↾ [ κ ν,j , κ σ m ) is ≤ κ ν,j -closed. Hence, as in Lemma 2 and Corollary 3,one can show that the product ∏ m < ω P σ m preserves cardinals, cofinalities and the GCH .Similarly, ∏ m < ω P σ m preserves cardinals, cofinalities and the GCH .17he next lemma implies that countable products ∏ m < ω G ∗ ( g σ m i m ) are V -genericover ∏ m < ω P σ m : Lemma 9.
Consider a sequence ( a m ∣ m < ω ) of pairwise disjoint sets such thatfor all m < ω , the following holds: a m is a subset of κ σ m for some < σ m < γ , andfor all κ ν,j < κ σ m , it follows that ∣ a m ∩ [ κ ν,j , κ ν,j + )∣ = , i.e. a m hits every interval [ κ ν,j , κ ν,j + ) ⊆ κ σ m in exactly one point. Then ∏ m < ω G ∗ ( a m ) ∶ = { ( p ( m ) ∣ m < ω ) ∣ ∀ m < ω p ( m ) ∈ G ∗ ( a m ) } is a V -generic filter on ∏ m < ω P σ m .Proof. Let D ⊆ ∏ m < ω P σ m be an open dense set in V . We show that D ∩ ∏ m < ω G ∗ ( a m ) ≠ ∅ . Let D ∶ = { p = ( p ∗ , ( p σi , a σi ) σ,i , ( p σ ) σ ) ∈ P ∣ ( p ∗ ( a m ) ∣ m < ω ) ∈ D } . It suffices to prove that D is dense in P . Assume p = ( p ∗ , ( p σi , a σi ) σ,i , ( p σ ) σ ) ∈ P given, and denote by ( q m ∣ m < ω ) an extension of ( p ∗ ( a m ) ∣ m < ω ) in D . Wehave to construct p ≤ p such that p ∗ ( a m ) ⊇ q m for all m < ω .Consider an interval [ κ ν,j , κ ν,j + ) ⊆ κ γ . In the case that ( dom p ∪ ⋃ m < ω dom q m ) ∩ [ κ ν,j , κ ν,j + ) = ∅ , let δ ν,j ∶ = κ ν,j . Otherwise, we pick δ ν,j ∈ [ κ ν,j , κ ν,j + ) such thatfirstly, ( dom p ∪ ⋃ m < ω dom q m ) ∩ [ κ ν,j , κ ν,j + ) ⊆ [ κ ν,j , δ ν,j ) ;secondly, for all m < ω , it follows that a m ∩ [ κ ν,j , κ ν,j + ) ⊆ [ κ ν,j , δ ν,j ) ; and thirdly, a σi ∩ [ κ ν,j , κ ν,j + ) ⊆ [ κ ν,j , δ ν,j ) for all ( σ, i ) ∈ supp p . This is possible, since the sets a m ∩ [ κ ν,j , κ ν,j + ) and a σi ∩ [ κ ν,j , κ ν,j + ) are singletons or empty, all the domainsdom p ∩ [ κ ν,j , κ ν,j + ) and dom q m ∩ [ κ ν,j , κ ν,j + ) for m < ω are bounded below κ ν,j + , and κ ν,j + is always a successor cardinal.Let dom p ∶ = ⋃ ν,j [ κ ν,j , δ ν,j ) . Then dom p is bounded below all regular κ ν, , since this holds true for dom p and ⋃ m < ω dom q m . We define p ∗ on ⋃ ν,j [ κ ν,j , δ ν,j ) as follows: Consider an in-terval [ κ ν,j , δ ν,j ) ≠ ∅ and ξ , ζ ∈ [ κ ν,j , δ ν,j ) . For ( ξ, ζ ) ∈ dom p × dom p , let p ∗ ( ξ, ζ ) ∶ = p ∗ ( ξ, ζ ) . If { ξ } = a m ∩ [ κ ν,j , κ ν,j + ) for some m < ω and ζ ∈ dom q m ,we set p ∗ ( ξ, ζ ) ∶ = q m ( ζ ) . This is not a contradiction towards p ∗ ↾ [ κ ν,j , δ ν,j ) ⊇ p ∗ ↾ [ κ ν,j , δ ν,j ) , since q m ⊇ p ∗ ( a m ) for all m < ω . Also, the a m are pairwise dis-joint, so for any ξ ∈ [ κ ν,j , δ ν,j ) , there is at most one m with ξ ∈ a m . For all theremaining ( ξ, ζ ) ∈ dom p ∗ , we can set p ∗ ( ξ, ζ ) ∈ { , } arbitrarily. This defines p ∗ on ⋃ ν,j [ κ ν,j , δ ν,j ) .For all ( σ, i ) ∈ supp p ∶ = supp p , we set a σi ∶ = a σi , and define p σi ⊇ p σi on the corre-sponding domain ⋃ κ ν,j < κ σ [ κ ν,j , δ ν,j ) according to the linking property : Whenever ζ ∈ ( dom p ∖ dom p ) ∩ [ κ ν,j , κ ν,j + ) and a σi ∩ [ κ ν,j , κ ν,j + ) = { ξ } , then ξ ∈ dom p p σi ( ζ ) ∶ = p ∗ ( ξ, ζ ) . For the ζ ∈ dom p σi ∖ dom p σi remaining, wecan define p σi ( ζ ) arbitrarily. This completes the construction of p .Let p ∶ = p . It is not difficult to check that p ≤ p indeed is a condition in P with p ∗ ( a m ) ⊇ q m for all m < ω . Hence, ( p ∗ ( a m ) ∣ m < ω ) ∈ D , and p ∈ D as desired.In particular, for (( σ m , i m ) ∣ m < ω ) a sequence of pairwise distinct pairs as beforewith σ m ∈ Lim, i m < α σ m for all m < ω , it follows that ∏ m < ω G ∗ ( g σ m i m ) is a V -genericfilter over ∏ m < ω P σ m .Similarly, one can show: Lemma 10.
Let (( σ m , i m ) ∣ m < ω ) denote a sequence of pairwise distinct pairswith < σ m < γ , i m < α σ m for all m < ω . Then ∏ m < ω G σ m i m ∶ = { ( p ( m ) ∣ m < ω ) ∣ ∀ m < ω p ( m ) ∈ G σ m i m } is a V -generic filter on ∏ m < ω P σ m . For constructing our symmetric submodel N , we will define a group A of P -automorphisms and a normal filter F on A . More accurately: In our setting,an automorphism π ∈ A will not be defined on P itself, but only on a dense subset D π ⊆ P . We call such π ∶ D π → D π a partial automorphism . Hence, the set A isnot quite a group, but has a very similar structure:For any π , σ ∈ A with π ∶ D π → D π , σ ∶ D σ → D σ and p ∈ D π ∩ D σ , the image σ ( p ) will be an element of D π ∩ D σ as well; and A will contain a map ν ∶ D ν → D ν suchthat D ν = D π ∩ D σ and ν = π ○ σ on D ν . (We will call ν the concatenation π ○ σ .)Moreover, for any π ∈ A , there will be a map ν in A with D ν = D π such that π ○ ν = ν ○ π = id D ν = id D π is the identity on D ν = D π . (We call ν the inverse π − .)There will also be an identity element id ∈ A , which is the identity map on itsdomain D id , where D id ⊇ D π for all π ∈ A .This does not quite give a group structure: For instance, for any π ∈ A , the con-catenation π ○ π − = π − ○ π = id D π is not really the identity element id , whichusually has a larger domain D id .In this setting, the standard approach would be using Boolean-valued models forthe construction of the symmetric submodel N : Any automorphism π ∶ D π → D π can be uniquely extended to an automorphism of the complete Boolean algebra B ( P ) , and thereby induces an automorphism of the Boolean valued model V B ( P ) .Then one can consider the group consisting of these extended automorphisms,define a normal filter and construct the corresponding symmetric submodel asdescribed in [Jec73, Chapter 5]. 19e try to avoid Boolean valued models here, and work with partial orders andautomorphisms π ∶ D π → D π on dense subsets D π ⊆ P instead.We will have a collection D of dense subsets D ⊆ P with certain properties, anda collection A of partial automorphisms π ∶ D π → D π with D π ∈ D for any π ∈ A .Whenever D ∈ D is fixed, the automorphisms { π ∈ A ∣ D π = D } will form a groupthat we denote by A D . Moreover, for any D , D ′ ∈ D with D ⊆ D ′ and π ∈ A D ′ , wewill have π [ D ] = D , and the restriction π ↾ D is an element of A D . Hence, thereare canonical homomorphisms φ D ′ D ∶ A D ′ → A D , π ↦ π ↾ D for any D , D ′ ∈ D with D ⊆ D ′ . This gives a directed system, and we can take the direct limit A ∶ = lim Ð → A D = ⊔ A D / ∼ with the following equivalence relation “ ∼ ”: Whenever π ∈ A D and π ′ ∈ A D ′ , then π ∼ π ′ iff there exists D ′′ ∈ A , D ′′ ⊆ D ∩ D ′ , such that π and π ′ agree on D ′′ . Sincefor any D , D ′ ∈ A , the intersection D ∩ D ′ will be contained in D as well, and P is separative, this will be the case if and only if π and π ′ agree on the intersection D ∩ D ′ .For π ∈ A , we denote by [ π ] its equivalence class: [ π ] ∶ = { σ ∈ A ∣ σ ∼ π } = { σ ∈ A ∣ π ↾ ( D π ∩ D σ ) = σ ↾ ( D π ∩ D σ )} . Then A = { [ π ] ∣ π ∈ A } becomes a group as follows: For π , σ ∈ A , let [ π ] ○ [ σ ] ∶ = [ ν ] ,where ν ∈ A with D ν = D π ∩ D σ and ν ( p ) = π ( σ ( p )) for all p ∈ D π ∩ D σ . Such ν will always exists by our construction of A , and [ ν ] is well-defined: If [ π ] = [ π ′ ] , [ σ ] = [ σ ′ ] and ν , ν ′ as above, then for all p ∈ ( D π ∩ D σ ) ∩ ( D π ′ ∩ D σ ′ ) , it followsthat ν ( p ) = π ( σ ( p )) = π ′ ( σ ′ ( p )) = ν ′ ( p ) . Hence, [ ν ] = [ ν ′ ] .The identity element id is the identity map on its domain D id ∈ D , with D id ⊇ D π for all π ∈ A (then [ π ] ○ [ id ] = [ id ] ○ [ π ] = [ π ] for all π ∈ A follows).Finally, for π ∈ A , let [ π ] − ∶ = [ ν ] , where ν ∈ A with D ν = D π and ν = π − on D π ,i.e. ν ( π ( p )) = π ( ν ( p )) = p for all p ∈ D π . Again, such ν will always exists by ourconstruction of A , and [ ν ] is well-defined: Whenever [ π ] = [ π ′ ] and ν , ν ′ as above,it follows that ν ( p ) = ν ′ ( p ) must hold for all p ∈ D π ∩ D π ′ = D ν ∩ D ν ′ ; hence, [ ν ] = [ ν ′ ] . Moreover, [ π ] ○ [ ν ] = [ ν ] ○ [ π ] = [ id D π ] = [ id ] .Hence, A is a group. Later on, we will define a collection of A -subgroups generat-ing a normal filter F on A , giving rise to our notion of symmetry.However, we first have to extend our partial automorphisms π ∈ A to the namespace N ame ( P ) .For any D ∈ D , we define a hierarchy Name α ( P ) D recursively: • Name ( P ) D ∶ = ∅ Name α + ( P ) D ∶ = { ˙ x ∈ Name ( P ) ∣ ˙ x ⊆ Name α ( P ) D × D } , and • Name λ ( P ) D ∶ = ⋃ α < λ Name α ( P ) D for λ a limit ordinal.Let Name ( P ) D ∶ = ⋃ α ∈ Ord
Name α ( P ) D . In other words: Name ( P ) D is the collection of all P -names ˙ x where only conditions p ∈ D occur.Whenever π ∈ A , π ∶ D π → D π , then the image π ˙ x can be defined es as usual aslong as ˙ x ∈ Name ( P ) D π . In the case that ˙ x is a P -name with ˙ x ∉ Name ( P ) D π ,however, it is not clear how to apply π , so the name ˙ x has to be modified.Given D ∈ D , we define recursively for ˙ x ∈ Name ( P ) : x D ∶ = {( y D , p ) ∣ ˙ y ∈ dom ˙ x , p ∈ D , p ⊩ ˙ y ∈ ˙ x } . Then x D ∈ Name ( P ) D with ˙ x G = ( x D ) G for any G a V -generic filter on P .We will call a P -name ˙ x symmetric if the collection of all [ π ] with πx D π = x D π iscontained in our normal filter F .Hereby, we have to make sure that this definition does not depend on which rep-resentative of [ π ] we choose: In Lemma 14 and 15, we prove that whenever π and π ′ belong to the same equivalence class [ π ] = [ π ′ ] , then πx D π = x D π holds if andonly if π ′ x D π ′ = x D π ′ .Then we set N ∶ = V ( G ) ∶ = { ˙ x G ∣ ˙ x ∈ HS } , where HS denotes the class of all hereditarily symmetric P -names, defined recur-sively as usual: For every ˙ x ∈ Name ( P ) , we have ˙ x ∈ HS if ˙ x is symmetric and dom ˙ x ⊆ HS .We have to verify that also with this modified notion of symmetry, N = V ( G ) is amodel of ZF . This will be done in Chapter 5. A . We start with constructing A , our collection of partial P -automorphisms with theproperties mentioned in Chapter 4.1.We will have A = A × A , where A is a collection of partial P -automorphisms,and A is a collection of partial P -automorphisms.Every π ∈ A will be an order-preserving bijection π ∶ D π → D π , where D π iscontained in our collection D , defined as follows:21et D denote the collection of all sets D ⊆ P given by • a countable support supp D ⊆ {( σ, i ) ∣ σ ∈ Lim , i < α σ } , and • a domain dom D ∶ = ⋃ ν < γ , j < cf κ ν + [ κ ν,j , δ ν,j ) such that δ ν,j ∈ [ κ ν,j , κ ν,j + ) forall ν < γ , j < cf κ ν + ; and for all regular κ ν, , it follows that dom D ∩ κ ν, isbounded below κ ν, ,such that D is the set of all p = ( p ∗ , ( p σi , a σi ) σ,i ) ∈ P with • supp p ⊇ supp D , dom p ⊇ dom D , and • for all intervals [ κ ν,j , κ ν,j + ) with dom p ∩ [ κ ν,j , κ ν,j + ) ≠ ∅ , it follows that ⋃ ( σ,i )∈ supp p a σi ∩ [ κ ν,j , κ ν,j + ) ⊆ dom p. In other words, D is the collection of all p ∈ P , the domain and support of whichcovers a certain domain and support given by D ; with the additional propertythat all the linking ordinals { ξ } = a σi ∩ [ κ ν,j , κ ν,j + ) contained in any interval [ κ ν,j , κ ν,j + ) hit by dom p , are already contained in dom p .It is not difficult to see that any D ∈ D is dense in P . The sets D ∈ D are notopen dense; but whenever p , q ∈ P with p ∈ D and q ≤ p such that supp q = supp p ,then by the linking property , it follows that also q ∈ D .Whenever D , D ′ ∈ D , then the intersection D ∩ D ′ is contained in D , as well,with supp ( D ∩ D ′ ) = supp D ∪ supp D ′ , dom ( D ∩ D ′ ) = dom D ∪ dom D ′ .We now describe the two types of partial P -automorphisms that will generate A :Our first goal is that for any two conditions p , q ∈ P with the same “shape”, i.e.dom p = dom q , supp p = supp q and ⋃ a σi = ⋃ b σi , there is an automorphism π ∈ A with π p = q . This homogeneity property will be achieved by giving the maps π ∈ A a lot of freedom regarding what can happen on supp π and dom π .For κ ν,j < κ γ , let supp π ( ν, j ) ∶ = {( σ, i ) ∈ supp π ∣ κ ν,j < κ σ } . Regarding the linking ordinals, we want that for any p ∈ D π , p = ( p ∗ , ( p σi , a σi ) σ,i ) with πp = p ′ = (( p ′ ) ∗ , (( p ′ ) σi , ( a ′ ) σi ) σ,i ) , the sets of linking ordinals for p and p ′ arethe same, i.e. ⋃ a σi = ⋃ ( a ′ ) σi . In other words, for any interval [ κ ν,j , κ ν,j + ) , thelinking ordinals ξ ∈ [ κ ν,j , κ ν,j + ) will be exchanged between the coordinates ( σ, i ) ∈ supp π ( ν, j ) , which is described by an isomorphism F π ( ν, j ) ∶ supp π ( ν, j ) → supp π ( ν, j ) . 22egarding the ( p ′ ) σi for ( σ, i ) ∈ supp π , there will be for every ζ ∈ [ κ ν,j , κ ν,j + ) ∩ dom π a bijection π ( ζ ) ∶ supp π ( ν,j ) → supp π ( ν,j ) with ( ( p ′ ) σi ( ζ ) ∣ ( σ, i ) ∈ supp π ( ν, j ) ) ∶ = π ( ζ )( p σi ( ζ ) ∣ ( σ, i ) ∈ supp π ( ν, j ) ) . Concerning p ′∗ , we will have a similar construction for the p ′∗ ( ξ, ζ ) in the casethat ζ ∈ dom π and ξ is a linking ordinal contained in ⋃ a σi . For all ( ξ, ζ ) ∈ dom π ∩ [ κ ν,j , κ ν,j + ) , we will have a bijection π ∗ ( ξ, ζ ) ∶ →
2, with p ′∗ ( ξ, ζ ) = π ∗ ( ξ, ζ )( p ∗ ( ξ, ζ )) whenever ξ , ζ ∈ dom π and ξ ∉ ⋃ a σi .Our second goal is that for any interval [ κ ν,j , κ ν,j + ) and ( σ, i ) , ( λ, k ) ∈ supp π ( ν, j ) ,there is an isomorphism π ∈ A such that ( π G ) λk ∩ [ κ ν,j , κ ν,j + ) = G σi ∩ [ κ ν,j , κ ν,j + ) .Thus, every π ∈ A will be equipped with bijections G π ( ν, j ) ∶ supp π ( ν, j ) → supp π ( ν, j ) for every κ ν,j , such that the following holds: For every p ∈ D π , p ′ ∶ = πp and ζ ∈ dom p ∖ dom π , ( σ, i ) ∈ supp π ( ν, j ) , we have ( p ′ ) σi ( ζ ) = p λk ( ζ ) with ( λ, k ) ∶ = G π ( ν, j )( σ, i ) .Whenever ζ ∈ dom π and ( σ, i ) ∈ supp π ( ν, j ) , the values ( p ′ ) σi ( ζ ) are describedby the maps π ( ζ ) mentioned in context with “our first goal” above, which al-lows for setting ( p ′ ) σi ( ζ ) ∶ = p λk ( ζ ) for any pair ( σ, i ) , ( λ, k ) ∈ supp π ( ν, j ) with ( λ, k ) = G π ( ν, j )( σ, i ) .Roughly speaking, A will be generated by these two types of isomorphism. Re-garding the construction of p ′∗ , some extra care is needed concerning the values p ′∗ ( ξ, ζ ) for ζ ∉ dom π and ξ ∈ ⋃ a σi a linking ordinal, since we have to make surethat the maps π ∈ A are order-preserving: Whenever p, q ∈ D π with q ≤ p , thenalso q ′ ≤ p ′ must hold; in particular, whenever { ξ σi } ∶ = a σi ∩ [ κ ν,j , κ ν,j + ) is a linkingordinal and ζ ∈ dom q ∖ dom p (hence, ζ ∉ dom π ), then { ξ σi } = ( a ′ ) λk ∩ [ κ ν,j , κ ν,j + ) with ( σ, i ) = F π ( ν, j )( λ, k ) , and q ′∗ ( ξ σi , ζ ) = ( q ′ ) λk ( ζ ) by the linking property for q ′ ≤ p ′ . Moreover, ( q ′ ) λk ( ζ ) = q µl ( ζ ) with ( µ, l ) = G π ( ν, j )( λ, k ) , and q µl ( ζ ) = q ∗ ( ξ µl , ζ ) with ξ µl = a µl ∩ [ κ ν,j , κ ν,j + ) by the linking property for q ≤ p . Hence, q ′∗ ( ξ σi , ζ ) = q ∗ ( ξ µl , ζ ) must hold, where ( µ, l ) = G π ( ν, j ) ○ ( F π ( ν, j )) − ( σ, i ) .This gives rise to the following definition: Definition 11.
Let A consist of all automorphisms π ∶ D π → D π such thatthere are • a countable set supp π ⊆ {( σ, i ) ∣ σ ∈ Lim , i < α σ } (for κ ν,j < κ γ , we set supp π ( ν, j ) ∶ = {( σ, i ) ∈ supp π ∣ κ ν,j < κ γ } ), • a domain dom π = ⋃ ν < γ , j < cf κ ν + [ κ ν,j , δ ν,j ) such that δ ν,j ∈ [ κ ν,j , κ ν,j + ) forall ν < γ , j < cf κ ν + ; and for all regular κ ν, , it follows that dom π ∩ κ ν, isbounded below κ ν, (for κ ν,j < κ γ , we set dom π ( ν, j ) ∶ = dom π ∩ [ κ ν,j , κ ν,j + ) ), uch that D π = { p = ( p ∗ , ( p σi , a σi ) σ,i ) ∈ P ∣ supp p ⊇ supp π , dom p ⊇ dom π , and ∀ κ ν,j < κ γ ∶ ( dom p ∩ [ κ ν,j , κ ν,j + ) ≠ ∅ ⇒ ⋃ ( σ,i )∈ supp p a σi ∩ [ κ ν,j , κ ν,j + ) ⊆ dom p ) } ; moreover, there are • for all ν < γ , j < cf κ ν + , a bijection F π ( ν, j ) ∶ supp π ( ν, j ) → supp π ( ν, j ) (which will be in charge of permuting the linking ordinals as mentionedabove),and a bijection G π ( ν, j ) ∶ supp π ( ν, j ) → supp π ( ν, j ) (which will be in charge of permuting the verticals p σi outside dom π on theinterval [ κ ν,j , κ ν,j + ) ), • for all ν < γ , j < cf κ ν + and ζ ∈ [ κ ν,j , κ ν,j + ) ∩ dom π , a bijection π ( ζ ) ∶ supp π ( ν,j ) → supp π ( ν,j ) (this map will be in charge of setting the values ( πp ) σi ( ζ ) for ( σ, i ) ∈ supp π ( ν, j ) , ζ ∈ dom π ), • for all ν < γ , j < cf κ ν + , ζ ∈ [ κ ν,j , κ ν,j + ) ∩ dom π , and ( ξ σi ∣ ( σ, i ) ∈ supp π ( ν, j )) ∈ ( dom π ( ν, j )) supp π ( ν,j ) a sequence of pairwise distinct ordinals, a bijection ( π ) ∗ ( ζ )( ξ σi ∣ ( σ, i ) ∈ supp π ( ν, j )) ∶ supp π ( ν,j ) → supp π ( ν,j ) (which will be in charge of setting the values ( πp ) ∗ ( ξ σi , ζ ) for { ξ σi } = a σi ∩ [ κ ν,j , κ ν,j + ) a linking ordinal and ζ ∈ [ κ ν,j , κ ν,j + ) ∩ dom π ), • for all ν < γ , j < cf κ ν + and ( ξ, ζ ) ∈ [ κ ν,j , κ ν,j + ) , a bijection ( π ) ∗ ( ξ, ζ ) ∶ → such that π ∗ ( ξ, ζ ) is the identity whenever ( ξ, ζ ) ∉ ( dom π ) (which will be in charge of the values ( πp ) ∗ ( ξ, ζ ) in the case that ξ ∉ ⋃ σ,i a σi is not a linking ordinal); hich defines for p ∈ D π , p = ( p ∗ , ( p σi , a σi ) σ,i ) , the image πp = ∶ p ′ = ( p ′∗ , (( p ′ ) σi , ( a ′ ) σi ) σ,i ) as follows:We will have supp p ′ = supp p , dom p ′ = dom p . Moreover: • Concerning the linking ordinals, for all ( σ, i ) ∈ supp p ′ = supp p and κ ν,j < κ σ : – ( a ′ ) σi ∩ [ κ ν,j , κ ν,j + ) = a σi ∩ [ κ ν,j , κ ν,j + ) for ( σ, i ) ∉ supp π ( ν, j ) , – ( a ′ ) σi ∩ [ κ ν,j , κ ν,j + ) = a λk ∩ [ κ ν,j , κ ν,j + ) with ( λ, k ) = F π ( ν, j )( σ, i ) inthe case that ( σ, i ) ∈ supp π ( ν, j ) . • Concerning the ( p ′ ) σi with ( σ, i ) ∈ supp π , – for ζ ∈ dom π , (( p ′ ) σi ( ζ ) ∣ ( σ, i ) ∈ supp π ( ν, j )) = π ( ζ )( p σi ( ζ ) ∣ ( σ, i ) ∈ supp π ( ν, j )) , – and in the case that ζ ∉ dom π , we will have ( p ′ ) σi ( ζ ) = p λk ( ζ ) with ( λ, k ) = G π ( ν, j )( σ, i ) . • Whenever ( σ, i ) ∉ supp π , then ( p ′ ) σi = p σi . • We now turn to p ′∗ . Consider an interval [ κ ν,j , κ ν,j + ) . For any ( σ, i ) ∈ supp π ( ν, j ) , let { ξ σi } ∶ = a σi ∩ [ κ ν,j , κ ν,j + ) . For ζ ∈ [ κ ν,j , κ ν,j + ) ∩ dom π ,we will have ( p ′∗ ( ξ σi , ζ ) ∣ ( σ, i ) ∈ supp π ( ν, j )) = ( π ) ∗ ( ζ )( ξ σi ∣ ( σ, i ) ∈ supp π ( ν, j ))( p ∗ ( ξ σi , ζ ) ∣ ( σ, i ) ∈ supp π ( ν, j )) . In the case that ζ ∈ [ κ ν,j , κ ν,j + ) ∩ ( dom p ∖ dom π ) , we will have for ( σ, i ) ∈ supp π ( ν, j ) : p ′∗ ( ξ σi , ζ ) ∶ = p ∗ ( ξ λk , ζ ) , where ( λ, k ) = G π ( ν, j ) ○ ( F π ( ν, j )) − ( σ, i ) .Finally, if ( ξ, ζ ) ∈ ( dom p ) with ξ, ζ ∈ [ κ ν,j , κ ν,j + ) such that ξ ∉ ⋃ σ,i a σi , then p ′∗ ( ξ, ζ ) = ( π ) ∗ ( ξ, ζ )( p ∗ ( ξ, ζ )) . For any π ∈ A , we have D π ∈ D with supp D π ∶ = supp π and dom D π ∶ = dom π .Moreover, whenever p is a condition in D π , then p ′ ∶ = π p ∈ P is well-definedwith p ′ ∈ D π , since supp p ′ = supp p , dom p ′ = dom p , and ⋃ σ,i a σi = ⋃ σ,i ( a ′ ) σi byconstruction.Here we use that π is only defined on D π and not on the entire forcing P , sincewe have to make sure that in our construction of the p ′∗ ( ξ σi , ζ ) for ζ ∉ dom π , wedo not run out of dom p .It is not difficult to see that for any p , q ∈ D π with q ≤ p , also q ′ ≤ p ′ holds. Thelinking property follows readily from our definition of the p ′∗ ( ξ σi , ζ ) for ζ ∉ dom π .25henever π ∈ A and D ∈ D with D ⊆ D π , it follows that the map π ∶ = π ↾ D is contained in A , as well. Here we have to use that the maps π do not disturbthe conditions’ domain or support, and merely permute the linking ordinals. Inparticular, whenever p ∈ D , it follows that the image π p is contained in D , as well.It remains to verify that A can be endowed with a group structure. More precisely:We will show that for any D ∈ D , the collection ( A ) D ∶ = { π ∈ A ∣ D π = D } givesa group; and then take the direct limit A ∶ = lim Ð→ ( A ) D .Firstly, for any π ∈ A , it is not difficult to write down a map ν ∈ A with D ν = D π such that ν is the inverse of π :Let supp ν ∶ = supp π and dom ν ∶ = dom π . For any κ ν,j < κ γ , we set F ν ( ν, j ) ∶ = ( F π ( ν, j )) − , G ν ( ν, j ) ∶ = ( G π ( ν, j )) − ; and for ζ ∈ [ κ ν,j , κ ν,j + ) , we let ν ( ζ ) ∶ = ( π ( ζ )) − . Regarding ( ν ) ∗ we use the following notation: For sets I , J with a bijection b ∶ I → J and a sequence ( x j ∣ j ∈ J ) , we denote by b ( x j ∣ j ∈ J ) the induced sequence parametrized by I : b ( x j ∣ j ∈ J ) ∶ = ( y i ∣ i ∈ I ) with y i ∶ = x b ( i ) for all i ∈ I . Whenever ζ ∈ [ κ ν,j , κ ν,j + ) ∩ dom π , and ( ξ σi ∣ ( σ, i ) ∈ supp π ( ν, j )) ∈ ( dom π ( ν, j )) supp π ( ν,j ) is a sequence of pairwise distinct ordinals, we set ( ν ) ∗ ( ξ σi ∣ ( σ, i ) ∈ supp π ( ν, j )) ∶ = F π ( ν, j ) ○ [( π ) ∗ ( F π ( ν, j ) − ( ξ σi ∣ ( σ, i ) ∈ supp π ( ν, j )) )] − ○ F π ( ν, j ) − , which is a bijection on 2 supp π ( ν,j ) .For ( ξ, ζ ) ∈ [ κ ν,j , κ ν,j + ) , let ( ν ) ∗ ( ξ, ζ ) ∶ = (( π ) ∗ ( ξ, ζ )) − .It is not difficult to verify that indeed, π ( ν ( p )) = ν ( π ( p )) = p holds for all p ∈ D π = D ν .Secondly, one has to make sure that for any π ∶ D π → D π , σ ∶ D σ → D σ in A ,there is a map τ ∈ A with D τ = D π ∩ D σ such that τ ( p ) = π ( σ ( p )) holds forall p ∈ D τ .Setting dom τ ∶ = dom π ∪ dom σ , supp τ ∶ = supp π ∪ supp σ , one can write down τ explicitly (similarly as for the inverse map), and verify that τ ( p ) = π ( σ ( p )) holds for all p ∈ D τ .Finally, the identity element id , which is the identity map on its domain D id ∶ = { p ∈ P ∣ ∀ κ ν,j < κ γ ∶ ( dom p ∩ [ κ ν,j , κ ν,j + ) ≠ ∅ ⇒ ⋃ ( σ,i )∈ supp p a σi ∩ [ κ ν,j , κ ν,j + ) ⊆ dom p ) } ,
26s contained in A as well with D id ⊇ D π for all π ∈ A .It follows that for any D ∈ D , the collection ( A ) D ∶ = { π ∈ A ∣ D π = D } givesa group with the identity element id D ∶ = id ↾ D ; and whenever D , D ′ ∈ D with D ⊆ D ′ , then there is a canonical homomorphism φ D ′ D ∶ ( A ) D ′ → ( A ) D thatmaps any π ∈ ( A ) D ′ to its restriction π ↾ D . We take the direct limit A ∶ = lim Ð→ ( A ) D = ⊔ ( A ) D / ∼ , with π ∼ π ′ iff π ( p ) = π ′ ( p ) holds for all p ∈ D π ∩ D π ′ . Then A becomes agroup with concatenation and the inverse elements as described in Chapter 4.1,and the identity element id ∶ = [ id ] as defined above.Now, we turn to P and define A , our collection of partial P -isomorphisms. Ev-ery π ∈ A will be a bijection π ∶ D π → D π with a dense set D π ∈ D , where D is defined as follows:Let D denote the collection of all D ⊆ P given by: • a countable support supp D ⊆ Succ , and • for every σ ∈ supp D , κ σ = κ σ + , a domain dom D ( σ ) = dom x D ( σ ) × dom y D ( σ ) ⊆ α σ × [ κ σ , κ σ ) with ∣ dom π ( σ )∣ < κ σ ,such that D = { p ∈ P ∣ supp p ⊇ supp D ∧ ∀ σ ∈ supp D dom p σ ⊇ dom D ( σ )} . Then every set D ∈ D is open dense; and whenever D , D ′ ∈ D , then the intersec-tion D ∩ D ′ is contained in D as well, with supp ( D ∩ D ′ ) = supp D ∪ supp D ′ , and dom x ( D ∩ D ′ )( σ ) = dom x D ( σ ) ∪ dom x D ′ ( σ ) , dom y ( D ∩ D ′ )( σ ) = dom y D ( σ ) ∪ dom y D ′ ( σ ) for all σ ∈ supp ( D ∩ D ′ ) .We now describe the two types of partial P -isomorphisms that will generate A :Like for A , our first goal is that for any p, q ∈ P which have “the same shape”, i.e. supp p = supp q and dom p σ = dom q σ for every σ ∈ supp p , there is an isomorphism π ∈ A with π p = q . These isomorphisms will be of the following form: For every σ ∈ supp π , we will have a collection of π ( σ )( i, ζ ) ∶ → ( i, ζ ) ∈ dom π ( σ ) ,such that for any p ∈ D π , the map π changes the value of p σ ( i, ζ ) if and only if π ( σ )( i, ζ ) ≠ id . In other words, ( π p ) σ ( i, ζ ) = π ( σ )( i, ζ )( p σ ( i, ζ )) .This allows for constructing an isomorphism π with π p = q for any pair ofconditions p , q that have the same supports and domains: One can simply set π ( σ )( i, ζ ) = id if p σ ( i, ζ ) = q σ ( i, ζ ) , and π ( σ )( i, ζ ) ≠ id in the case that p σ ( i, ζ ) ≠ q σ ( i, ζ ) .Secondly, for every pair of generic κ σ -subsets G σi and G σi ′ for σ ∈ Succ and i, i ′ < α σ ,we want an isomorphism π ∈ A such that πG σi = G σi ′ . Therefore, we include into A π = ( π ( σ ) ∣ σ ∈ supp π ) such that for every σ ∈ supp π , there isa bijection f π ( σ ) on a countable set supp π ( σ ) ⊆ α σ ; and π is defined as follows:Whenever p ∈ D π , then ( π p ) σ ( i, ζ ) = p σ ( f π ( σ )( i ) , ζ ) for all ( i, ζ ) ∈ dom p σ .Then πG σi = G σf π ( σ )( i ) .Roughly speaking, A will be generated by these two types of isomorphisms. Inorder to retain a group structure, the values ( π p ) σ ( i, ζ ) for ( i, ζ ) ∈ dom π ( σ ) and i ∈ supp π ( σ ) have to be treated separately: For every ζ ∈ dom y π ( σ ) , therewill be a bijection π ( ζ ) ∶ supp π ( σ ) → supp π ( σ ) such that (( π p ) σ ( i, ζ ) ∣ i ∈ supp π ( σ )) = π ( ζ )( p σ ( i, ζ ) ∣ i ∈ supp π ( σ )) .This gives the following definition: Definition 12. A consists of all isomorphisms π ∶ D π → D π , π = ( π ( σ ) ∣ σ ∈ supp π ) with countable support supp π ⊆ Succ , such that for all σ ∈ supp π , κ σ = κ σ + , there are • a countable set supp π ( σ ) ⊆ α σ with a bijection f π ( σ ) ∶ supp π ( σ ) → supp π ( σ ) , • a domain dom π ( σ ) = dom x π ( σ ) × dom y π ( σ ) ⊆ α σ × [ κ σ , κ σ ) with ∣ dom π ( σ )∣ < κ σ , such that supp π ( σ ) ⊆ dom x π ( σ ) , • for every ( i, ζ ) ∈ α σ × [ κ σ , κ σ ) a bijection π ( σ )( i, ζ ) ∶ → , with π ( σ )( i, ζ ) = id whenever ( i, ζ ) ∉ dom π ( σ ) , and • for every ζ ∈ dom y π ( σ ) a bijection π ( ζ ) ∶ supp π ( σ ) → supp π ( σ ) with D π = { p ∈ P ∣ supp p ⊇ supp π ∧ ∀ σ ∈ supp π dom p σ ⊇ dom π ( σ )} , and for every p ∈ D π , the image π p is defined as follows:We will have supp ( π p ) = supp p with ( π p ) σ = p σ whenever σ ∉ supp π . Moreover,for σ ∈ supp π , • for every ( i, ζ ) ∈ dom p σ with i ∉ supp π ( σ ) , we have ( π p ) σ ( i, ζ ) = π ( σ )( i, ζ )( p σ ( i, ζ )) , • for every i ∈ supp π ( σ ) and ζ ∈ dom y p σ ∖ dom y π ( σ ) , ( π p ) σ ( i, ζ ) = p σ ( f π ( σ )( i ) , ζ ) , and • for all ζ ∈ dom y π ( σ ) , (( π p ) σ ( i, ζ ) ∣ i ∈ supp π ( σ )) = π ( ζ )( p σ ( i, ζ ) ∣ i ∈ supp π ( σ )) .
28n other words: Outside the domain dom π ( σ ) , we have a swap of the horizontallines p σ ( i, ⋅ ) for i ∈ supp π ( σ ) , according to f π ( σ ) . If ζ ∈ dom π ( σ ) , then thevalues ( π p ) σ ( i, ζ ) for i ∈ supp π ( σ ) are given by the map π ( ζ ) . Any remainingvalue ( π p ) σ ( i, ζ ) with i ∉ supp π ( σ ) is equal to p σ ( ζ, i ) or not, depending onwhether π ( σ )( i, ζ ) ∶ → D π in order to make sure that dom ( π p ) σ = dom p σ . Inparticular, we do not want to run out of dom x p σ when permuting the p σ ( i, ⋅ ) for i ∈ supp π ( σ ) .It is not difficult to see that any map π ∶ D π → D π as in Definition 12 is order-preserving.Whenever π ∈ A and D ∈ D with D ⊆ D π , then the map π ∶ = π ↾ D is con-tained in A , as well. Here we have to use that the maps π do not disturb theconditions’ support or domain. In particular, whenever p ∈ D , it follows that π p is contained in D , as well.It remains to verify that A can be endowed with a group structure, which happenssimilarly as for A :Firstly, for any π ∈ A , π ∶ D π → D π , one can write down a map ν ∈ A with D ν = D π such that ν is the inverse of π .Secondly, whenever π , σ ∈ A , π ∶ D π → D π , σ ∶ D σ → D σ , one can explicitlywrite down a map τ ∈ A with D τ = D π ∩ D σ such that τ ( p ) = π ( σ ( p )) holdsfor all p ∈ D τ .Thirdly, A contains the identity element id , which is the identity on its domain D id = P .As before, we set ( A ) D ∶ = { π ∈ A ∣ D π = D } for D ∈ D , and take the directlimit A ∶ = lim Ð→ ( A ) D = ⊔ ( A ) D / ∼ , with π ∼ π ′ iff π ( p ) = π ′ ( p ) holds for all p ∈ D π ∩ D π ′ . Then A becomes agroup with the identity element id ∶ = [ id ] .Now, we can define our group A : Definition 13.
Let
D ∶ = D × D and A ∶ = A × A ; i.e. A is the collection ofall π = ( π , π ) with π ∈ A and π ∈ A as defined above, and domain D π = D π × D π ∈ D . Let A ∶ = A × A . This is a group with the identity element id = ( id , id ) . For D ∈ D , we define Name ( P ) D as in Chapter 4.1. For D ∈ D and ˙ x ∈ Name ( P ) ,we define recursively: x D ∶ = {( y D , p ) ∣ ˙ y ∈ dom ˙ x , p ∈ D , p ⊩ ˙ y ∈ ˙ x } . x D ∈ Name ( P ) D ; and for any G a V -generic filter on P , it follows that ( x D ) G = ˙ x G .It is not difficult to check that whenever D , D ′ ∈ D and ˙ x ∈ Name ( P ) , then x DD ′ = x D ′ .The next lemma is important to establish a notion of symmetry that is coherentwith the equivalence relation ∼ : Lemma 14.
Let π , π ′ ∈ A with π ∼ π ′ , i.e. π ↾ ( D π ∩ D π ′ ) = π ′ ↾ ( D π ∩ D π ′ ) .Then for ˙ x ∈ Name ( P ) , it follows that πx D π = x D π if and only if π ′ x D π ′ = x D π ′ . We prove the following more general statement by induction over α : Lemma 15.
Let π , π ′ ∈ A with π ∼ π ′ , i.e. π ↾ ( D π ∩ D π ′ ) = π ′ ↾ ( D π ∩ D π ′ ) ,and α ∈ Ord . Then for any ˙ y , ˙ z ∈ Name ( P ) with rk ˙ y = rk ˙ z = α , it follows that πy D π = z D π if and only if π ′ y D π ′ = z D π ′ .Proof. W.l.o.g. we can assume that D π ′ ⊆ D π , since the map ̃ π ∶ = π ↾ ( D π ∩ D π ′ ) = π ′ ↾ ( D π ∩ D π ′ ) is contained in A as well, with D ̃ π = D π ∩ D π ′ .Consider α ∈ Ord, and assume that the statement is true for all β < α . Let ˙ y ,˙ z ∈ Name ( P ) with rk ˙ y = rk ˙ z = α .“ ⇒ ”: First, assume that πy D π = z D π . We only prove z D π ′ ⊆ π ′ y D π ′ ; the otherinclusion is similar.Let ( x D π ′ , p ) ∈ z D π ′ , i.e. ˙ x ∈ dom ˙ z , p ∈ D π ′ , and p ⊩ ˙ x ∈ ˙ z . Then also p ∈ D π holds. Hence, ( x D π , p ) ∈ z D π , and z D π = πy D π by assumption; sothere must be ˙ u ∈ dom ˙ y with x D π = πu D π . Setting q ∶ = π − p , it follows that q ⊩ u D π ∈ y D π and q ⊩ ˙ u ∈ ˙ y .Since rk ˙ u = rk ˙ x < α , our inductive assumption implies that x D π ′ = π ′ u D π ′ holds. Hence, ( x D π ′ , p ) = ( πu D π ′ , πq ) , which is contained in π ′ y D π ′ , since˙ u ∈ dom ˙ y , q ∈ D π ′ (since p ∈ D π ′ and q = π − p ), and q ⊩ ˙ u ∈ ˙ y .“ ⇐ ”: Now, we assume that π ′ y D π ′ = z D π ′ . As before, we only prove the inclusion z D π ⊆ πy D π .Consider ( x D π , p ) ∈ z D π , i.e. ˙ x ∈ dom ˙ z , p ∈ D π and p ⊩ ˙ x ∈ ˙ z . Let ̃ p ≤ p with ̃ p ∈ D π ′ . Then ( x D π ′ , ̃ p ) ∈ z D π ′ = π ′ y D π ′ , so there must be ˙ u ∈ dom ˙ y with x D π ′ = π ′ u D π ′ . By the inductive assumption, it follows that x D π = πu D π ,since rk ˙ u = rk ˙ x < α . Let q ∶ = π − p . We have to show that ( πu D π , πq ) ∈ πy D π .Since ˙ u ∈ dom ˙ y and q ∈ D π , it suffices to verify that q ⊩ ˙ u ∈ ˙ y . We provethat whenever r ≤ q , r ∈ D π ′ , then r ⊩ ˙ u ∈ ˙ y . Consider r ≤ q with r ∈ D π ′ .Then πr ∈ D π ′ , and πr ≤ p implies that πr ⊩ ˙ x ∈ ˙ z . Hence, ( x D π ′ , πr ) ∈ z D π ′ ,and z D π ′ = π ′ y D π ′ by assumption. Now, ( π ′ u D π ′ , π ′ r ) = ( x D π ′ , πr ) ∈ π ′ y D π ′ implies that r ⊩ u D π ′ ∈ y D π ′ ; hence, r ⊩ ˙ u ∈ ˙ y as desired.30 .3 Constructing F . Now, we define our collection of A -subgroups that will generate a normal filter F on A , establishing our notion of symmetry.We will introduce two different types of subgroups.Firstly, for any 0 < η < γ , i < α η (with η ∈ Lim or η ∈ Succ), let
F ix ( η, i ) ∶ = { [ π ] ∈ A ∣ ∀ p ∈ D π ( πp ) ηi = p ηi } . Whenever π ∼ π ′ , it follows that ( πp ) ηi = p ηi for all p ∈ D π if and only if ( π ′ p ) ηi = p ηi for all p ∈ D π ′ . Hence, F ix ( η, i ) is well-defined, and clearly, any F ix ( η, i ) is asubgroup of A .By including F ix ( η, i ) into our filter F , we make sure that any canonical name˙ G ηi for the i -th generic κ η -subset G ηi is hereditarily symmetric, since π ˙ G ηi = ˙ G ηi forall π ∈ F ix ( η, i ) . Hence, our eventual model N will contain any generic κ η -subset G ηi .Now, we turn to the second type of A -subgroup. For any 0 < λ < γ and k < α λ (with λ ∈ Lim or λ ∈ Succ), we need in N a surjection s ∶ ℘ ( κ λ ) → k in order to makesure that θ N ( κ λ ) ≥ α λ . However, the sequence ( G λi ∣ i < α λ ) must not be includedinto N , since θ N ( κ λ ) ≤ α λ , so N must not contain a surjection s ∶ ℘ ( κ λ ) → α λ .The idea (which appears in [GK12] in a slightly different setting, and in [FK16]similar as here) is that for any 0 < λ < γ and k < α λ , we define a “cloud” aroundeach G λi for i ≤ k , denoted by ( ̃ G λi ) ( k ) , and make sure that the “sequence of clouds” ( ( ̃ G λi ) ( k ) ∣ i < k ) makes its way into N .When defining these subgroups, we have to treat limit cardinals and successorcardinals separately.For λ ∈ Lim, k < α λ , let H λk ∶ = { [ π ] ∈ A ∣ ∃ κ ν, < κ λ ∀ κ ν,j ∈ [ κ ν, , κ λ ) ∀ i ≤ k ∶ (( λ, i ) ∉ supp π ( ν, j ) ∨ G π ( ν, j )( λ, i ) = ( λ, i )) } . It is not difficult to verify that any H λk is well-defined and indeed a subgroup of A .Roughly speaking, H λk contains all [ π ] ∈ A such that above some κ ν, < κ λ , thereis no permutation of the vertical lines P λi ↾ [ κ ν, , κ λ ) for i ≤ k .This implies that for any i, j < k with i ≠ j and [ π ] ∈ H λk , it is not possible that πG λi = G λj . Hence, for any i < k , we can define a “cloud” around G λi as follows: ( ˙ ̃ G λi ) ( k ) ∶ = {( πG λi D π , ) ∣ [ π ] ∈ H λk } . ( ̃ G λi ) ( k ) ∶ = (( ˙ ̃ G λi ) ( k ) ) G , it follows that ( ̃ G λi ) ( k ) is the orbit of G λi under H λk ;so two distinct orbits ( ̃ G λi ) ( k ) and ( ̃ G λj ) ( k ) for i ≠ j are disjoint. The sequence (( ̃ G λi ) ( k ) ∣ i < k ) , which has a canonical symmetric name stabilized by all π with [ π ] ∈ H λk , gives a surjection s ∶ ℘ ( κ λ ) → k in N (see Chapter 6.1).Now, we consider the case that λ ∈ Succ. For k < α λ , let H λk ∶ = { [ π ] ∈ A ∣ ∀ i ≤ k ( i ∉ supp π ( λ ) ∨ f π ( λ )( i ) = i ) } . Again, one can easily check that H λk is well-defined and indeed an A -subgroup.Whenever [ π ] is contained in H λk , then π does not interchange any G λi and G λj for i, j < k in the case that i ≠ j . Thus, as for λ ∈ Lim, we can define “clouds” ( ̃ G λi ) ( k ) for i ≤ k and obtain a surjection s ∶ ℘ ( κ η ) → k in N (see Chapter 6.1).We are now ready to define our normal filter F on A . Note that the F ix ( η, i ) and H λk are not normal A -subgroups: For instance, if [ π ] ∈ F ix ( η, i ) for some η ∈ Lim, i < α η , and σ ∈ A with G σ ( ν, j )( η, i ) = ( η, i ′ ) for all κ ν,j , < κ η such that [ π ] ∉ F ix ( η, i ′ ) , then in general, [ σ ] − [ π ][ σ ] is not contained in F ix ( η, i ) .However, it is not difficult to verify: Lemma 16. • For all σ ∈ A , and η ∈ Lim , i < α η , [ σ ] F ix ( η, i )[ σ ] − ⊇ F ix ( η, i ) ∩ ⋂ { F ix ( η m , i m ) ∣ m < ω, ( η m , i m ) ∈ supp σ } . In the case that σ ∈ A , and η ∈ Succ , i < α η , [ σ ] F ix ( η, i )[ σ ] − ⊇ F ix ( η, i ) ∩ ⋂ { F ix ( η, i m ) ∣ m < ω, i m ∈ supp σ ( η )} . • For σ ∈ A and λ ∈ Lim , k < α λ , [ σ ] H λk [ σ ] − ⊇ H λk ∩ ⋂ { F ix ( η m , i m ) ∣ m < ω, ( η m , i m ) ∈ supp σ } . In the case that λ ∈ Succ , k < α λ , [ σ ] H λk [ σ ] − ⊇ H λk ∩ ⋂ { F ix ( λ, i m ) ∣ m < ω, i m ∈ supp σ ( λ )} . Hence, it follows that countable intersections of the A -subgroups F ix ( η, i ) and H λk generate a normal filter on A . Definition 17.
Let F denote the filter on A defined as follows:A subgroup B ⊆ A is contained in F if there are (( η m , i m ) ∣ m < ω ) , (( λ m , k m ) ∣ m < ω ) with B ⊇ ⋂ m < ω F ix ( η m , i m ) ∩ ⋂ m < ω H λ m k m . F is a normal filter on A ; which is countablyclosed.Now, we can use F to establish our notion of symmetry. Definition 18. A P -name ˙ x is symmetric if { [ π ] ∈ A ∣ πx D π = x D π } ∈ F . Recursively, a name ˙ x is hereditarily symmetric , x ∈ HS , if ˙ x is symmetric, and ˙ y is hereditarily symmetric for all ˙ y ∈ dom ˙ x . By Lemma 14, this is well-defined, since for π ∼ π ′ and ˙ x ∈ Name ( P ) , it followsthat πx D π = x D π if and only if π ′ x D π ′ = x D π ′ .We will use the following properties: If ˙ x ∈ HS and π ∈ A , then firstly, it is notdifficult to verify that also x D π ∈ HS holds, and secondly, πx D π ∈ HS . For thesecond claim, one can check that whenever σ ∈ A with σx D σ = x D σ , then ( πσπ − ) πx D π D πσπ − = πx D π D πσπ − , and then use the normality of F . Let G be a V -generic filter on P . Our symmetric extension is N ∶ = V ( G ) = { ˙ x G ∣ ˙ x ∈ HS } . Since we do not use the standard method for constructing symmetric extensions,we first have to make sure that N ⊧ ZF .The symmetric forcing relation “ ⊩ s ” can be defined in the ground model as usu-ally, and the forcing theorem holds.Whenever ˙ x , ˙ y ∈ HS and p ∈ P , then p ⊩ s ˙ y ∈ ˙ x if and only if p ⊩ ˙ y ∈ ˙ x (with theordinary forcing relation ⊩ ) and p ⊩ s ˙ x = ˙ y if and only if p ⊩ ˙ x = ˙ y . In particular,for any ˙ x ∈ HS and D ∈ D , we have x D = {( y D , p ) ∣ ˙ y ∈ dom ˙ x , p ∈ D , p ⊩ s ˙ y ∈ ˙ x } . Since V ⊆ V ( G ) ⊆ V [ G ] and V ( G ) is transitive, it follows that V ( G ) satisfies theaxioms of Emptyset , F oundation , Extensionality and
Inf inity . The proofs ofthe axioms of
P airing , U nion and
Separation are similar to the proofs for thestandard construction, with some extra care needed to make sure that all the in-volved names are indeed symmetric.We give a proof of
P ower Set and
Replacement .33 emma 19. V ( G ) ⊧ P ower Set .Proof.
Consider X ∈ N , X = ˙ X G with ˙ X ∈ HS . We have to show that ℘ N ( X ) ∈ N .Let ˙ B ∶ = {( ˙ Y , p ) ∣ ˙ Y ∈ HS , ˙ Y ⊆ dom ˙ X × P , p ⊩ s ˙ Y ⊆ ˙ X } . Then ˙ B G = ℘ N ( X ) , since for any Y ∈ N with Y ⊆ X , there exists a name ˙ Y ∈ HS ,˙ Y G = Y , such that ˙ Y ⊆ dom ˙ X × P .It remains to make sure that the name ˙ B is symmetric. Consider π ∈ A with πX D π = X . Then B D π = { ( Y D π , p ) ∣ ˙ Y ∈ HS , ˙ Y ⊆ dom ˙ X × P , p ⊩ s ˙ Y ∈ ˙ B , p ∈ D π } . It is not difficult to check that B D π = { ( Y D π , p ) ∣ ˙ Y ∈ HS , ˙ Y ⊆ dom ˙ X × P , p ⊩ s ˙ Y ⊆ ˙ X , p ∈ D π } , since for any p ∈ D π and ˙ Y ∈ HS , ˙ Y ⊆ dom ˙ X × P , it follows that p ⊩ s ˙ Y ∈ ˙ B if andonly if p ⊩ s ˙ Y ⊆ ˙ X . Hence, πB D π = { ( πY D π , πp ) ∣ ˙ Y ∈ HS , ˙ Y ⊆ dom ˙ X × P , πp ⊩ s πY D π ⊆ πX D π , πp ∈ D π } . It remains to show that B D π = πB D π ; then { [ π ] ∈ A ∣ πB D π = B D π } ⊇ { [ π ] ∈ A ∣ πX D π = X D π } ∈ F as desired.For the inclusion B D π ⊆ πB D π , consider ( Y D π , p ) ∈ B D π as above. It suffices toconstruct ˙ Y ∈ HS , ˙ Y ⊆ dom ˙ X × P with πY D π = Y D π . Then setting p ∶ = π − p , itfollows that ( Y D π , p ) = ( πY D π , πp ) ∈ πB D π , since p ⊩ s Y D π ⊆ X D π and πX D π = X D π gives πp ⊩ s πY D π ⊆ πX D π .Let ˙ Y ∶ = { ( ˙ z, p ) ∣ ˙ z ∈ dom ˙ X , p ∈ D π , πz D π ∈ dom Y D π , πp ⊩ s πz D π ∈ Y D π } . Then πY D π = {( πz D π , πp )∣ ˙ z ∈ dom ˙ X , p ∈ D π , πz D π ∈ dom Y D π , p ⊩ s ˙ z ∈ ˙ Y } . We first show that whenever ˙ z ∈ dom ˙ X , p ∈ D π and πz D π ∈ dom Y D π as above,then p ⊩ s ˙ z ∈ ˙ Y if and only if πp ⊩ s πz D π ∈ Y D π .“ ⇒ ”: If πp ⊩ s πz D π ∈ Y D π , it follows that ( ˙ z, p ) ∈ ˙ Y ; hence, p ⊩ s ˙ z ∈ ˙ Y as desired.“ ⇐ ”: Now, assume that p ⊩ s ˙ z ∈ ˙ Y . Let H be a V -generic filter on P with πp ∈ H . We have to show that ( πz D π ) H ∈ ( Y D π ) H . Let H ′ ∶ = π − H . Then ( πz D π ) H = ˙ z H ′ , and p ∈ H ′ . Hence, ˙ z H ′ ∈ ˙ Y H ′ implies that there mustbe ( ˙ u, r ) ∈ ˙ Y with ˙ u H ′ = ˙ z H ′ and r ∈ H ′ . Then πr ⊩ s πu D π ∈ Y D π byconstruction of ˙ Y . Since πr ∈ H , it follows that ( πu D π ) H ∈ ( Y D π ) H , with ( πu D π ) H = ˙ u H ′ = ˙ z H ′ = ( πz D π ) H as desired.34ence, πY D π = {( πz D π , πp )∣ ˙ z ∈ dom ˙ X , p ∈ D π , πz D π ∈ dom Y D π , πp ⊩ s πz D π ∈ Y D π } . We have to make sure that πY D π = Y D π . The inclusion πY D π ⊆ Y D π is clear. Re-garding Y D π ⊆ πY D π , consider ( u D π , q ) ∈ Y D π with ˙ u ∈ dom ˙ Y ⊆ dom ˙ X and q ∈ D π such that q ⊩ s ˙ u ∈ ˙ Y . From u D π ∈ dom X D π = dom πX D π , it follows that there mustbe ˙ v ∈ dom ˙ X with u D π = πv D π . Let r ∶ = π − q . Then ( u D π , q ) = ( πv D π , πr ) ∈ πY D π ,since q ⊩ s ˙ u ∈ ˙ Y implies πr ⊩ s πv D π ∈ Y D π as desired.Thus, we have constructed ˙ Y ⊆ dom ˙ X × P with πY D π = Y D π . It remains to makesure that ˙ Y ∈ HS . Firstly, ˙ Y ⊆ dom ˙ X × P , so dom ˙ Y ⊆ HS . Secondly, for any σ ∈ A with σY D σ = Y D σ , it follows that ( π − σπ ) Y D π − σπ = ( π − σπ ) Y D π D π − σπ = ( π − σπ ) π − Y D π D π − σπ , and since σY D σ = Y D σ , one can easily check that ( π − σπ ) π − Y D π D π − σπ = π − Y D π D π − σπ , and π − Y D π D π − σπ = Y D π D π − σπ = Y D π − σπ . Since the name ˙ Y is symmetric, it follows by normality of F that ˙ Y is symmetric,as well. Hence, ˙ Y has all the desired properties; and it follows that B D π ⊆ πB D π .The inclusion πB D π ⊆ B D π is similar. Lemma 20. V ( G ) ⊧ Replacement .Proof.
Consider a ∈ N such that N ⊧ ∀ x ∈ a ∃ y ϕ ( x, y ) . We have to show thatthere is b ∈ N with N ⊧ ∀ x ∈ a ∃ y ∈ b ϕ ( x, y ) . Let a = ˙ a G with ˙ a ∈ HS . We proceed like in the proof of Replacement in ordinaryforcing extensions. For ˙ x ∈ dom ˙ a and p ∈ P , let α ( ˙ x, p ) ∶ = min { α ∣ ∃ ˙ w ∈ Name α ( P ) ∩ HS ∶ p ⊩ s ( ϕ ( ˙ x, ˙ w ) ∧ ˙ x ∈ ˙ a )} if such α exists, and α ( ˙ x, p ) ∶ =
0, else.By
Replacement in V , take β ∈ Ord with β ≥ sup { α ( ˙ x, p ) ∣ ˙ x ∈ dom ˙ a , p ∈ P } . Let˙ b ∶ = {( ˙ y, ) ∣ ˙ y ∈ Name β ( P ) ∩ HS } , b ∶ = ˙ b G . Then for all x ∈ a , it follows that there exists y ∈ b with N ⊧ ϕ ( x, y ) .It remains to show that the name ˙ b is symmetric. Let π ∈ A . Then b D π = {( y D π , q ) ∣ ˙ y ∈ Name β ( P ) ∩ HS , q ∈ D π } , and πb D π = {( πy D π , πq ) ∣ ˙ y ∈ Name β ( P ) ∩ HS , q ∈ D π } . We will show that πb D π = b D π .Since it is not possible to apply π to arbitrary P -names ˙ y with ˙ y ∉ Name ( P ) D π , weconstruct an alternative ̃ π which is sufficient for our purposes here. Recursively,we define for P -names ˙ y : ̃ π ( ˙ y ) ∶ = { (̃ π ( ˙ z ) , πq ) ∣ ∃ ( ˙ z, q ) ∈ ˙ y , q ≤ q , q ∈ D π } . Then for all ˙ y ∈ Name β ( P ) , it follows that ̃ π ( ˙ y ) ∈ Name β ( P ) , as well.Whenever H is a V -generic filter on P , H ′ ∶ = π − H and ˙ y ∈ Name ( P ) , then (̃ π ( ˙ y )) H = ˙ y H ′ , and one can easily check that πy D π = ̃ π ( ˙ y ) D π . Moreover, whenever σ ∈ A with σy D σ = y D σ , then setting τ ∶ = πσπ − , it followsrecursively that τ ̃ π ( ˙ y ) D τ = ̃ π ( ˙ y ) D τ . Hence, { [ τ ] ∈ A ∣ τ ̃ π ( ˙ y ) D τ = ̃ π ( ˙ y ) D τ } ⊇ { [ π ][ σ ][ π ] − ∣ [ σ ] ∈ A , σy D σ = y D σ } . In the case that ˙ y is symmetric, i.e. { [ σ ] ∈ A ∣ σy D σ = y D σ } ∈ F , it followsby normality that also { [ τ ] ∈ A ∣ τ ̃ π ( ˙ y ) D τ = ̃ π ( ˙ y ) D τ } ∈ F . Hence, ̃ π ( ˙ y ) ∈ HS whenever ˙ y ∈ HS .Now, πb D π = b D π follows: For the inclusion “ ⊆ ”, consider ( πy D π , πq ) ∈ πb D π with˙ y ∈ Name β ( P ) ∩ HS , q ∈ D π . Then also πq ∈ D π , and πy D π = ̃ π ( ˙ y ) D π , where ̃ π ( ˙ y ) ∈ Name β ( P ) ∩ HS ; so ( πy D π , πq ) = (̃ π ( ˙ y ) D π , πq ) ∈ b D π follows. The inclusion“ ⊇ ” is similar.Hence, ˙ b ∈ HS as desired.Thus, our symmetric extension N is a model of ZF . Since our forcing P is count-ably closed (Proposition 7), and our normal filter F generating N is countablyclosed, it follows that N ⊧ DC (see for example [Kar14, Lemma 1]). Moreover, N ⊧ AX (see [She10, p.3 and p.15]): For any cardinal λ , we have ([ λ ] ℵ ) N = [ λ ] ℵ ) V ; so the set [ λ ] ℵ can be well-ordered in N , using the according well-ordering of [ λ ] ℵ in V .Next, we want to show that N preserves all V -cardinals; which will follow from thefact that any set of ordinals X ⊆ α , X ∈ N , can be captured in a “mild” V -genericextension by a forcing notion as in Lemma 9 and Lemma 10.This Approximation Lemma demonstrates how our symmetric extension N can beapproximated from within by fairly nice V -generic extensions. Later on, this willbe a crucial step in keeping control over the values θ N ( κ η ) . Lemma 21 (Approximation Lemma) . Consider X ∈ N , X ⊆ α with X = ˙ X G suchthat πX D π = X D π holds for π ∈ A with [ π ] contained in the intersection ⋂ m < ω F ix ( η m , i m ) ∩ ⋂ m < ω F ix ( η m , i m ) ∩ ⋂ m < ω H λ m k m ∩ ⋂ m < ω H λ m k m , where (( η m , i m ) ∣ m < ω ) , (( η m , i m ) ∣ m < ω ) , (( λ m , k m ) ∣ m < ω ) and (( λ m , k m ) ∣ m < ω ) denote sequences with η m ∈ Lim , i m < α η m ; η m ∈ Succ , i m < α η m for all m < ω ;and λ m ∈ Lim , k m < α λ m ; λ m ∈ Succ , k m < α λ m for all m < ω .Then X ∈ V [ ∏ m < ω G η m i m × ∏ m < ω G η m i m ] . Proof.
Let X ′ ∶ = { β < α ∣ ∃ p = ( p ∗ , ( p σi , a σi ) σ,i , ( p σ ) σ ) ∶ p ⊩ s β ∈ ˙ X , ∀ m ∶ ( η m , i m ) ∈ supp p , ∀ m ∶ a η m i m = g η m i m , ( p η m i m ) m < ω ∈ ∏ m < ω G η m i m , ( p η i i m ) m < ω ∈ ∏ m < ω G η m i m } . Then X ′ ∈ V [ ∏ m < ω G η m i m × ∏ m < ω G η m i m ] , since the sequence ( g η m i m ) m < ω is contained in V . It remains to show that X = X ′ .The inclusion X ⊆ X ′ follows from the forcing theorem. Concerning “ ⊇ ” , assumetowards a contradiction there was β ∈ X ′ ∖ X . Take p as above with ( η m , i m ) ∈ supp p for all m < ω , and p ⊩ s β ∈ ˙ X , ∀ m ∶ a η m i m = g η m i m , ( p η m i m ) m < ω ∈ ∏ m < ω G η m i m , ( p η i i m ) m < ω ∈ ∏ m < ω G η m i m . Since β ∉ X , we can take p ′ ∈ G , p ′ = ( p ′∗ , (( p ′ ) σi , ( a ′ ) σi ) σ,i , (( p ′ ) σ ) σ ) with p ′ ⊩ s β ∉ ˙ X , such that ( η m , i m ) ∈ supp p ′ for all m < ω .First, we want to extend p and p ′ and obtain conditions p ≤ p , p ′ ≤ p ′ , p = ( p ∗ , ( p σi , a σi ) σ,i , ( p σ ) σ ) , p ′ = ( p ′∗ , (( p ′ ) σi , ( a ′ ) σi ) σ,i , (( p ′ ) σ ) σ ) such that the followingholds: • ∀ m < ω p η m i m = ( p ′ ) η m i m , a η m i m = ( a ′ ) η m i m ∀ m < ω p η m i m = ( p ′ ) η m i m • dom p = dom p ′ • supp p = supp p ′ • ⋃ ( σ,i )∈ supp p a σi = ⋃ ( σ,i )∈ supp p ′ ( a ′ ) σi • ∀ ( ν, j ) ∶ dom p ∩ [ κ ν,j , κ ν,j + ) ≠ ∅ → ( ⋃ σ,i a σi ∩ [ κ ν,j , κ ν,j + )) ⊆ dom p , ∀ ( ν, j ) ∶ dom p ′ ∩ [ κ ν,j , κ ν,j + ) ≠ ∅ → ( ⋃ σ,i ( a ′ ) σi ∩ [ κ ν,j , κ ν,j + )) ⊆ dom p ′ • supp p = supp p ′ • ∀ σ ∈ supp p = supp p ′ ∶ dom p ( σ ) = dom p ′ ( σ ) .We will now describe how p and p ′ can be constructed. First, we need a setsupp ∶ = supp p = supp p ′ . Consider s ∶ = sup { κ σ ∣ σ ∈ Lim , ∃ i < α σ ∶ ( σ, i ) ∈ supp p ∪ supp p ′ } . Then by closure of the sequence ( κ σ ∣ < σ < γ ) , it follows that s = κ γ forsome γ ≤ γ . If γ = γ , then cf κ γ = ω and we can take (( σ k , l k ) ∣ k < ω ) with σ k ∈ Lim, l k < α σ k for all k < ω such that ( κ σ k ∣ k < ω ) is cofinal in κ γ , and ( σ k , l k ) ∉ supp p ∪ supp p ′ for all k < ω . Letsupp ∶ = supp p ∶ = supp p ′ ∶ = supp p ∪ supp p ′ ∪ {( σ k , l k ) ∣ k < ω } . If γ < γ , we can set σ k ∶ = γ ∈ Lim for all k < ω and take ( l k ∣ k < ω ) such that l k < α σ k with ( σ k , l k ) = ( γ, l k ) ∉ supp p ∪ supp p ′ for all k < ω . Letsupp ∶ = supp p ∶ = supp p ′ ∶ = supp p ∪ supp p ′ ∪ {( σ k , l k ) ∣ k < ω } as before.The next step is to define the linking ordinals. Take a set X ⊆ κ γ such thatfor all intervals [ κ ν,j , κ ν,j + ) ⊆ κ γ , it follows that ∣ X ∩ [ κ ν,j , κ ν,j + )∣ = ℵ ; and X ∩ ( ⋃ ( σ,i )∈ supp p a σi ∪ ⋃ ( σ,i )∈ supp p ′ ( a ′ ) σi ) = ∅ . Let X ∶ = X ∪ ⋃ σ,i a σi ∪ ⋃ σ,i ( a ′ ) σi . Our aim is to construct p and p ′ such that ⋃ σ,i a σi = ⋃ σ,i ( a ′ ) σi = X .Consider an interval [ κ ν,j , κ ν,j + ) ⊆ κ γ . For every ( σ, i ) ∈ supp p with κ σ > κ ν,j ,we let a σi ∩ [ κ ν,j , κ ν,j + ) ∶ = a σi ∩ [ κ ν,j , κ ν,j + ) .Define { ξ k ( ν, j ) ∣ k < ω } ∶ = ( X ∩ [ κ ν,j , κ ν,j + )) ∖ ⋃ σ,i a σi . This set has cardinality ℵ by construction of X .38oreover, let { ( σ k ( ν, j ) , l k ( ν, j )) ∣ k < ω } = ∶ {( σ, i ) ∈ supp p ∖ supp p ∣ κ σ > κ ν,j } . This set also has cardinality ℵ by construction of supp p = supp .Now, for any k < ω , let a σ k ( ν,j ) l k ( ν,j ) ∩ [ κ ν,j , κ ν,j + ) ∶ = { ξ k ( ν, j )} . Together with same construction for p ′ , we obtain the linking ordinals a σi , ( a ′ ) σi for ( σ, i ) ∈ supp = supp p = supp ( p ′ ) such that the independence property holds,and ⋃ σ,i a σi = ⋃ σ,i ( a ′ ) σi = X .Next, we construct dom ∶ = dom p = dom ( p ′ ) ∶ = ⋃ ν,j [ κ ν,j , δ ν,j ) as follows: Con-sider an interval [ κ ν,j , κ ν,j + ) ⊆ κ γ . In the case that dom p ∩ [ κ ν,j , κ ν,j + ) = dom ( p ′ ) ∩ [ κ ν,j , κ ν,j + ) = ∅ , let δ ν,j ∶ = κ ν,j . Otherwise, take δ ν,j ∈ [ κ ν,j , κ ν,j + ) such that ( dom p ∪ dom p ′ ∪ X ) ∩ [ κ ν,j , κ ν,j + ) ⊆ [ κ ν,j , δ ν,j ) . (This is possiblesince the set X ∩ [ κ ν,j , κ ν,j + ) is countable, and any κ ν,j + is a successor cardinal.)Let dom ∶ = dom p ∶ = dom p ′ ∶ = ⋃ ν,j [ κ ν,j , δ ν,j ) . This set is bounded below all regular κ ν, by construction, since dom p and dom p ′ are bounded below all regular κ ν, .Now, for ( σ, i ) ∈ supp p , let p σi ∶ dom p σi → p σi = dom p ∩ κ σ , such that p σi ⊇ p σi for all ( σ, i ) ∈ supp p , and in the casethat ( σ, i ) = ( η m , i m ) for some m < ω , we additionally require that p η m i m ⊇ ( p ′ ) η m i m .This is possible, since p ′ ∈ G and p η m i m ∈ G η m i m , so p η m i m and ( p ′ ) η m i m are compatible.We define p ∗ on the according domain ⋃ ν,j [ κ ν,j , δ ν,j ) such that p ∗ ⊇ p ∗ , and the linking property holds for p ≤ p : Consider an interval [ κ ν,j , κ ν,j + ) with δ ν,j > κ ν,j .For ζ ∈ ( dom p ∖ dom p ) ∩ [ κ ν,j , κ ν,j + ) and { ξ } ∶ = a σi ∩ [ κ ν,j , κ ν,j + ) for some ( σ, i ) ∈ supp p , it follows by construction that ξ ∈ dom p . Let p ∗ ( ξ, ζ ) ∶ = p σi ( ζ ) .For all ξ , ζ ∈ dom p ∩ [ κ ν,j , κ ν,j + ) , we set p ∗ ( ξ, ζ ) ∶ = p ∗ ( ξ, ζ ) ; and p ∗ ( ξ, ζ ) ∈ { , } arbitrary for the ξ , ζ ∈ dom p remaining.Concerning p ′ , we set ( p ′ ) η m i m = p η m i m for all m < ω . Then ( p ′ ) η m i m ⊇ ( p ′ ) η m i m byconstruction. For the ( σ, i ) ∈ supp ( p ′ ) = supp p remaining, we can set ( p ′ ) σi arbitrarily on the given domain such that ( p ′ ) σi ⊇ ( p ′ ) σi .Finally, we let ( p ′ ) ∗ ⊇ ( p ′ ) ∗ according to the linking property for p ′ ≤ p ′ (sameconstruction as for p ∗ ).It follows that p ≤ p and p ′ ≤ p ′ , and p and p ′ have all the required properties.The construction of p ≤ p and p ′ ≤ p ′ is similar.Our aim is to write down an isomorphism π ∈ A with the following properties:39 p ∈ D π with πp = p ′ , • π ∈ ⋂ m < ω F ix ( η m , i m ) ∩ ⋂ m < ω F ix ( η m , i m ) ∩ ⋂ m < ω H λ m k m ∩ ⋂ m < ω H λ m k m (then πX D π = X D π follows).From p ⊩ s β ∈ ˙ X , we will then obtain πp ⊩ s β ∈ πX D π ; hence, p ′ ⊩ s β ∈ X D π . Thiswill be a contradiction towards p ′ ⊩ s β ∉ ˙ X .We start with π . Let dom π ∶ = dom p = dom p ′ , and supp π ∶ = supp p = supp p ′ . • Consider an interval [ κ ν,j , κ ν,j + ) . We define F π ( ν, j ) ∶ supp π ( ν, j ) → supp π ( ν, j ) as follows: Let F π ( ν, j )( σ, i ) ∶ = ( λ, k ) in the case that ( a ′ ) σi ∩ [ κ ν,j , κ ν,j + ) = a λk ∩ [ κ ν,j , κ ν,j + ) . This is well-defined by the independenceproperty , and since we have arranged ⋃ σ,i a σi = ⋃ σ,i ( a ′ ) σi . • For every interval [ κ ν,j , κ ν,j + ) , let G π ( ν, j )( σ, i ) = ( σ, i ) for all ( σ, i ) ∈ supp π ( ν, j ) .(These maps G π ( ν, j ) will be the only parameters of π which are not de-termined by the requirement that π p = p ′ . However, in order to make surethat π ∈ ⋂ m < ω F ix ( η m , i m ) ∩ ⋂ m < ω H λ m k m , we firstly need G π ( ν, j )( η m , i m ) = ( η m , i m ) for all m < ω ; and secondly, whenever m < ω and i ≤ k m , we needthat G π ( ν, j )( λ m , i ) = ( λ m , i ) for all κ ν,j above a certain κ ν, .) • For ζ ∈ [ κ ν,j , κ ν,j + ) ∩ dom π , we define π ( ζ ) ∶ supp π ( ν,j ) → supp π ( ν,j ) asfollows: For ( ǫ ( σ,i ) ∣ ( σ, i ) ∈ supp π ( ν, j )) ∈ supp π ( ν,j ) given, let π ( ζ )( ǫ ( σ,i ) ∣( σ, i ) ∈ supp π ( ν, j )) ∶ = (̃ ǫ ( σ,i ) ∣ ( σ, i ) ∈ supp π ( ν, j )) such that ̃ ǫ ( σ,i ) = ǫ ( σ,i ) whenever p σi ( ζ ) = ( p ′ ) σi ( ζ ) , and ̃ ǫ ( σ,i ) ≠ ǫ ( σ,i ) in the case that p σi ( ζ ) ≠ ( p ′ ) σi ( ζ ) . • Let now ζ ∈ dom π ∩ [ κ ν,j , κ ν,j + ) , and ( ξ σi ( ν, j ) ∣ ( σ, i ) ∈ supp π ( ν, j )) ∈ dom π ( ν, j ) supp π ( ν,j ) . The map π ∗ ( ζ )( ξ σi ( ν, j ) ∣ ( σ, i ) ∈ supp π ( ν, j )) ∶ supp π ( ν,j ) → supp π ( ν,j ) is defined as follows: A sequence ( ǫ ( σ,i ) ∣ ( σ, i ) ∈ supp π ( ν, j )) is mapped to (̃ ǫ ( σ,i ) ∣ ( σ, i ) ∈ supp π ( ν, j )) with ̃ ǫ ( σ,i ) = ǫ ( σ,i ) if p ∗ ( ξ σi ( ν, j ) , ζ ) = p ′∗ ( ξ σi ( ν, j ) , ζ ) , and ̃ ǫ ( σ,i ) ≠ ǫ ( σ,i ) in the case that p ∗ ( ξ σi ( ν, j ) , ζ ) ≠ p ′∗ ( ξ σi ( ν, j ) , ζ ) . • For ( ξ, ζ ) ∈ [ κ ν,j , κ ν,j + ) , the map π ∗ ( ξ, ζ ) ∶ → π ∗ ( ξ, ζ ) = id in the case that ( ξ, ζ ) ∉ ( dom π ( ν, j )) . If ξ, ζ ∈ dom π ( ν, j ) ,let π ∗ ( ξ, ζ ) = id if p ∗ ( ξ, ζ ) = p ′∗ ( ξ, ζ ) , and π ∗ ( ξ, ζ ) ≠ id in the case that p ∗ ( ξ, ζ ) ≠ p ′∗ ( ξ, ζ ) .This defines π . Directly by construction, it follows that π p = p ′ : Let π p = ∶ (( πp ) ∗ , (( πp ) σi , ( πa ) σi ) σ,i , ( πp σ ) σ ) . Then for any ( σ, i ) ∈ supp ( π p ) = supp p and κ ν,j < κ σ , we have ( πa ) σi ∩ [ κ ν,j , κ ν,j + ) = a λk ∩ [ κ ν,j , κ ν,j + ) , where ( λ, k ) = F π ( ν, j )( σ, i ) ; hence, a λk ∩ [ κ ν,j , κ ν,j + ) = ( a ′ ) σi ∩ [ κ ν,j , κ ν,j + ) as desired.40or any ζ ∈ dom p , it follows by definition of π ( ζ ) that (( πp ) σi ( ζ ) ∣ ( σ, i ) ∈ supp π ( ν, j )) = (( p ′ ) σi ( ζ ) ∣ ( σ, i ) ∈ supp π ( ν, j )) , and similarly, ( πp ) ∗ ( ξ, ζ ) = p ′∗ ( ξ, ζ ) for all ( ξ, ζ ) ∈ dom ( πp ) ∗ = dom p ∗ .Hence, π p = p ′ .It remains to verify that π ∈ ⋂ m F ix ( η m , i m ) ∩ ⋂ m H λ m k m . Consider a condition r ∈ D π and let r ′ ∶ = π r . Take an interval [ κ ν,j , κ ν,j + ) ⊆ κ γ . Then for any m < ω with ( η m , i m ) ∈ supp π ( ν, j ) and ζ ∈ dom π ( ν, j ) , it follows that ( r ′ ) η m i m ( ζ ) = r η m i m ( ζ ) by construction of the map π ( ζ ) , since we have arranged p η m i m ( ζ ) = ( p ′ ) η m i m ( ζ ) .In the case that ζ ∈ [ κ ν,j , κ ν,j + ) with ζ ∈ dom r ∖ dom π , it follows for m < ω that ( r ′ ) η m i m ( ζ ) = r λk ( ζ ) , where ( λ, k ) = G π ( ν, j )( η m , i m ) = ( η m , i m ) as desired.Hence, ( r ′ ) η m i m = r η m i m for all m < ω . Since r ∈ D π was arbitrary, it follows that π ∈ ⋂ m < ω F ix ( η m , i m ) .Similarly, π ∈ ⋂ m H λ m k m follows from the fact that G π ( ν, j ) = id for all intervals [ κ ν,j , κ ν,j + ) ⊆ κ γ .Now, we turn to the map π .Let supp π ∶ = supp p = supp p ′ , and dom π ( σ ) ∶ = dom p ( σ ) = dom p ′ ( σ ) for σ ∈ supp π . We set supp π ( σ ) ∶ = ∅ for all σ ∈ supp π . Then we only have to de-fine maps π ( σ )( i, ζ ) ∶ → σ ∈ supp π , ( i, ζ ) ∈ dom π ( σ ) : Let π ( σ )( i, ζ ) = id if p ( σ )( i, ζ ) = p ′ ( σ )( i, ζ ) , and π ( σ ) ≠ id in the case that p ( σ )( i, ζ ) ≠ p ′ ( σ )( i, ζ ) .Clearly, π p = p ′ . Moreover, π ∈ ⋂ m F ix ( η m , i m ) : Let m < ω and r ∈ D π with η m ∈ supp r and i m ∈ dom x r ( η m ) . In the case that η m ∈ supp π , it followsfor any ζ ∈ dom y r ( η m ) that ( πr )( η m )( i m , ζ ) = π ( η m )( i m , ζ )( r ( η m )( i m , ζ )) = r ( η m )( i m , ζ ) by construction of π , since we have arranged that p ′ ( η m )( i m , ζ ) = p ( η m )( i m , ζ ) whenever ( i m , ζ ) ∈ dom p ( η m ) = dom p ′ ( η m ) = dom π ( η m ) . If η m ∉ supp π , then ( πr )( η m ) = r ( η m ) by construction.Finally, π ∈ ⋂ m H λ m k m follows from the fact that supp π ( λ ) = λ ∈ supp π .Hence, the map π has all the desired properties.This finishes the proof of X = X ′ , and X = X ′ ∈ V [ ∏ m < ω G η m i m × ∏ m < ω G η m i m ] follows.It is not difficult to see that with the exception of the maps G π ( ν, j ) , all theparameters describing π are given by the requirement that πp = p ′ . We call anisomorphism π ∈ A of this form a standard isomorphism for πp = p ′ .With the same proofs as for Lemma 2 and 3, one can show:41 emma 22. Let (( σ m , i m ) ∣ m < ω ) , (( σ m , i m ) ∣ m < ω ) with σ m ∈ Lim , i m < α σ m ,and σ m ∈ Succ , i m < α σ m for all m < ω . Then ∏ m < ω P σ m × ∏ m < ω P σ m preservescardinals, cofinalities and the GCH . Hence, the
Approximation Lemma
21 implies:
Corollary 23.
Cardinals and cofinalities are V - N -absolute. We will now take a closer look at the intermediate generic extensions introducedin the
Approximation Lemma
21. Firstly, we replace the generic filters G σ m i m by G ∗ ( g σ m i m ) , and secondly, we factor at κ η (or κ η + ). Definition 24.
For < η < γ , we say that (( a m ) m < ω , ( σ m , i m ) m < ω ) is an η -goodpair if the following hold: • ( a m ∣ m < ω ) is a sequence of pairwise disjoint κ η -subsets, such that for all m < ω and κ ν, < κ η , it follows that ∣ a m ∩ [ κ ν, , κ ν, + )∣ = , • for all m < ω , we have σ m ∈ Succ with σ m ≤ η , i m < α σ m , • if m ≠ m ′ , then ( σ m , i m ) ≠ ( σ m ′ , i m ′ ) . As in Lemma 9 and Lemma 10, it follows that for any η -good pair (( a m ) m < ω , ( σ m , i m ) m < ω ) , ∏ m < ω G ∗ ( a m ) × ∏ m < ω G σ m i m is a V -generic filter on ∏ m < ω ( P η ) ω × ∏ m < ω P σ m . Proposition 25.
Let < η < γ and X ∈ N with X ⊆ κ η . If κ η + > κ + η (or κ η = κ γ with γ = γ + ), it follows that there is an η -good pair (( a m ) m < ω , ( σ m , i m ) m < ω ) with X ∈ V [ ∏ m < ω G ∗ ( a m ) × ∏ m < ω G σ m i m ] . Proof.
By the
Approximation Lemma
21, there are sequences (( σ m , i m ) ∣ m < ω ) , (( σ m , i m ) ∣ m < ω ) of pairwise distinct pairs with σ m ∈ Lim , i m < α σ m ; σ m ∈ Succ , i m < α σ m for all m < ω , such that X ∈ V [ ∏ m < ω G σ m i m × ∏ m < ω G σ m i m ] . The sequence of linking ordinals ( g σ m i m ∣ m < ω ) is contained in V , and by the linking property , it follows that V [ ∏ m < ω G σ m i m ] = V [ ∏ m < ω G ∗ ( g σ m i m )] .Hence, X ∈ V [ ∏ m < ω G ∗ ( g σ m i m ) × ∏ m < ω G σ m i m ] . The forcing ∏ m < ω P σ m × ∏ m < ω P σ m can be factored as ( ∏ m < ω P σ m ↾ κ η × ∏ σ m ≤ η P σ m ) × ( ∏ m < ω P σ m ↾ [ κ η , κ σ m ) × ∏ σ m > η P σ m ) , ≤ κ + η by the GCH in V , and the “upperpart” is ≤ κ + η -closed: If κ η + is a limit cardinal, this follows from the fact that κ η,j + ≥ κ ++ η,j for all j < cf κ η + by construction (in particular, κ η, ≥ κ ++ η ); and if κ η + is a successor cardinal, we use our assumption that κ η + > κ + η . Hence, X ∈ V [ ∏ m < ω G ∗ ( g σ m i m ∩ κ η ) × ∏ σ m ≤ η G σ m i m ] . Setting a m ∶ = g σ m i m ∩ κ η for m < ω , it follows by the independence property that (( a m ) m < ω , ( σ m , i m ) m < ω , σ m ≤ η ) is an η -good pair with X ∈ V [ ∏ m < ω G ∗ ( a m ) × ∏ σ m ≤ η G σ m i m ] . In the case that κ η + = κ + η , we use our notion of an η -almost good pair , whichis defined like an η -good pair, with the exception that for an η -almost good pair (( a m ) m < ω , ( σ m , i m ) m < ω ) , we have a m ⊆ κ η + for all m < ω . Definition 26.
For < η < γ with κ η + = κ + η , we say that (( a m ) m < ω , ( σ m , i m ) m < ω ) is an η -almost good pair if the following hold: • ( a m ∣ m < ω ) is a sequence of pairwise disjoint κ η + -subsets, such that for all m < ω and κ ν, < κ η + , it follows that ∣ a m ∩ [ κ ν, , κ ν, + )∣ = , • for all m , we have σ m ∈ Succ with σ m ≤ η , and i m < α σ m , • if m ≠ m ′ , then ( σ m , i m ) ≠ ( σ m ′ , i m ′ ) . The counterpart of Proposition 25 states:
Proposition 27.
Let < η < γ and X ∈ N with X ⊆ κ η . In the case that κ η + = κ + η ,there is an η -almost good pair (( a m ) m < ω , ( σ m , i m ) m < ω ) with X ∈ V [ ∏ m < ω G ∗ ( a m ) × ∏ m < ω G σ m i m × G η + ] . Proof.
We follow the proof of Proposition 25 with a slightly different factorization:Let X ∈ V [ ∏ m < ω G ∗ ( g σ m i m ) × ∏ m < ω G σ m i m ] as before with σ m ∈ Lim , i m < α σ m ; σ m ∈ Succ , i m < α σ m for all m < ω . The forcing ∏ m < ω P σ m i m × ∏ m < ω P σ m i m can be factored as ( ∏ m < ω P σ m ↾ κ η + × ∏ σ m ≤ η + P σ m ) × ( ∏ m < ω P σ m ↾ [ κ η + , κ σ m ) × ∏ σ m > η + P σ m ) , where the “lower part” has cardinality ≤ κ η + by the GCH in V (since κ η + = κ + η ),and the “upper part” is ≤ κ η + -closed. Hence, X ∈ V [ ∏ m < ω G ∗ ( g σ m i m ∩ κ η + ) × ∏ σ m ≤ η + G σ m i m ] ⊆ V [ ∏ m < ω G ∗ ( g σ m i m ∩ κ η + ) × ∏ σ m ≤ η G σ m i m × G η + ] . a m ∶ = g σ m i m ∩ κ η + for m < ω , it follows that (( a m ) m < ω , ( σ m , i m ) m < ω , σ m ≤ η ) isan η -almost good pair with X ∈ V [ ∏ m < ω G ∗ ( a m ) × ∏ σ m ≤ η G σ m i m × G η + ] as desired. ∀ η θ N ( κ η ) = α η . It remains to make sure that in our ZF -model N , the values θ N ( κ η ) are as desired.Firstly, in Chapter 6.1, 6.2 and 6.3, we will show that θ N ( κ η ) = α η holds for all0 < η < γ . After that, in Chapter 6.4 and 6.5, we will see that for any cardinal λ ∈ ( κ η , κ η + ) in a “gap”, or λ ≥ κ γ = sup { κ η ∣ < η < γ } , the value θ N ( λ ) is thesmallest possible.By our remarks from Chapter 2, this justifies our assumption from the beginningthat the sequence ( α η ∣ < η < γ ) is strictly increasing. ∀ η θ N ( κ η ) ≥ α η . Using the subgroups H ηk , it is not difficult to see that for all k < α η , there existsin N a surjection s ∶ ℘ ( κ η ) → k . Proposition 28.
Let < η < γ . Then θ N ( κ η ) ≥ α η .Proof. Let k < α η . We construct in N a surjection s ∶ ℘ ( κ η ) → k . As alreadyoutlined in Chapter 4.3, we define around each G ηi with i < k a “cloud” as follows: ( ̃ G ηi ) ( k ) ∶ = ( ( ˙ ̃ G ηi ) ( k ) ) G , where ( ˙ ̃ G ηi ) ( k ) ∶ = { ( πG ηi D π , ) ∣ [ π ] ∈ H ηk } ;and we take the following canonical name for the i -th generic κ η -subset:˙ G ηi ∶ = { ( a, p ) ∣ p ∈ P , ∃ ζ < κ η ∃ ǫ ∈ { , } ∶ a = OR P ( ˇ ζ, ˇ ǫ ) ∧ p ηi ( ζ ) = ǫ } . Roughly speaking, (̃ G ηi ) ( k ) is the orbit of G ηi under the A -subgroup H ηk ; hence, itscanonical name ( ˙ ̃ G ηi ) ( k ) is fixed by all automorphisms in H ηk .More precisely:Let σ ∈ A with [ σ ] ∈ H ηk . Then ( ˙ ̃ G ηi ) ( k ) D σ = { ( πG ηi D π D σ , p ) ∣ [ π ] ∈ H ηk , p ∈ D σ } . π , πG ηi D π D σ = { ( a D σ , p ) ∣ p ∈ D σ , p ⊩ s a ∈ πG ηi D π , ∃ ζ < κ η ∃ ǫ ∈ { , } ∶ a = OR P ( ˇ ζ, ˇ ǫ ) } , since for any a = OR P ( ˇ ζ, ˇ ǫ ) as above, it follows that πa D π D σ = a D π D σ = a D σ .Now, it is not difficult to see that p ∈ D σ with p ⊩ s a ∈ πG ηi D π if and only if p ∈ D σ and for all q ≤ p with q ∈ D π ∩ D σ and ζ ∈ dom q , it follows that ( π − q ) ηi ( ζ ) = ǫ .Also, σa D σ = a D σ holds for all σ .Hence, σ πG ηi D π D σ = { ( σa D σ , σp ) ∣ p ∈ D σ , ∃ ζ < κ η ∃ ǫ ∈ { , } a = OR P ( ˇ ζ, ˇ ǫ ) , ∀ q ∈ D π ∩ D σ ( ( q ≤ p ∧ ζ ∈ dom q ) ⇒ ( π − q ) ηi ( ζ ) = ǫ ) } = { ( a D σ , p ) ∣ p ∈ D σ , ∃ ζ < κ η ∃ ǫ ∈ { , } a = OR P ( ˇ ζ, ˇ ǫ ) , ∀ q ∈ D π ∩ D σ ( ( q ≤ p ∧ ζ ∈ dom q ) ⇒ ( π − σ − q ) ηi ( ζ ) = ǫ ) } . Setting τ ∶ = σπ , it follows that σπG ηi D π D σ = τ G ηi D τ D σ . Now, any element of σ ( ˙ ̃ G ηi ) ( k ) D σ is of the form ( σπG ηi D π D σ , σp ) with [ π ] ∈ H ηk and p ∈ D σ . Since ( σπG ηi D π D σ , σp ) = ( τ G ηi D τ D σ , p ) , where τ ∶ = σπ and p ∶ = σp satisfy [ τ ] ∈ H ηk and p ∈ D σ , it follows that ( σπG ηi D π D σ , σp ) ∈ ( ˙ ̃ G ηi ) ( k ) D σ . Hence, σ ( ˙ ̃ G ηi ) ( k ) D σ ⊆ ( ˙ ̃ G ηi ) ( k ) D σ . The inclusion “ ⊇ ” is similar.Thus, ( ( ˙ ̃ G ηi ) ( k ) ∣ i < k ) ∶ = { ( OR P ( ˇ i, ( ˙ ̃ G ηi ) ( k ) ) , ) ∣ i < k } ,
45s a name for the sequence (( ̃ G ηi ) ( k ) ∣ i < k ) that is stabilized by all σ with [ σ ] ∈ H ηk .Hence, (( ̃ G ηi ) ( k ) ∣ i < k ) ∈ N .Now, we can define in N a surjection s ∶ ℘ ( κ η ) → k as follows: For X ∈ N , X ⊆ κ η ,let s ( X ) ∶ = i in the case that X ∈ ( ̃ G ηi ) ( k ) if such i exists, and s ( x ) ∶ =
0, else.The surjectivity of s is clear, since G ηi ∈ N for all i < k with s ( G ηi ) = i . It remainsto show that s is well-defined; i.e. for any i, i ′ < k with i ≠ i ′ , it follows that ( ̃ G ηi ) ( k ) ∩ ( ̃ G ηi ′ ) ( k ) = ∅ .First, let η ∈ Lim , and take i , i ′ < k with i ≠ i ′ .The point is that the automorphisms in H ηk do not permute the vertical lines P ηi ↾ [ κ ν, , κ η ) and P ηi ′ ↾ [ κ ν, , κ η ) above some κ ν, < κ η . Thus, the orbits of G ηi and G ηi ′ under H ηk must be disjoint:Assume towards a contradiction there was X ∈ ( ̃ G ηi ) ( k ) ∩ ( ̃ G ηi ′ ) ( k ) . Then we have ( πG ηi D π ) G = ( τ G ηi ′ D τ ) G for some π, τ with [ π ] ∈ H ηk and [ τ ] ∈ H ηk . Hence, ( π − G ) ηi = ( τ − G ) ηi ′ . Take κ ν, < κ η such that for all κ ν,j ∈ [ κ ν, , κ η ) and l < k , it follows that G π ( ν, j )( η, l ) = ( η, l ) whenever ( η, l ) ∈ supp π ( ν, j ) , and G τ ( ν, j )( η, l ) = ( η, l ) whenever ( η, l ) ∈ supp τ ( ν, j ) .By genericity, take q ∈ G with q ∈ D π ∩ D τ such that there is ζ ∈ dom q ∖ ( dom π ∩ dom τ ) , ζ ∈ [ κ ν, , κ η ) with q ηi ( ζ ) ≠ q ηi ′ ( ζ ) .W.l.o.g., let q ηi ( ζ ) = q ηi ′ ( ζ ) =
0. With r ∶ = π − q , r ′ ∶ = τ − q , it follows by construc-tion of the isomorphism that r ηi ( ζ ) = q ηi ( η ) = ( r ′ ) ηi ′ ( ζ ) = q ηi ′ ( ζ ) =
0, whichwould contradict ( π − G ) ηi = ( τ − G ) ηi ′ .Hence, s ∶ ℘ ( κ η ) → k is a well-defined surjection in N .The case η ∈ Succ is similar. ∀ η ( κ η + > κ + η Ð → θ N ( κ η ) ≤ α η ) . Let 0 < η < γ . Throughout this Chapter 6.2, we assume that κ η + > κ + η . Then Proposition 25 can be applied.In Chapter 6.3, we discuss the case that κ η + = κ + η , where the proof can be struc-tured the very same way; except that the intermediate generic extensions wherethe κ η -subsets in N are located are given by Proposition 27. Thus, we will haveto take care of an extra factor G η + in our products describing these intermediategeneric extensions, which will lead to a couple of modifications. In Chapter 6.3,46e take a brief look at each step in the proof presented here, and go through themajor changes.Assume towards a contradiction that there was a surjective function f ∶ ℘ ( κ η ) → α η in N . Let f = ˙ f G with ˙ f ∈ HS , such that πf D π = f D π holds for all π ∈ A with [ π ] contained in the intersection ⋂ m < ω F ix ( η m , i m ) ∩ ⋂ m < ω H λ m k m ( A ˙ f ) . By Proposition 25, it follows that any X ∈ dom f is of the form X = ˙ X ∏ m < ω G ∗ ( a m ) × ∏ m < ω G σmim , where (( a m ) m < ω , ( σ m , i m ) m < ω ) is an η -good pair.Our proof will be structured as follows: We pick some β < α η large enough for theintersection ( A ˙ f ) (we give a definition of this term on the next page) and considera map f β , which will be obtained from f by restricting its domain to those X thatare contained in a generic extension V [ ∏ m < ω G ∗ ( a m ) × ∏ m < ω G σ m i m ] for an η -good pair (( a m ) m < ω , ( σ m , i m ) m < ω ) such that i m < β for all m < ω .We wonder if this restricted function f β could still be surjective onto α η .The main steps of our proof (similar as in [FK16, Chapter 5]) can be outlined asfollows:First, we assume that also f β ∶ dom f β → α η was surjective onto α η .A) We define a forcing notion P β ↾ ( η + ) , which will be obtained from P byessentially “cutting off ” at height η + β . We show that there isa projection of forcing posets ρ β ∶ P → P β ↾ ( η + ) . Then the V -generic filter G on P induces a V -generic filter G β ↾ ( η + ) on P β ↾ ( η + ) .B) We show that f β is contained in an intermediate generic extension similarto V [ G β ↾ ( η + )] .C) We prove that the forcing P β ↾ ( η + ) preserves cardinals ≥ α η .D) We construct in V [ G β ↾ ( η + )] a set ̃ ℘ ( κ η ) ⊇ dom f β with an injection ι ∶ ̃ ℘ ( κ η ) ↪ β .Then D) together with B) and C) gives the desired contradiction.Hence, f β ∶ dom f β → α η must not be surjective.47) We consider α < α η with α ∈ rg f ∖ rg f β , and use an isomorphism argumentto obtain a contradiction, again.We see that either case, whether f β was surjective or not, leads into a contradic-tion. Thus, our initial assumption must be wrong, and we can finally conclude: There is no surjective function f ∶ ℘ ( κ η ) → α η .Before we start with Chapter 6.2 A), we first define our term large enough for theintersection ( A ˙ f ) : Definition 29.
A limit ordinal ̃ β < α η is large enough for the intersection ( A ˙ f ) if the following hold: • ̃ β > κ + η • ̃ β > sup { i m ∣ η m ≤ η } • ̃ β > sup { k m ∣ λ m ≤ η } (We use that α η ≥ κ ++ η , and cf α η > ω .)Fix a limit ordinal ̃ β < α η large enough for the intersection ( A ˙ f ) , and let β ∶ = ̃ β + κ + η (addition of ordinals).The restriction f β is defined as follows: Definition 30. f β ∶ = { ( X, α ) ∈ f ∣ ∃ (( a m ) m < ω , ( σ m , i m ) m < ω ) η -good pair ∶ ( ∀ m i m < β ) ∧ ∃ ˙ X ∈ Name (( P η ) ω × ∏ m < ω P σ m ) X = ˙ X ∏ m < ω G ∗ ( a m ) × ∏ m < ω G σmim } . First, we assume towards a contradiction that f β ∶ dom f β → α η is surjective . A) Constructing P β ↾ ( η + ) . Our aim is to construct a forcing notion P β ↾ ( η + ) that is obtained from P byessentially “cutting off” at height η and width β ; i.e. only the cardinals κ σ for σ ≤ η should be considered, and for any such κ σ , we add at most β -many new κ σ -subsets G σi .Regarding our V -generic filter G on P , we need that the restriction G β ↾ ( η + ) ∶ = G ↾ ( P β ↾ ( η + )) is a V -generic filter on P β ↾ ( η + ) , which will be guaranteed bymaking sure that the canonical map ρ β ∶ P → P β ↾ ( η + ) , p ↦ p β ↾ ( η + ) is aprojection of forcing posets.A first attempt to define P β ↾ ( η + ) could be the following:48or p ∈ P , let p β ↾ ( η + ) = ( p ∗ ↾ κ η , ( p σi , a σi ) σ ≤ η,i < min { α σ ,β } , ( p σ ↾ ( min { α σ , β } × dom y p σ ) σ ≤ η ) denote the canonical restriction; and set P β ↾ ( η + ) ∶ = { p β ↾ ( η + ) ∣ p ∈ P } . But then, G β ↾ ( η + ) ∶ = { p β ↾ ( η + ) ∣ p ∈ G } would not be a V -generic filter on P β ↾ ( η + ) : Consider a linking ordinal ξ ∈ g σi for some ( σ, i ) , such that η < σ < γ , i < α σ holds; or σ ≤ η , β ≤ i < α σ . The set D ∶ = { p ∈ P β ↾ ( η + ) ∣ ξ ∈ ⋃ σ ≤ η,i < β a σi } is dense in P β ↾ ( η + ) ; but D ∩ G β ↾ ( η + ) = ∅ by the independence property .Hence, G β ↾ ( η + ) can not be a V -generic filter on P β ↾ ( η + ) .This shows that the conditions in P β ↾ ( η + ) should contain some informationabout which linking ordinals are “forbidden” for ⋃ σ ≤ η,i < β a σi , being already occu-pied by some index ( σ, i ) with σ > η or i ≥ β .Thus, for p ∈ P , we add to p β ↾ ( η + ) a new coordinate X p , which is essentiallythe union of all a σi ∩ κ η for σ > η or i ≥ β . Then X p is a subset of κ η that hits anyinterval [ κ ν,j , κ ν,j + ) in at most countably many points.Let ̃ η ∶ = sup { σ < η ∣ σ ∈ Lim } . By closure of the sequence ( κ σ ∣ < σ < γ ) , it followsthat ̃ η ∈ Lim with ̃ η = max { σ ≤ η ∣ η ∈ Lim } , and κ ̃ η = sup { κ σ ∣ σ ∈ Lim , σ < ̃ η } .W.l.o.g. we restrict to the case that β < α ̃ η or Lim ∩ ( η, γ ) ≠ ∅ ;which is the same as requiring that there exist coordinates ( σ, i ) with σ ∈ Lim ,and σ > η or i ≥ β . (Otherwise, the forcing P β ↾ ( η + ) already contains allcoordinates ( σ, i ) with σ ∈ Lim , and there are no “forbidden” linking ordinals.In that case, we can indeed set P β ↾ ( η + ) ∶ = { ( p ∗ ↾ κ η , ( p σi , a σi ) σ ≤ η,i < β , ( p σ ↾ ( β × dom y p σ )) σ ≤ η ) ∣ p ∈ P } , and obtain that G β ↾ ( η + ) is a V -generic filter on P β ↾ ( η + ) . )For a condition p ∈ P , let X p ∶ = ⋃ { a σi ∩ κ ̃ η ∣ σ ∈ Lim with ( σ > η or i ≥ β ) } , and p β ↾ ( η + ) ∶ = ( p ∗ ↾ κ η , ( p σi , a σi ) σ ≤̃ η,i < β , ( p σ ↾ ( β × dom y p σ )) σ ≤ η , X p ) . For reasons of homogeneity, we include into P β ↾ ( η + ) only those conditions p β ↾ ( η + ) for which the set X p hits every interval [ κ ν,j , κ ν,j + ) ⊆ κ ̃ η in countablymany points, which is the same as requiring ∣{( σ, i ) ∈ supp p ∣ σ > ̃ η or i ≥ β }∣ = ℵ .49 efinition 31. P β ↾ ( η + ) ∶ = { p β ↾ ( η + ) ∣ p ∈ P , ∣{( σ, i ) ∈ supp p ∣ σ > η or i ≥ β }∣ = ℵ } ∪ { βη + } , with βη + as the maximal element.For conditions p β ↾ ( η + ) , q β ↾ ( η + ) in P β ↾ ( η + ) ∖ { βη + } , let q β ↾ ( η + ) ≤ βη + p β ↾ ( η + ) if X q ⊇ X p , and ( q ∗ ↾ κ η , ( q σi , b σi ) σ ≤ η,i < β , ( q σ ↾ ( β × dom y q σ ) σ ≤ η ) ≤ ( p ∗ ↾ κ η , ( p σi , a σi ) σ ≤ η,i < β , ( p σ ↾ ( β × dom y p σ ) σ ≤ η ) regarded as conditions in P . In other words: P β ↾ ( η + ) is the collection of all ( p ∗ , ( p σi , a σi ) σ ≤ η,i < β , ( p σ ) σ ≤ η , X p ) such that • p ∶ = ( p ∗ , ( p σi , a σi ) σ ≤ η,i < β , ( p σ ) σ ≤ η ) is a condition in P with dom p ⊆ κ η , supp p ⊆ {( σ, i ) ∣ σ ≤ η, i < β } , and supp p ⊆ η + ∀ σ ∈ supp p ∶ dom x p σ ⊆ β , • X p ⊆ κ ̃ η with ∀ [ κ ν,j , κ ν,j + ) ⊆ κ ̃ η ∣ X p ∩ [ κ ν,j , κ ν,j + )∣ = ℵ , and X p ∩ ⋃ σ ≤ η , i < β a σi = ∅ .For p , q ∈ P with q ≤ p and ∣{( σ, i ) ∈ supp p ∣ σ > η or i ≥ β }∣ = ℵ , it follows that q β ↾ ( η + ) ≤ p β ↾ ( η + ) . Definition 32. G β ↾ ( η + ) ∶ = { p ∈ P β ↾ ( η + ) ∣ ∃ p ∈ G ∶ ∣{( σ, i ) ∈ supp p ∣ σ > η or i ≥ β }∣ = ℵ ,p β ↾ ( η + ) ≤ βη + p } . We will now show that G β ↾ ( η + ) is a V -generic filter on P β ↾ ( η + ) .Let P ⊆ P denote the collection of all p ∈ P with the property that ∣ {( σ, i ) ∈ supp p ∣ σ > η ∨ i ≥ β } ∣ = ℵ , together with the maximal element . Then P is adense subforcing of P . Proposition 33.
The map ρ β ∶ P → P β ↾ ( η + ) with p ↦ p β ↾ ( η + ) in the casethat ∣ {( σ, i ) ∈ supp p ∣ σ > η ∨ i ≥ β } ∣ = ℵ , and ↦ βη + , is a projection of forcingposets: • ρ β ( ) = βη + , • if p , q ∈ P with q ≤ p , it follows that ρ β ( q ) ≤ βη + ρ β ( p ) , • for any p ∈ P and q ∈ P β ↾ ( η + ) with q ≤ βη + ρ β ( p ) , there exists q ∈ P suchthat q ≤ p and ρ β ( q ) ≤ q .Hence, G β ↾ ( η + ) is a V -generic filter on P β ↾ ( η + ) . roof. Clearly, the map ρ β as defined above is order-preserving with ρ β ( ) = βη + .Consider p = ( p ∗ , ( p σi , a σi ) σ,i , ( p σ ) σ ) ∈ P and q = ( q ∗ ↾ κ η , ( q σi , b σi ) σ ≤ η,i < β , ( q σ ) σ ≤ η , X q ) ∈ P β ↾ ( η + ) with q ≤ βη + ρ β ( p ) = p β ↾ ( η + ) . Then ( q ∗ ↾ κ η , ( q σi , a σi ) σ ≤ η,i < β ) ≤ ( p ∗ ↾ κ η , ( p σi , a σi ) σ ≤ η,i < β ) in P , ( q σ ) σ ≤ η ≤ ( p σ ↾ ( β × dom y p σ )) σ ≤ η in P , and X q ⊇ ⋃ { a σi ∩ κ ̃ η ∣ σ > η ∨ i ≥ β } . We have to construct q ∈ P , q = ( q ∗ , ( q σi , b σi ) σ,i , ( q σ ) σ ) , with q ≤ p and ρ β ( q ) = q β ↾ ( η + ) ≤ βη + q .We start with q : • In order to achieve X q ⊇ X q , we will enlarge supp p ∪ supp q by countablymany (( ˆ η, m k ) ∣ k < ω ) , where ˆ η > η or m k ≥ β for all k < ω , and arrange thatany ξ ∈ X q ∖ X p occurs as a linking ordinal in some b ˆ ηm k .More precisely: Let supp q ∶ = supp p ∪ supp q ∪ supp ∗ , where supp ∗ ∶ = {( ˆ η, m k ) ∣ k < ω } such that ( ˆ η, m k ) ∉ supp p ∪ supp q for all k < ω , andsince we are working in the case that β < α ̃ η or ( η, γ ) ∩ Lim ≠ ∅ , we cantake either ˆ η ∶ = ̃ η and m k ∈ ( β, α ̃ η ) for all k < ω ; or ˆ η ∈ ( η, γ ) ∩ Lim . Thenfor all ( ˆ η, m k ) , it follows that ˆ η > η or m k ≥ β . • Next, we define the linking ordinals b σi for ( σ, i ) ∈ supp q such that X q ⊇ X q .For ( σ, i ) ∈ supp q , we let b σi ∶ = b σi ⊇ a σi ; and in the case that ( σ, i ) ∈ supp p ∖ supp q , we set b σi ∶ = a σi . Finally, we define ( b ˆ ηm k ∣ k < ω ) with the followingproperties: – as usual, every b ˆ ηm k is a subset of κ ˆ η that hits every interval [ κ ν,j , κ ν,j + ) ⊆ κ ˆ η in exactly one point , – ⋃ { b ˆ ηm k ∩ κ ̃ η ∣ k < ω } ⊇ X q ∖ X p , – b ˆ ηm k ∩ b σi = ∅ for all k < ω and ( σ, i ) ∈ supp q , – b ˆ ηm k ∩ a σi = ∅ for all k < ω and ( σ, i ) ∈ supp p ∖ supp q (since q ≤ βη + p β ↾ ( η + ) , it follows that in this case, σ > η or i ≥ β ) , – b ˆ ηm k ∩ b ˆ ηm k ′ = ∅ whenever k ≠ k ′ .This is possible, since X q ∩ b σi = ∅ for any ( σ, i ) ∈ supp q by constructionof P β ↾ ( η + ) ; and whenever ( σ, i ) ∈ supp p ∖ supp q , then σ > η or i ≥ β implies a σi ⊆ X p ; thus ( X q ∖ X p ) ∩ a σi = ∅ . • We now define dom q . For any interval [ κ ν,j , κ ν,j + ) ⊆ κ η , take δ ν,j ∈ [ κ ν,j , κ ν,j + ) as follows: In the case that dom q ∩ [ κ ν,j , κ ν,j + ) = ∅ , let δ ν,j ∶ = κ ν,j . If dom q ∩ [ κ ν,j , κ ν,j + ) ≠ ∅ , we take δ ν,j ∈ ( κ ν,j , κ ν,j + ) such that ⋃ { b σi ∣ ( σ, i ) ∈ supp q } ∩ [ κ ν,j , κ ν,j + ) ⊆ [ κ ν,j , δ ν,j ) and dom q ∩ [ κ ν,j , κ ν,j + ) ⊆ κ ν,j , δ ν,j ) . Since dom q is bounded below all regular cardinals κ ν, , this isalso true for ⋃ { [ κ ν,j , δ ν,j ) ∣ κ ν,j < κ η } . Letdom q ∩ κ η ∶ = ⋃ { [ κ ν,j , δ ν,j ) ∣ κ ν,j < κ η } , and dom q ∩ [ κ η , κ γ ) ∶ = dom p ∩ [ κ η , κ γ ) . • We take q ∗ ↾ κ η ⊇ q ∗ ↾ κ η arbitrary on the given domain; and q ∗ ↾ [ κ η , κ γ ) ∶ = p ∗ ↾ [ κ η , κ γ ) . • It remains to define q σi for ( σ, i ) ∈ supp q .For ( σ, i ) ∈ supp q , we define q σi ⊇ q σi on the given domain ⋃ κ ν,j < κ σ [ κ ν,j , δ ν,j ) according to the linking property : Consider an interval [ κ ν,j , κ ν,j + ) with δ ν,j > κ ν,j . For any ζ ∈ ( dom q ∖ dom q ) ∩ [ κ ν,j , κ ν,j + ) , set q σi ( ζ ) ∶ = q ∗ ( ξ, ζ ) ,where { ξ } ∶ = b σi ∩ [ κ ν,j , κ ν,j + ) = b σi ∩ [ κ ν,j , κ ν,j + ) . (Note that ξ ∈ dom q by construction). For ( σ, i ) ∈ supp p ∖ supp q , we set q σi ↾ [ κ η , κ γ ) ∶ = p σi ↾ [ κ η , κ γ ) , and define q σi ↾ κ η ⊇ p σi ↾ κ η on the given domain according to the linking property as before.Finally, q ˆ ηm k for k < ω can be arbitrary on the given domain.Then q = ( q ∗ , ( q σi , b σi ) σ,i ) is a condition in P . In particular, the independenceproperty holds for the linking ordinals b σi : Firstly, by construction of ( b ˆ ηm k ∣ k < ω ) ,it follows that b ˆ ηm k ∩ b σi = ∅ for any ( σ, i ) ∈ supp q ∪ supp p . Secondly, whenever ( σ , i ) ∈ supp q and ( σ , i ) ∈ supp p ∖ supp q , then σ > η or i ≥ β ; hence, b σ i = a σ i ⊆ X p ⊆ X q . Since b σ i ∩ X q = b σ i ∩ X q = ∅ , this implies b σ i ∩ b σ i = ∅ asdesired. Thus, the independence property holds for q .Moreover, ( ρ β ( q )) = ( q ∗ ↾ κ η , ( q σi , b σi ) σ ≤ η,i < β , X q ) ≤ q by construction; in particu-lar, X q ⊆ X q : Consider ξ ∈ X q . In the case that ξ ∈ X p , it follows that ξ ∈ a σi forsome ( σ, i ) ∈ supp p with σ > η , or i ≥ β . Then ( σ, i ) ∈ supp p ∖ supp q ; hence, b σi = a σi , and it follows that ξ ∈ b σi ⊆ X q as desired. In the case that ξ ∈ X q ∖ X p , wehave ξ ∈ b ˆ ηm k for some k < ω ; so again, ξ ∈ X q as desired.Finally, q ≤ p by construction; and it follows that q has all the desired properties.The construction of q is similar. Thus, the map ρ β ∶ P → P β ↾ ( η + ) as definedabove, is indeed a projection of forcing posets.It follows that G β ↾ ( η + ) is a V -generic filter on P β ↾ ( η + ) : For genericity,consider an open dense set D ⊆ P β ↾ ( η + ) . It suffices to show that the set D ∶ = { p ∈ P ∣ p β ↾ ( η + ) ∈ D } is dense in P . Take a condition p ∈ P , and let p ≤ p with p ∈ P . Since D ⊆ P β ↾ ( η + ) is dense, there exists q ∈ P β ↾ ( η + ) with q ≤ βη + p β ↾ ( η + ) . By what we have just shown, we there exists q ≤ p with q β ↾ ( η + ) ≤ q . Then q is an extension of p in D as desired.52 ) Capturing f β . In this section, we will show that the map f β is contained in a generic extensionsimilar to V [ G β ↾ ( η + )] .Recall that we are working in the case that κ η + > κ + η , and β < α ̃ η or (̃ η, γ ) ∩ Lim ≠ ∅ ,where ̃ η ∶ = max { σ < η ∣ σ ∈ Lim } .Recall that any X ∈ dom f is of the form X = ˙ X ∏ m < ω G ∗ ( a m ) × ∏ m < ω G σmim , where ˙ X ∈ Name (( P η ) ω ) × ∏ m < ω P σ m i m ) and (( a m ) m < ω , ( σ m , i m ) m < ω ) is an η -goodpair. Moreover, f β ∶ = { ( X, α ) ∈ f ∣ ∃ (( a m ) m < ω , ( σ m , i m ) m < ω ) η -good pair ∶ ( ∀ m i m < β ) ∧ ∃ ˙ X ∈ Name (( P η ) ω × ∏ m < ω P σ m ) X = ˙ X ∏ m < ω G ∗ ( a m ) × ∏ m < ω G σmim } . Fix an η -good pair ̺ = (( a m ) m < ω , ( σ m , i m ) m < ω ) . We use recursion over the Name (( P η ) ω × ∏ m < ω P σ m ) -hierarchy to define a map τ ̺ ∶ Name (( P η ) ω × ∏ m < ω P σ m ) → Name ( P ) that maps any name ˙ Y ∈ Name (( P η ) ω × ∏ m < ω P σ m ) to a name τ ̺ ( ˙ Y ) ∈ Name ( P ) such that ˙ Y ∏ m < ω G ∗ ( a m ) × ∏ m < ω G σmim = ( τ ̺ ( ˙ Y )) G . Definition 34.
For an η -good pair ̺ = (( a m ) m < ω , ( σ m , i m ) m < ω ) , we define recur-sively for ˙ Y ∈ Name (( P η ) ω × ∏ m < ω P σ m ) : τ ̺ ( ˙ Y ) ∶ = { ( τ ̺ ( ˙ Z ) , q ) ∣ q ∈ P , ∃ ( ˙ Z , (( p ∗ ( a m )) m < ω , ( p σ m i m ) m < ω ) ) ∈ ˙ Y ∶∀ m ( q ∗ ( a m ) ⊇ p ∗ ( a m ) , q σ m i m ⊇ p σ m i m ) } . It is not difficult to check that indeed, ˙ Y ∏ m < ω G ∗ ( a m ) × ∏ m < ω G σmim = ( τ ̺ ( ˙ Y )) G holdsfor all ˙ Y ∈ Name (( P η ) ω × ∏ m < ω P σ m ) .Now, we define a map ( f β ) ′ ⊇ f β , which is contained in an intermediate genericextension similar to V [ G β ↾ ( η + )] . We will then use an isomorphism argumentto show that actually, ( f β ) ′ = f β .Recall that f = ˙ f G , where πf D π = f D π whenever [ π ] contained in the intersection ⋂ m < ω F ix ( η m , i m ) ∩ ⋂ m < ω H λ m k m denoted by ( A ˙ f ) .The idea is that we include into P β ↾ ( η + ) the verticals P η m i m for η m ∈ Lim, η m > η .Below κ η , the linking property will be important, so we also have to include thelinking ordinals a η m i m ∩ κ η . 53or a condition p ∈ P , we set ̃ X p ∶ = ⋃ { a σi ∩ κ ̃ η ∣ σ ∈ Lim , ( σ, i ) ≠ ( η m , i m ) for all m < ω , ( σ > η or i ≥ β )} . Then ̃ X p is similar to X p , but excludes the linking ordinals a η m i m for η m ∈ Lim .For reasons of notational convenience and better clarity, we introduce the followingad-hoc notation:Let ( p β ↾ ( η + )) ( η m ,i m ) m < ω ∶ = ( p ∗ ↾ κ η , ( p σi , a σi ) σ ≤ η , i < β , ( p η m i m ↾ κ η , a η m i m ∩ κ η ) m < ω , η m > η , ( p σ ↾ ( β × dom y p σ )) σ ≤ η , ̃ X p ) . Then ( p β ↾ ( η + )) ( η m ,i m ) m < ω can be obtained from p β ↾ ( η + ) by using ̃ X p insteadof X p , and including ( p η m i m ↾ κ η , a η m i m ∩ κ η ) for η m ∈ Lim with η m > η . (Note thatfor η m ≤ η , it follows that i m < β , so ( p η m i m , a η m i m ) is already part of the condition p β ↾ ( η + ) .)We are now ready to define our forcing notion ( P β ↾ ( η + )) ( η m ,i m ) m < ω . The orderrelation is defined similarly as for the forcing notion P β ↾ ( η + ) ; but additionally,we require for ( q β ↾ ( η + )) ( η m ,i m ) m < ω ≤ ( p β ↾ ( η + )) ( η m ,i m ) m < ω that the linkingproperty below κ η holds for all ( η m , i m ) with η m ∈ Lim, η m > η . Definition 35.
Let ( P β ↾ ( η + )) ( η m ,i m ) m < ω denote the collection of all ( p β ↾ ( η + )) ( η m ,i m ) m < ω such that p ∈ P (i.e. p ∈ P with ∣{( σ, i ) ∈ supp p ∣ σ > η or i ≥ β }∣ = ℵ ); together with ( βη + ) ( η m ,i m ) m < ω as the maximal element.For conditions p , q ∈ P , let ( q β ↾ ( η + )) ( η m ,i m ) m < ω ≤ ( p β ↾ ( η + )) ( η m ,i m ) m < ω if • ̃ X q ⊇ ̃ X p , • ( q ∗ ↾ κ η , ( q σi , b σi ) σ ≤ η,i < β , ( q σ ↾ ( β × dom y q σ )) σ ≤ η ) ≤ ( p ∗ ↾ κ η , ( p σi , a σi ) σ ≤ η,i < β , ( p σ ↾ ( β × dom y p σ )) σ ≤ η ) regarded as conditions in P , • ∀ η m > η ∶ q η m i m ↾ κ η ⊇ p η m i m ↾ κ η , • ∀ η m > η , ( η m , i m ) ∈ supp p ∶ b η m i m = a η m i m , • for all intervals [ κ ν,j , κ ν,j + ) ⊆ κ η and η m > η with a η m i m ∩ [ κ ν,j , κ ν,j + ) = { ξ } ,it follows that q η m i m ( ζ ) = q ∗ ( ξ, ζ ) whenever ζ ∈ ( dom q ∖ dom p ) ∩ [ κ ν,j , κ ν,j + ) . Finally, for constructing our intermediate generic extension for capturing f β , wealso have to include the verticals P η m ↾ [ κ η , κ η m ) for η m > η .This gives a product ( P β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω P η m ↾ [ κ η , κ η m ) , ( ( p β ↾ ( η + )) ( η m ,i m ) m < ω , ( p η m i m ↾ [ κ η , κ η m )) m < ω ) such that p ∈ P (i.e. p ∈ P with ∣{( σ, i ) ∈ supp p ∣ σ > η or i ≥ β }∣ = ℵ ); togetherwith a maximal element ( βη + ) ( η m ,i m ) m < ω .Then ( G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω G η m i m ↾ [ κ η , κ η m ) is the set of all ( ( p β ↾ ( η + )) ( η m ,i m ) m < ω , ( p η m i m ↾ [ κ η , κ η m )) m < ω ) such that thereexists q ∈ G ∩ P with ( q β ↾ ( η + )) ( η m ,i m ) m < ω ≤ ( p β ↾ ( η + )) ( η m ,i m ) m < ω and q η m i m ↾ [ κ η , κ η m ) ⊇ p η m i m ↾ [ κ η , κ η m ) for all m < ω ; together with the maximal element ( βη + ) ( η m ,i m ) m < ω .In order to show that ( G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω G η m i m ↾ [ κ η , κ η m ) is a V -generic filter on ( P β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω P η m ↾ [ κ η , κ η m ) , we proceedsimilarly as in Proposition 33: Proposition 36.
The map ( ρ β ) ( η m ,i m ) m < ω ∶ P → ( P β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω P η m ↾ [ κ η , κ η m ) , p ↦ ( ( p β ↾ ( η + )) ( η m ,i m ) m < ω , ( p η m i m ↾ [ κ η , κ η m )) m < ω ) in the case that ∣{( σ, i ) ∈ supp p ∣ σ > η ∨ i ≥ β }∣ = ℵ , and ↦ ( βη + ) ( η m ,i m ) m < ω ,is a projection of forcing posets.Proof. We closely follow the proof of Proposition 33. Consider p ∈ P with ∣{( σ, i ) ∈ supp p ∣ σ > η ∨ i ≥ β }∣ = ℵ , and a condition q = ( q ∗ ↾ κ η , ( q σi , b σi ) σ ≤ η,i < β , ( q η m i m ↾ κ η , b η m i m ∩ κ η ) m < ω,η m > η , ( q σ ) σ ≤ η , ̃ X q , ( q η m i m ↾ [ κ η , κ η m )) m < ω ) in ( P β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω P η m ↾ [ κ η , κ η m ) with q ≤ ( ρ β ) ( η m ,i m ) m < ω ( p ) . We have to construct q ≤ p , q = ( q ∗ , ( q σi , b σi ) σ,i , ( q σ ) σ ) ,such that ( ρ β ) ( η m ,i m ) m < ω ( q ) ≤ q .We start with q . • Similarly as in Proposition 33, we construct supp ∗ = {( ˇ η, m k ) ∣ k < ω } suchthat ˇ η > η or m k ≥ β , and ( ˇ η, m k ) ∉ supp p ∪ supp q for all k < ω ; withthe additional property that for all k < ω , we have ( ˇ η, m k ) ∉ {( η m , i m ) ∣ m < ω , η m ∈ Lim } . We set supp q = supp p ∪ supp q ∪ supp ∗ ∪ {( η m , i m ) ∣ m < ω , η m ∈ Lim } . 55 Next, we define the linking ordinals b σi for ( σ, i ) ∈ supp q , such that ̃ X q ⊇ ̃ X q holds:First, we consider the case that ( σ, i ) ∉ {( η m , i m ) ∣ m < ω , η m ∈ Lim } . For ( σ, i ) ∈ supp q , we let b σi ∶ = b σi ⊇ a σi , and b σi ∶ = a σi in the case that ( σ, i ) ∈ supp p ∖ supp q .We construct ( b ˇ ηm k ∣ k < ω ) as in Proposition 33.After that, we define the linking ordinals ( b η m i m ∣ m < ω , η m ∈ Lim ) with thefollowing properties: – As usual, every b η m i m is a subset of κ η m that hits any interval [ κ ν,j , κ ν,j + ) ⊆ κ η m in exactly one point. – The b η m i m are pairwise disjoint, and b η m i m ∩ b σi = ∅ for every m < ω and ( σ, i ) ∈ supp q with ( σ, i ) ≠ ( η m , i m ) . – For every ( η m , i m ) ∈ supp p , we set b η m i m ∶ = a η m i m ; for every ( η m , i m ) ∈ supp q ∖ supp p with η m ≤ η , we set b η m i m ∶ = b η m i m ; and whenever ( η m , i m ) ∈ supp q ∖ supp p with η m > η , we let b η m i m ⊇ b η m i m ∩ κ η .This concludes our construction of the linking ordinals b σi . • We define dom q = ⋃ ν,j [ κ ν,j , δ ν,j ) as follows: Let dom ∶ = dom p ∪ dom q ∪ ⋃ η m ∈ Lim dom q η m i m ↾ [ κ η , κ η m ) . For every interval [ κ ν,j , κ ν,j + ) with dom ∩ [ κ ν,j , κ ν,j + ) = ∅ , we set δ ν,j ∶ = κ ν,j ; and whenever dom ∩ [ κ ν,j , κ ν,j + ) ≠ ∅ ,we pick δ ν,j ∈ ( κ ν,j , κ ν,j + ) with the property that dom ∩ [ κ ν,j , κ ν,j + ) ⊆ [ κ ν,j , δ ν,j ) , and b σi ∩ [ κ ν,j , κ ν,j + ) ⊆ [ κ ν,j , δ ν,j ) for all ( σ, i ) ∈ supp q .Since dom p , dom q and the domains dom q η m i m ↾ [ κ η , κ η m ) are boundedbelow all regular cardinals, this is also true for dom and dom q . • We take q ∗ ↾ κ η ⊇ q ∗ ↾ κ η arbitrary on the given domain.The verticals q σi ↾ κ η for ( σ, i ) ∈ ( supp q ∪ supp p ) ∖ {( η m , i m ) ∣ m < ω , η m ∈ Lim } can be defined according to the linking property as in Proposition 33.The verticals q ˆ ηm k ↾ κ η with ( ˆ η, m k ) ∈ supp ∗ can be set arbitrarily on thegiven domain.Now, consider ( η m , i m ) with η m ∈ Lim. In the case that ( η m , i m ) ∈ supp q with η m ≤ η , we can proceed as before, and define q η m i m ⊇ q η m i m according to the linking property as in Proposition 33.Concerning the verticals q η m i m ↾ κ η for ( η m , i m ) ∈ supp q with η m > η , wedefine q η m i m ↾ [ κ ν,j , κ ν,j + ) ⊇ q η m i m ↾ [ κ ν,j , κ ν,j + ) on intervals [ κ ν,j , κ ν,j + ) ⊆ κ η according to the linking property , and use that we have incorporated thelinking ordinals b η m i m ∩ κ η into our forcing notion ( P β ↾ ( η + )) ( η m ,i m ) m < ω :For ζ ∈ ( dom q ∖ dom q ) ∩ [ κ ν,j , κ ν,j + ) , we set q η m i m ( ζ ) ∶ = q ∗ ( ξ, ζ ) , where56 ξ } = b η m i m ∩ [ κ ν,j , κ ν,j + ) = b η m i m ∩ [ κ ν,j , κ ν,j + ) . (Note that ξ ∈ dom q byconstruction.)In the case that ( η m , i m ) ∉ supp q , it follows that also ( η m , i m ) ∉ supp p ,and we can set q η m i m ↾ κ η arbitrarily on the given domain. • Next, consider an interval [ κ ν,j , κ ν,j + ) ⊆ [ κ η , κ γ ) . We first set the verti-cals q σi ↾ [ κ ν,j , κ ν,j + ) for ( σ, i ) ∈ supp q , σ > η , on the given domain, withthe property that q η m i m ↾ [ κ ν,j , κ ν,j + ) ⊇ q η m i m ↾ [ κ ν,j , κ ν,j + ) for all m < ω with ( η m , i m ) ∈ supp q , and q σi ↾ [ κ ν,j , κ ν,j + ) ⊇ p σi ↾ [ κ ν,j , κ ν,j + ) whenever ( σ, i ) ∈ supp p . After that, we define q ∗ ↾ [ κ ν,j , κ ν,j + ) ⊇ p ∗ ↾ [ κ ν,j , κ ν,j + ) accord-ing to the linking property : Whenever ζ ∈ ( dom q ∖ dom p ) ∩ [ κ ν,j , κ ν,j + ) and { ξ } = a σi ∩ [ κ ν,j , κ ν,j + ) = b σi ∩ [ κ ν,j , κ ν,j + ) for some ( σ, i ) ∈ supp p , then q ∗ ( ξ, ζ ) ∶ = q σi ( ζ ) . Otherwise, q ∗ ( ξ, ζ ) can be set arbitrarily.This defines q . The construction of q is similar; and it is not difficult to see that q ≤ p with ( ρ β ) ( η m ,i m ) m < ω ( q ) ≤ q .Hence, ( ρ β ) ( η m ,i m ) m < ω is a projection of forcing posets.Thus, it follows that ( G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω G η m i m ↾ [ κ η , κ η m ) is a V -genericfilter on the forcing notion ( P β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω P η m ↾ [ κ η , κ η m ) .The aim of Chapter 6.2 B) is to show that f β is contained in the intermediate V -generic extension V [ ( G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω G η m i m ↾ [ κ η , κ η m ) ] . Definition 37.
Let ( f β ) ′ denote the set of all ( X, α ) for which there exists an η -good pair ̺ = (( a m ) m < ω , ( σ m , i m ) m < ω ) with i m < β for all m < ω such that X = ˙ X ∏ m G ∗ ( a m ) × ∏ m G σmim , and there is a condition p ∈ P with • ∣{( σ, i ) ∈ supp p ∣ σ > η or i ≥ β }∣ = ℵ , • p ⊩ s ( τ ̺ ( ˙ X ) , α ) ∈ ˙ f • ( ( p β ↾ ( η + )) ( η m ,i m ) m < ω , ( p η m i m ↾ [ κ η , κ η m )) m < ω ) ∈ ( G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω G η m i m ↾ [ κ η , κ η m ) , • ∀ η m ∈ Lim ∶ ( η m , i m ) ∈ supp p with a η m i m = g η m i m . Then ( f β ) ′ ∈ V [ ( G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω G η m i m ↾ [ κ η , κ η m ) ] , since the se-quence ( g η m i m ∣ m < ω ) is contained in the ground model V .We will now use an isomorphism argument and show that f β = ( f β ) ′ . Proposition 38. f β = ( f β ) ′ . roof. By the forcing theorem, it follows that ( f β ) ′ ⊇ f β . Assume towards acontradiction, there was ( X, α ) ∈ ( f β ) ′ ∖ f β . Let X = ˙ X ∏ m < ω G ∗ ( a m ) × ∏ m < ω G σmim for an η -good pair ̺ = (( a m ) m < ω , ( σ m , i m ) m < ω ) with i m < β for all m < ω . Take p ∈ P as in Definition 37 with p ⊩ s ( τ ̺ ( ˙ X ) , α ) ∈ ˙ f ; and since ( X, α ) ∉ f β , we cantake p ′ ∈ G with p ′ ⊩ s ( τ ̺ ( ˙ X ) , α ) ∉ ˙ f and ( η m , i m ) ∈ supp p ′ for all η m ∈ Lim .Our first step will be to extend the conditions p and p ′ and obtain p ≤ p , p ′ ≤ p ′ suchthat p and p ′ have “the same shape” similarly as in the Approximation Lemma
21; but additionally, p β ↾ ( η + ) = ( p ′ ) β ↾ ( η + ) holds, and p η m i m = ( p ′ ) η m i m for all m < ω , and a η m i m = ( a ′ ) η m i m for all m < ω with η m ∈ Lim .After that, we construct an isomorphism π such that firstly, πp = p ′ ; secondly, π should not disturb the forcing P β ↾ ( η + ) (which will imply π τ ̺ ( ˙ X ) D π = τ ̺ ( ˙ X ) D π );and thirdly [ π ] should be contained in the intersection ⋂ m F ix ( η m , i m ) ∩ ⋂ m H λ m k m (which implies πf D π = f D π ).Then from p ⊩ s ( τ ̺ ( ˙ X ) , α ) ∈ ˙ f it follows πp ⊩ s ( π τ ̺ ( ˙ X ) D π , α ) ∈ πf D π . Togetherwith p ′ ⊩ s ( τ ̺ ( ˙ X ) D π , α ) ∉ f D π , this gives our desired contradiction.In order to make such an isomorphism π possible, the extensions p ≤ p and p ′ ≤ p ′ will satisfy the following properties: • supp ∶ = supp p = supp p ′ • dom ∶ = dom p = dom p ′ • ⋃ a ∶ = ⋃ ( σ,i )∈ supp a σi = ⋃ ( σ,i )∈ supp ( a ′ ) σi • ∀ ν , j ∶ ( dom ∩ [ κ ν,j , κ ν,j + ) ≠ ∅ ⇒ ⋃ a ∩ [ κ ν,j , κ ν,j + ) ⊆ dom ) • supp ∶ = supp p = supp p ′ • ∀ σ ∈ supp ∶ dom ( σ ) ∶ = dom p σ = dom ( p ′ ) σ .Additionally, we want: • ∀ m < ω ∶ p η m i m = ( p ′ ) η m i m • ∀ m < ω , η m ∈ Lim ∶ a η m i m = ( a ′ ) η m i m • p β ↾ ( η + ) = ( p ′ ) β ↾ ( η + ) , i.e. – p ∗ ↾ κ η = p ′∗ ↾ κ η – ∀ σ ∈ Lim , σ ≤ η , i < min { α σ , β } ∶ p σi = ( p ′ ) σi , a σi = ( a ′ ) σi – ∀ σ ∈ Succ σ ≤ η ∶ p σ ↾ ( β × dom y p σ ) = ( p ′ ) σ ↾ ( β × dom y ( p ′ ) σ ) .58hen it follows that ̃ X p = ̃ X p ′ .Note that a η m i m = ( a ′ ) η m i m for η m ∈ Lim follows automatically, since a η m i m = ( a ′ ) η m i m = g η m i m by assumption.Now, we construct the conditions p and p ′ .We start with the linking ordinals a σi and ( a ′ ) σi , with our aim that ⋃ σ,i a σi = ⋃ σ,i ( a ′ ) σi = ∶ ⋃ a . We closely follows our construction from the ApproximationLemma
21; but now, some extra care is needed, since we additionally have tomake sure that a σi = ( a ′ ) σi holds for all σ ≤ η , i < β .Similarly as in the Approximation Lemma
21, let s ∶ = κ δ ∶ = sup { κ σ ∣ σ ∈ Lim , ∃ i < α σ ( σ, i ) ∈ supp p ∪ supp p ′ } . Recall that we are assuming β < α ̃ η or Lim ∩ (̃ η, γ ) ≠ ∅ , where ̃ η ∶ = max { σ ≤ η ∣ σ ∈ Lim } .In the case that κ δ = κ γ , we set γ ∶ = δ and take (( σ k , l k ) ∣ k < ω ) such thatsup { κ σ k ∣ k < ω } = κ γ = κ γ , and ( σ k , l k ) ∉ supp p ∪ supp p ′ for all k < ω , with theadditional property that σ k > ̃ η or l k ≥ β for all k < ω .If κ δ < κ γ and Lim ∩ (̃ η, γ ) ≠ ∅ , let γ ∈ Lim ∩ (̃ η, γ ) with γ ≥ δ , and take (( σ k , l k ) ∣ k < ω ) such that ( σ k , l k ) = ( γ, l k ) ∉ supp p ∪ supp p ′ for all k < ω .Finally, if κ δ < κ γ and Lim ∩ (̃ η, γ ) = ∅ , then β < α ̃ η follows. In this case, let γ ∶ = ̃ η ≥ δ , and take (( σ k , l k ) ∣ k < ω ) with ( σ k , l k ) = ( γ, l k ) ∉ supp p ∪ supp p ′ forall k < ω ; with the additional property that l k ≥ β for all k < ω .Let supp ∶ = supp p ∶ = supp p ′ ∶ = supp p ∪ supp p ′ ∪ {( σ k , l k ) ∣ k < ω } . We now construct the linking ordinals a σi . For any ( σ, i ) ∈ supp p , we set a σi ∶ = a σi ;and whenever ( σ, i ) ∈ supp p ′ ∖ supp p with σ ≤ η , i < β , then a σi ∶ = ( a ′ ) σi .Now, take a set Z ⊆ κ γ such that for all intervals [ κ ν,j , κ ν,j + ) ⊆ κ γ , we have ∣ Z ∩ [ κ ν,j , κ ν,j + )∣ = ℵ , and Z ∩ ( ⋃ ( σ,i )∈ supp p a σi ∪ ⋃ ( σ,i )∈ supp p ′ ( a ′ ) σi ) = ∅ . Let Z ∶ = Z ∪ ⋃ σ,i a σi ∪ ⋃ σ,i ( a ′ ) σi . Our aim is to construct p and p ′ with ⋃ σ,i a σi = ⋃ σ,i ( a ′ ) σi = ⋃ a ∶ = Z .Fix an interval [ κ ν,j , κ ν,j + ) ⊆ κ γ . Let Z ν,j ∶ = ( ⋃ { a σi ∣ ( σ, i ) ∈ supp p } ∪ ⋃ {( a ′ ) σi ∣ ( σ, i ) ∈ supp p ′ , σ ≤ η, i < β } ) ∩∩ [ κ ν,j , κ ν,j + ) { ξ k ( ν, j ) ∣ k < ω } ∶ = ( Z ∩ [ κ ν,j , κ ν,j + )) ∖ Z ν,j . This set has cardinality ℵ by construction of Z .Now, let {( σ k , l k ) ∣ k < ω } = ∶ {( σ, i ) ∈ supp p ∖ supp p ∣ κ ν,j < κ σ and ( σ > η or i ≥ β )} . This set also has cardinality ℵ by construction of supp p . Now, for any k < ω ,we let a σ k l k ∩ [ κ ν,j , κ ν,j + ) ∶ = { ξ k ( ν, j )} . We apply the same construction to the linking ordinals ( a ′ ) σi for ( σ, i ) ∈ supp p ′ = supp . It is not difficult to see that ⋃ σ,i a σi = ⋃ σ,i ( a ′ ) σi = ⋃ a = Z , the independenceproperty holds, and a σi = ( a ′ ) σi whenever σ ≤ η , i < β .Next, take dom ∶ = dom p = dom p ′ = ⋃ ν,j [ κ ν,j , δ ν,j ) with the property that firstly,dom p ∪ dom p ′ ⊆ dom , and secondly, for every interval [ κ ν,j , κ ν,j + ) ⊆ κ γ withdom ∩ [ κ ν,j , κ ν,j + ) ≠ ∅ , it follows that Z ∩ [ κ ν,j , κ ν,j + ) ⊆ dom .It remains to construct p ∗ , p ′∗ , and p σi , ( p ′ ) σi for ( σ, i ) ∈ supp .First, we consider an interval [ κ ν,j , κ ν,j + ) ⊆ κ η .We start with the construction of p ∗ ↾ [ κ ν,j , κ ν,j + ) = p ′∗ ↾ [ κ ν,j , κ ν,j + ) .Let ξ , ζ ∈ [ κ ν,j , κ ν,j + ) ∩ dom . • In the case that ( ξ, ζ ) ∈ dom p × dom p , we set p ′∗ ( ξ, ζ ) ∶ = p ∗ ( ξ, ζ ) ∶ = p ∗ ( ξ, ζ ) . • If ( ξ, ζ ) ∈ dom p ′ × dom p ′ , then p ′∗ ( ξ, ζ ) ∶ = p ∗ ( ξ, ζ ) ∶ = p ′∗ ( ξ, ζ ) .For ( ξ, ζ ) ∈ ( dom p × dom p ) ∩ ( dom p ′ × dom p ′ ) , this is not a contradiction,since p ∗ ↾ κ η and p ′∗ ↾ κ η are compatible. • If ζ ∈ dom p ∖ dom p ′ and ξ ∉ dom p , we proceed as follows: In the casethat { ξ } = a σi ∩ [ κ ν,j , κ ν,j + ) for some ( σ, i ) ∈ supp p with σ ≤ η , i < β or ( σ, i ) ∈ {( η m , i m ) ∣ m < ω } , we set p ′∗ ( ξ, ζ ) ∶ = p ∗ ( ξ, ζ ) ∶ = p σi ( ζ ) . Otherwise, weset p ′∗ ( ξ, ζ ) = p ∗ ( ξ, ζ ) arbitrarily. • In the case that ζ ∈ dom p ′ ∖ dom p and ξ ∉ dom p ′ , we proceed as before:If { ξ } = ( a ′ ) σi ∩ [ κ ν,j , κ ν,j + ) for some ( σ, i ) ∈ supp p ′ with σ ≤ η , i < β or ( σ, i ) ∈ {( η m , i m ) ∣ m < ω } , then p ′∗ ( ξ, ζ ) ∶ = p ∗ ( ξ, ζ ) ∶ = ( p ′ ) σi ( ζ ) . Otherwise,we set p ′∗ ( ξ, ζ ) = p ∗ ( ξ, ζ ) arbitrarily. • In all other cases, p ′∗ ( ξ, ζ ) = p ∗ ( ξ, ζ ) can be set arbitrarily.60his defines p ∗ ↾ [ κ ν,j , κ ν,j + ) = p ′∗ ↾ [ κ ν,j , κ ν,j + ) .Now, consider ( σ, i ) ∈ supp . We define p σi and ( p ′ ) σi on the interval [ κ ν,j , κ ν,j + ) ⊆ κ η as follows: • For ( σ, i ) ∈ supp p , we define p σi ↾ [ κ ν,j , κ ν,j + ) ⊇ p σi ↾ [ κ ν,j , κ ν,j + ) ac-cording to the linking property : Let { ξ } ∶ = a σi ∩ [ κ ν,j , κ ν,j + ) and con-sider ζ ∈ [ κ ν,j , κ ν,j + ) ∩ dom . If ζ ∈ dom p , we set p σi ( ζ ) ∶ = p σi ( ζ ) ; and p σi ( ζ ) ∶ = p ∗ ( ξ, ζ ) in the case that ζ ∈ dom ∖ dom p . (Note that ξ ∈ dom follows by construction.) • In the case that ( σ, i ) ∈ supp p ′ , we define ( p ′ ) σi ↾ [ κ ν,j , κ ν,j + ) ⊇ ( p ′ ) σi ↾ [ κ ν,j , κ ν,j + ) according to the linking property as before: Let { ξ } ∶ = ( a ′ ) σi ∩ [ κ ν,j , κ ν,j + ) , and consider ζ ∈ [ κ ν,j , κ ν,j + ) ∩ dom . If ζ ∈ dom p ′ , we set ( p ′ ) σi ( ζ ) ∶ = ( p ′ ) σi ( ζ ) ; and ( p ′ ) σi ( ζ ) ∶ = ( p ′∗ )( ξ, ζ ) in the case that ζ ∈ dom ∖ dom p ′ . (Again, ξ ∈ dom by construction.) • For ( σ, i ) ∈ supp p ∖ supp p ′ , let ( p ′ ) σi ↾ [ κ ν,j , κ ν,j + ) ∶ = p σi ↾ [ κ ν,j , κ ν,j + ) . • For ( σ, i ) ∈ supp p ′ ∖ supp p , let p σi ↾ [ κ ν,j , κ ν,j + ) ∶ = ( p ′ ) σi ↾ [ κ ν,j , κ ν,j + ) . • If ( σ, i ) ∈ supp ∖ ( supp p ∪ supp p ′ ) , then p σi ↾ [ κ ν,j , κ ν,j + ) = ( p ′ ) σi ↾ [ κ ν,j , κ ν,j + ) can be set arbitrarily on the given domain.This defines all p σi and ( p ′ ) σi for ( σ, i ) ∈ supp on intervals [ κ ν,j , κ ν,j + ) ⊆ κ η .We now have to verify that p σi = ( p ′ ) σi for any ( σ, i ) ∈ supp with σ ≤ η , i < β . Weonly have to treat the case that ( σ, i ) ∈ supp p ∩ supp p ′ .Consider an interval [ κ ν,j , κ ν,j + ) ⊆ κ σ ⊆ κ η . Then p ′ ∈ G and ( p β ↾ ( η + )) ( η m ,i m ) m < ω ∈ ( G β ↾ ( η + )) ( η m ,i m ) m < ω implies that p σi and ( p ′ ) σi are compatible, and a σi ∩ [ κ ν,j , κ ν,j + ) = ( a ′ ) σi ∩ [ κ ν,j , κ ν,j + ) = ∶ { ξ } .Let ζ ∈ [ κ ν,j , κ ν,j + ) ∩ dom . • If ζ ∈ dom p ∩ dom p ′ , then p σi ( ζ ) = p σi ( ζ ) = ( p ′ ) σi ( ζ ) = ( p ′ ) σi ( ζ ) . • For ζ ∈ dom ∖ ( dom p ∪ dom p ′ ) , it follows that p σi ( ζ ) = p ∗ ( ξ, ζ ) = p ′∗ ( ξ, ζ ) = ( p ′ ) σi ( ζ ) by construction, since we have arranged p ∗ ↾ κ η = p ′∗ ↾ κ η . • Let now ζ ∈ dom p ∖ dom p ′ , ξ ∉ dom p . Then p σi ( ζ ) = p σi ( ζ ) , and ( p ′ ) σi ( ζ ) = p ′∗ ( ξ, ζ ) . Since p ′∗ ( ξ, ζ ) = p σi ( ζ ) by construction of p ′∗ , this gives p σi ( ζ ) = ( p ′ ) σi ( ζ ) as desired.The case that ζ ∈ dom p ′ ∖ dom p , ξ ∉ dom p ′ , can be treated similarly.61 If ζ ∈ dom p ∖ dom p ′ and ξ ∈ dom p , it follows that p σi ( ζ ) = p σi ( ζ ) and ( p ′ ) σi ( ζ ) = p ′∗ ( ξ, ζ ) as before; but in this case, we have set p ′∗ ( ξ, ζ ) ∶ = p ∗ ( ξ, ζ ) ,so it remains to verify that p σi ( ζ ) = p ∗ ( ξ, ζ ) .Since p ′ ∈ G , ( p β ↾ ( η + )) ( η m ,i m ) m < ω ∈ ( G β ↾ ( η + )) ( η m ,i m ) m < ω , we cantake q ∈ G with ( q β ↾ ( η + )) ( η m ,i m ) m < ω ≤ ( p β ↾ ( η + )) ( η m ,i m ) m < ω , andassume w.l.o.g. that q ≤ p ′ . Then q σi ( ζ ) = q ∗ ( ξ, ζ ) by the linking propertyfor q ≤ p ′ , since ( a ′ ) σi ∩ [ κ ν,j , κ ν,j + ) = { ξ } . Moreover, p σi ( ζ ) = q σi ( ζ ) and p ∗ ( ξ, ζ ) = q ∗ ( ξ, ζ ) , and we are done. • The remaining case is that ζ ∈ dom p ′ ∖ dom p and ξ ∈ dom p ′ . Then p σi ( ζ ) = p ∗ ( ξ, ζ ) = p ′∗ ( ξ, ζ ) and ( p ′ ) σi ( ζ ) = ( p ′ ) σi ( ζ ) , and it remains to ver-ify that ( p ′ ) σi ( ζ ) = p ′∗ ( ξ, ζ ) . As before, take q ∈ G with q ≤ p ′ and ( q β ↾ ( η + )) ( η m ,i m ) m < ω ≤ ( p β ↾ ( η + )) ( η m ,i m ) m < ω . The latter gives q σi ( ζ ) = q ∗ ( ξ, ζ ) by the linking property, since σ ≤ η , i < β , a σi ∩ [ κ ν,j , κ ν,j + ) = { ξ } and ζ ∈ dom q ∖ dom p . Moreover, from q ≤ p ′ it follows that ( p ′ ) σi ( ζ ) = q σi ( ζ ) and p ′∗ ( ξ, ζ ) = q ∗ ( ξ, ζ ) ; hence, ( p ′ ) σi ( ζ ) = p ′∗ ( ξ, ζ ) as desired.Thus, it follows that p σi = ( p ′ ) σi holds for all ( σ, i ) ∈ supp with σ ≤ η , i < β .If m < ω with η m ≤ η , then i m < β follows by construction of β . Hence, p η m i m = ( p ′ ) η m i m .It remains to make sure that whenever m < ω with η m > η , then p η m i m ↾ κ η = ( p ′ ) η m i m ↾ κ η holds; which can be shown similarly as p σi = ( p ′ ) σi in the case that σ ≤ η , i < β : We use that a η m i m = ( a ′ ) η m i m and p η m i m ( ζ ) = ( p ′ ) η m i m ( ζ ) for all m < ω and ζ ∈ dom p ∩ dom p ′ ; and now, it is important that we are using the forcing notion ( P β ↾ ( η + )) ( η m ,i m ) m < ω instead of P β ↾ ( η + ) ; since we need the linking propertybelow κ η for the ( η m , i m ) with η m > η .It remains to construct p ∗ ↾ [ κ η , κ γ ) , p ′∗ ↾ [ κ η , κ γ ) , and p σi ↾ [ κ η , κ γ ) , ( p ′ ) σi ↾ [ κ η , κ γ ) for all ( σ, i ) ∈ supp with σ > η . • For ( η m , i m ) with η m > η , we take p η m i m ↾ [ κ η , κ η m ) ⊇ p η m i m ↾ [ κ η , κ η m ) , ( p ′ ) η m i m ↾ [ κ η , κ η m ) ⊇ ( p ′ ) η m i m ↾ [ κ η , κ η m ) on the given domain, such that p η m i m ↾ [ κ η , κ η m ) = ( p ′ ) η m i m ↾ [ κ η , κ η m ) . This is possible, since p ′ ∈ G and ( p β ↾ ( η + )) ( η m ,i m ) m < ω ∈ ( G β ↾ ( η + )) ( η m ,i m ) m < ω ; so p η m i m and ( p ′ ) η m i m arecompatible for all m < ω . • For the ( σ, i ) ∈ supp remaining, we set p σi ↾ [ κ η , κ γ ) ⊇ p σi ↾ [ κ η , κ γ ) and ( p ′ ) σi ↾ [ κ η , κ γ ) ⊇ ( p ′ ) σi ↾ [ κ η , κ γ ) arbitrarily on the given domain. • Consider an interval [ κ ν,j , κ ν,j + ) ⊆ [ κ η , κ γ ) . We define p ∗ ↾ [ κ ν,j , κ ν,j + ) ⊇ p ∗ ↾ [ κ ν,j , κ ν,j + ) according to the linking property : Whenever ζ ∈ dom ∖ dom p and { ξ } = a σi ∩ [ κ ν,j , κ ν,j + ) for some ( σ, i ) ∈ supp p , then p ∗ ( ξ, ζ ) ∶ = p σi ( ζ ) .The construction of p ′∗ ↾ [ κ ν,j , κ ν,j + ) ⊇ p ′∗ ↾ [ κ ν,j , κ ν,j + ) is similar.62his completes our construction of p ≤ p and p ′ ≤ p ′ with all the desired prop-erties.Similarly, one can construct p ≤ p , p ′ ≤ p ′ such that supp ∶ = supp p = supp p ′ ,dom ( σ ) ∶ = dom p ( σ ) = dom p ′ ( σ ) for all σ ∈ supp ; and p σi = ( p ′ ) σi for all σ ≤ η , i < β with σ ∈ Succ , and p η m i m = ( p ′ ) η m i m for all m < ω with η m ∈ Succ .We now proceed similarly as in the
Approximation Lemma
21 and construct anisomorphism π such that π a standard isomorphism for πp = p ′ . This determinesall parameters of π except the maps G ( ν, j ) ∶ supp π ( ν, j ) → supp π ( ν, j ) , whichwill be defined as follows: Consider an interval [ κ ν,j , κ ν,j + ) . Recall that we havethe map F π ( ν, j ) ∶ supp π ( ν, j ) → supp π ( ν, j ) , which is in charge of permutingthe linking ordinals: We set F π ( ν, j )( σ, i ) ∶ = ( λ, k ) for ( a ′ ) σi ∩ [ κ ν,j , κ ν,j + ) = a λk ∩ [ κ ν,j , κ ν,j + ) . We define G π ( ν, j ) ∶ = F π ( ν, j ) for all κ ν,j < κ η , and G π ( ν, j ) ∶ = id whenever κ ν,j ≥ κ η .By construction, it follows that πp = p ′ . We will now check that [ π ] is containedin the intersection ⋂ m F ix ( η m , i m ) ∩ ⋂ m H λ m k m . • Consider m < ω with η m ∈ Lim and r ∈ D π , r ′ ∶ = πr , with ( η m , i m ) ∈ supp r .For an interval [ κ ν,j , κ ν,j + ) ⊆ κ η m and ζ ∈ dom π ∩ [ κ ν,j , κ ν,j + ) , it followsby construction of the map π ( ζ ) that ( r ′ ) η m i m ( ζ ) = r η m i m ( ζ ) holds; since p η m i m = ( p ′ ) η m i m .In the case that ζ ∈ [ κ ν,j , κ ν,j + ) ∩ ( dom r ∖ dom π ) , it follows that ( r ′ ) η m i m ( ζ ) = r λk ( ζ ) with ( λ, k ) = G π ( ν, j )( η m , i m ) . If κ ν,j < κ η , then ( λ, k ) = G π ( ν, j )( η m , i m ) = F π ( ν, j )( η m , i m ) = ( η m , i m ) , since a η m i m = ( a ′ ) η m i m . In the case that κ ν,j ≥ κ η ,we have G π ( ν, j ) = id ; so again, ( λ, k ) = ( η m , i m ) .Hence, r η m i m ( ζ ) = ( r ′ ) η m i m ( ζ ) holds for all ζ ∈ dom r ∩ κ η m .This proves [ π ] ∈ F ix ( η m , i m ) in the case that η m ∈ Lim . For η m ∈ Succ , weobtain [ π ] ∈ F ix ( η m , i m ) as in the Approximation Lemma . • Consider m < ω with λ m ∈ Lim . In the case that λ m > η , we have G π ( ν, j )( λ m , i ) = ( λ m , i ) for all κ ν,j ∈ [ κ η , κ λ m ) , and [ π ] ∈ H λ m k m follows. If λ m ≤ η , it followsthat k m < β by construction of β . Hence, whenever κ ν,j < κ λ m and i ≤ k m , wehave G π ( ν, j )( λ m , i ) = F π ( ν, j )( λ m , i ) = ( λ m , i ) ; since a λ m i = ( a ′ ) λ m i followsfrom λ m ≤ η , i < β .In the case that λ m ∈ Succ , we obtain [ π ] ∈ H λ m k m as in the ApproximationLemma .Thus, we have shown that [ π ] ∈ ⋂ m F ix ( η m , i m ) ∩ ⋂ m H λ m k m ; which implies πf D π = f D π . It remains to make sure that πτ ̺ ( ˙ X ) D π = τ ̺ ( ˙ X ) D π .63ecall that we have an η -good pair ̺ = (( a m ) m < ω , ( σ m , i m ) m < ω ) with i m < β forall m < ω , and ˙ X ∈ Name (( P η ) ω × ∏ m < ω P σ m ) with τ ̺ ( ˙ X ) = { ( τ ̺ ( ˙ Y ) , q ) ∣ q ∈ P , ∃ ( ˙ Y , (( p ∗ ( a m )) m < ω , ( p σ m i m ) m < ω ) ) ∈ ˙ X ∶∀ m ( q ∗ ( a m ) ⊇ p ∗ ( a m ) , q σ m i m ⊇ p σ m i m ) } . Then τ ̺ ( ˙ X ) D π = { ( τ ̺ ( ˙ Y ) D π , q ) ∣ q ∈ D π , ˙ Y ∈ dom ˙ X , q ⊩ s τ ̺ ( ˙ Y ) ∈ τ ̺ ( ˙ X ) } , and πτ ̺ ( ˙ X ) D π = { ( πτ ̺ ( ˙ Y ) D π , πq ) ∣ πq ∈ D π , ˙ Y ∈ dom ˙ X , q ⊩ s τ ̺ ( ˙ Y ) ∈ τ ̺ ( ˙ X ) } . We will now check that π is the identity on P β ↾ ( η + ) . More precisely: Let q ∈ D π , q = ( q ∗ , ( q σi , b σi ) σ,i , ( q σ ) σ ) with πq = q ′ = ( q ′∗ , (( q ′ ) σi , ( b ′ ) σi ) σ,i , (( q ′ ) σ ) σ ) . Weprove that q ′∗ ↾ κ η = q ∗ ↾ κ η ; moreover, ( q ′ ) σi = q σi , ( b ′ ) σi = b σi for all σ ≤ η , i < β with σ ∈ Lim , and ( q ′ ) σi = q σi for all σ ≤ η , i < β with σ ∈ Succ . • Since π is a standard isomorphism for πp = p ′ , it follows that q ′∗ ↾ κ η = q ∗ ↾ κ η for all q ∈ D π ; since firstly, p ∗ ↾ κ η = p ′∗ ↾ κ η , and secondly, G π ( ν, j ) = F π ( ν, j ) for all κ ν,j < κ η . The latter makes sure that q ′∗ ( ξ σi ( ν, j ) , ζ ) = q ∗ ( ξ σi ( ν, j ) , ζ ) whenever ζ ∈ dom q ∖ dom π , and { ξ σi ( ν, j )} ∶ = b σi ∩ [ κ ν,j , κ ν,j + ) for some ( σ, i ) ∈ supp π ( ν, j ) : We have q ′∗ ( ξ σi ( ν, j ) , ζ ) = q ∗ ( ξ λk ( ν, j ) , ζ ) with ( λ, k ) = G π ( ν, j ) ○ ( F π ( ν, j )) − ( σ, i ) ; so from G π ( ν, j ) = F π ( ν, j ) it followsthat q ′∗ ( ξ σi ( ν, j ) , ζ ) = q ∗ ( ξ σi ( ν, j ) , ζ ) as desired. • Let now ( σ, i ) ∈ supp π = supp p with σ ≤ η , i < β and σ ∈ Lim . Then a σi = ( a ′ ) σi ; hence, F π ( ν, j )( σ, i ) = ( σ, i ) for all κ ν,j < κ σ . This gives ( b ′ ) σi = b σi as desired. For ζ ∈ dom π = dom p , it follows from p σi = ( p ′ ) σi by constructionof π that ( q ′ ) σi ( ζ ) = q σi ( ζ ) holds. Finally, if ζ ∈ ( dom q ∖ dom π ) , and ζ is contained in an interval [ κ ν,j , κ ν,j + ) ⊆ κ σ , then ( q ′ ) σi ( ζ ) = q λk ( ζ ) with ( λ, k ) = G π ( ν, j )( σ, i ) = F π ( ν, j )( σ, i ) = ( σ, i ) as desired. Hence, it followsthat ( q ′ ) σi = q σi for all σ ≤ η , i < β . • In the case that σ ≤ η , i < β with σ ∈ Succ , we obtain ( q ′ ) σi = q σi from p σi = ( p ′ ) σi as in the Approximation Lemma π is the identity on P β ↾ ( η + ) .Now, it is not difficult to prove recursively that for every ˙ Z ∈ Name (( P η ) ω × ∏ m < ω P σ m ) the following holds: If H is a V -generic filter on P , then ( τ ̺ ( ˙ Z )) πH = ( τ ̺ ( ˙ Z )) H = ( τ ̺ ( ˙ Z )) π − H .This implies τ ̺ ( ˙ X ) D π = πτ ̺ ( ˙ X ) D π , since for every q ∈ D π and ˙ Y ∈ dom ˙ X , we have q ⊩ s τ ̺ ( ˙ Y ) ∈ τ ̺ ( ˙ X ) if and only if πq ⊩ s τ ̺ ( ˙ Y ) ∈ τ ̺ ( ˙ X ) holds.64umming up, this gives our desired contradiction: Since p ⊩ s ( τ ̺ ( ˙ X ) , α ) ∈ ˙ f , itfollows that πp ⊩ s ( πτ ̺ ( ˙ X ) D π , α ) ∈ πf D π ; hence, p ′ ⊩ s ( τ ̺ ( ˙ X ) D π , α ) ∈ f D π . Butthis contradicts p ′ ⊩ s ( τ ̺ ( ˙ X ) , α ) ∉ ˙ f .Thus, our assumption that ( X, α ) ∈ ( f β ) ′ ∖ f β was wrong, and it follows that ( f β ) ′ = f β as desired.Hence, f β ∈ V [ ( G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω G η m i m ↾ [ κ η , κ η m ) ] . C) ( P β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω P η m ↾ [ κ η , κ η m ) preserves cardinals ≥ α η . The next step is to show that cardinals ≥ α η are absolute between V and V [ G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω G η m i m ↾ [ κ η , κ η m )] .Recall that we are assuming GCH in our ground model V , which will be used im-plicitly throughout this Chapter 6.2 C): When we claim that a particular forcingnotion preserves cardinals, then we mean it preserves cardinals under the assump-tion that GCH holds, if not stated differently.First, we have a look at the cardinality of ( P β ↾ ( η + )) ( η m ,i m ) m < ω . Recall that β was an ordinal large enough for the intersection ( A ˙ f ) with κ + η < β < α η . Lemma 39. ∣( P β ↾ ( η + )) ( η m ,i m ) m < ω ∣ ≤ ∣ β ∣ + .Proof. The forcing notion ( P β ↾ ( η + )) ( η m ,i m ) m < ω is the set of all ( p ∗ ↾ κ η , ( p σi , a σi ) σ ≤ η,i < β , ( p σ ↾ ( β × dom y p σ )) σ ≤ η , ( p η m i m ↾ κ η , a η m i m ∩ κ η ) m < ω , η m > η , ̃ X p ) for p ∈ P with ∣{( σ, i ) ∈ supp p ∣ σ > η ∨ i ≥ β }∣ = ℵ , together with the max-imal element ( βη + ) ( η m ,i m ) m < ω . Since ̃ X p ⊆ κ η , there are only κ + η ≤ ∣ β ∣ -manypossibilities for ̃ X p ; and there are only ≤ κ + η ≤ ∣ β ∣ -many possibilities for p ∗ ↾ κ η and ( p η m i m ↾ κ η , a η m i m ∩ κ η ) m < ω . Concerning ( p σi , a σi ) σ ≤ η,i < β , there are ∣ β ∣ ℵ ≤ ∣ β ∣ + -many possibilities for the countable support; and with the support fixed, we have ( κ η ) ℵ ≤ κ + η ≤ ∣ β ∣ -many possibilities for countably many ( p σi , a σi ) with σ ≤ η , i < β .Finally, for ( p σ ↾ ( β × dom y p σ )) σ ≤ η , there are only ∣ η ∣ ω ≤ κ + η ≤ ∣ β ∣ -many pos-sibilities for the countable support; and with the countable support fixed, thereare ≤ ( ∣ β ∣ ⋅ κ η ) ℵ = ∣ β ∣ + -many possibilities for countably many p σ with dom p σ ⊆ β × κ σ ⊆ β × κ η . Hence, it follows that the forcing notion ( P β ↾ ( η + )) ( η m ,i m ) m < ω has cardinality ≤ ∣ β ∣ + . Corollary 40. If ∣ β ∣ + < α η , then ( P β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω P η m ↾ [ κ η , κ η m ) preserves cardinals ≥ α η . roof. With the same arguments as in Lemma 2, one can show that the forc-ing ∏ m < ω P η m ↾ [ κ η , κ η m ) preserves all cardinals. By Lemma 39 above, theforcing ( P β ↾ ( η + )) ( η m ,i m ) m < ω has cardinality ≤ ∣ β ∣ + (in V ; and hence, alsoin any ∏ m < ω P η m ↾ [ κ η , κ η m ) -generic extension). It follows that the product ( P β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω P η m ↾ [ κ η , κ η m ) preserves all cardinals ≥ ∣ β ∣ ++ .It remains to consider the case that ∣ β ∣ + = α η . Then by our assumptions on thesequence ( α η ∣ < η < γ ) (cf. Chapter 2), it follows that cf ∣ β ∣ > ω . Hence, GCH gives ∣ β ∣ ℵ = ∣ β ∣ < α η ; and by our proof of Lemma 39, it follows that all compo-nents of ( P β ↾ ( η + )) ( η m ,i m ) m < ω have cardinality ≤ ∣ β ∣ < α η ; with the exceptionof ( p σ ↾ ( β × dom y p σ )) σ ≤ η , where there might be ( ∣ β ∣ ⋅ κ η ) ℵ = ∣ β ∣ + = α η -manypossibilities.We now have to distinguish several cases depending on whether η is a limit ordinalor not, and depending on whether κ η is a limit cardinal or a successor cardinal(i.e. η ∈ Lim or η ∈ Succ ).We will have to separate one or two components P σ ↾ ( β × [ κ σ , κ σ )) , where σ ∈ Succ , σ ≤ η , κ σ = κ σ + , from the forcing notion ( P β ↾ ( η + )) ( η m ,i m ) m < ω ; andobtain a forcing (( P β ↾ ( η + )) ( η m ,i m ) m < ω ) ′ which has cardinality < α η , while theproduct of the remaining P σ ↾ ( β × [ κ σ , κ σ )) and ∏ m < ω P η m ↾ [ κ η , κ η m ) preservescardinals. Proposition 41.
The forcing notion ( P β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω P η m ↾ [ κ η , κ η m ) preserves all cardinals ≥ α η .Proof. By Corollary 40, we only have to treat the case that α η = ∣ β ∣ + . Thencf ∣ β ∣ > ω and ∣ β ∣ ℵ = ∣ β ∣ .First, we assume that η is a limit ordinal . Then by closure of the sequence ( κ σ ∣ < σ < γ ) , it follows that η ∈ Lim , i.e. κ η = sup { κ σ ∣ < σ < η } is a limitcardinal.Since the sequence ( α σ ∣ < σ < γ ) is strictly increasing (cf. Chapter 2), it followsthat α σ < ∣ β ∣ for all σ < η . Hence, for any σ ∈ Succ with σ < η , the forcingnotion P σ ↾ ( β × [ κ σ , κ σ )) = P σ ↾ ( α σ × [ κ σ , κ σ )) has cardinality ≤ α + σ ≤ ∣ β ∣ ;and we conclude that there are only ≤ ∣ η ∣ ℵ ⋅ ∣ β ∣ ℵ = ∣ β ∣ -many possibilities for ( p σ ↾ ( β × dom y p σ )) σ ≤ η .Hence, by the proof of Lemma 39, it follows that ( P β ↾ ( η + )) ( η m ,i m ) m < ω hascardinality ≤ ∣ β ∣ < α η . Like in Corollary 40, this implies that the product ( P β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω P η m ↾ [ κ η , κ η m ) preserves all cardinals ≥ ∣ β ∣ + = α η asdesired.The remaining case is that η is a successor ordinal . Let η = η + . We nowhave to distinguish four cases, depending on whether κ η and κ η are successor car-dinals or limit cardinals. 66f η ∈ Lim and η ∈ Lim , it follows for any P σ ↾ ( β × [ κ σ , κ σ )) with σ ≤ η , σ ∈ Succ that σ < η must hold; hence, α σ < α η < α η = ∣ β ∣ + , which implies α σ < ∣ β ∣ . Thus,the corresponding forcing notion P σ ↾ ( β × [ κ σ , κ σ )) = P σ ↾ ( α σ × [ κ σ , κ σ )) has car-dinality ≤ α + σ ≤ ∣ β ∣ ; and as before, it follows that the forcing ( P β ↾ ( η + )) ( η m ,i m ) m < ω has cardinality ≤ ∣ β ∣ ℵ = ∣ β ∣ . Like in Corollary 40, this implies that the product ( P β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω P η m ↾ [ κ η , κ η m ) preserves all cardinals ≥ ∣ β ∣ + = α η as desired.If η ∈ Lim and η ∈ Succ , we consider the forcing notion (( P β ↾ ( η + )) ( η m ,i m ) m < ω ) ′ ,which is obtained from ( P β ↾ ( η + )) ( η m ,i m ) m < ω by excluding P η ↾ ( β × [ κ η , κ η )) ;i.e. we consider ( p σ ↾ ( β × dom y p σ )) σ < η = ( p σ ↾ ( β × dom y p σ )) σ < η instead of ( p σ ↾ ( β × dom y p σ )) σ ≤ η . Then (( P β ↾ ( η + )) ( η m ,i m ) m < ω ) ′ has cardinality ≤ ∣ β ∣ as before; and it suffices to check that the remaining product P η ↾ ( β × [ κ η , κ η )) × ∏ m < ω P η m ↾ [ κ η , κ η m ) preserves all cardinals.The forcing notion ∏ m < ω P η m ↾ [ κ η , κ η m ) preserves cardinals. Moreover, ∏ m < ω P η m ↾ [ κ η , κ η m ) is ≤ κ η -closed. Hence, in any V -generic extension by ∏ m < ω P η m ↾ [ κ η , κ η m ) the following holds: Firstly, P η ↾ ( β × [ κ η , κ η )) is the same forcing no-tion as in V ; and secondly, P η ↾ ( β × [ κ η , κ η )) preserves cardinals, since 2 < κ η = κ η .Thus, it follows that the product P η ↾ ( β × [ κ η , κ η )) × ∏ m < ω P η m ↾ [ κ η , κ η m ) preserves all cardinals as desired.If η ∈ Succ and η ∈ Lim , we proceed similarly, but exclude P η ↾ ( β × [ κ η , κ η )) instead of P η ↾ ( β × [ κ η , κ η )) .If η ∈ Succ and η ∈ Succ , then both P η ↾ ( β × [ κ η , κ η )) and P η ↾ ( β × [ κ η , κ η )) have to be parted from ( P β ↾ ( η + )) ( η m ,i m ) m < ω . As before, it follows that firstly, theremaining forcing notion, denoted by (( P β ↾ ( η + )) ( η m ,i m ) m < ω ) ′′ , has cardinality ≤ ∣ β ∣ ; and secondly, the remaining product P η ↾ ( β × [ κ η , κ η )) × P η ↾ ( β × [ κ η , κ η )) × ∏ m < ω P η m ↾ [ κ η , κ η m ) preserves all cardinals.It follows that ( P β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω P η m ↾ [ κ η , κ η m ) preserves all car-dinals ≥ α η .This concludes our proof by cases. 67 ) A set ̃ ℘ ( κ η ) ⊇ dom f β with an injection ι ∶ ̃ ℘ ( κ η ) ↪ ∣ β ∣ ℵ . In this section, we construct in V [( G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω G η m i m ↾ [ κ η , κ η m )] a set ̃ ℘ ( κ η ) with ̃ ℘ ( κ η ) ⊇ dom f β , together with an injective function ι ∶ ̃ ℘ ( κ η ) ↪ (∣ β ∣ ℵ ) V < α η . Since f β is contained in V [( G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω G η m i m ↾ [ κ η , κ η m )] by Definition 37 and Proposition 38, and ( P β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω P η m ↾ [ κ η , κ η m ) preserves cardinals ≥ α η by Proposition 41, this will contra-dict our initial assumption that f β ∶ dom f β → α η was surjective.Fix an η -good pair ̺ = (( a m ) m < ω , ( σ m , i m ) m < ω ) . Then ∏ m G ∗ ( a m ) × ∏ m G σ m i m is a V -generic filter on ∏ m P η × ∏ m P σ m ; and as in Lemma 2, it follows that this forcingpreserves cardinals and the GCH . Hence, there is an injection χ ∶ ℘ ( κ η ) ↪ ( κ + η ) V in V [ ∏ m G ∗ ( a m ) × ∏ m G σ m i m ] .Let M β be the set of all η -good pairs (( a m ) m < ω , ( σ m , i m ) m < ω ) in V with the prop-erty that i m < β for all m < ω . Then M β has cardinality ≤ ( κ η ) ℵ ⋅ ∣ η ∣ ℵ ⋅ ∣ β ∣ ℵ ≤ ∣ β ∣ ℵ .First, we consider the case that ∣ β ∣ + = α η . Then cf ∣ β ∣ > ω ; hence, GCH gives ∣ β ∣ ℵ = ∣ β ∣ and there is an injection ψ ∶ M β ↪ ∣ β ∣ in V .By construction of f β (cf. Definition 30), it follows that any X ⊆ κ η with X ∈ dom f β is contained in a model V [ ∏ m G ∗ ( a m ) × ∏ m G σ m i m ] for some η -good pair (( a m ) m < ω , ( σ m , i m ) m < ω ) ∈ M β . Hence, dom f β is a subset of ̃ ℘ ( κ η ) ∶ = ⋃ { ℘ ( κ η ) ∩ V [ ∏ m G ∗ ( a m ) × ∏ m G σ m i m ] ∣ (( a m ) m < ω , ( σ m , i m ) m < ω ) ∈ M β } . The set ̃ ℘ ( κ η ) can be defined in V [( G β ↾ ( η + )) ( η m ,i m ) m < ω × G η m i m ↾ [ κ η , κ η m )] ,since for any (( a m ) m < ω , ( σ m , i m ) m < ω ) ∈ M β , we have a m ⊆ κ η , and σ m ≤ η , i m < β for all m < ω .For the rest of this section, we work in V [( G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω G η m i m ↾ [ κ η , κ η m )] , and construct there an injective function ι ∶ ̃ ℘ ( κ η ) ↪ ∣ β ∣ V .For a set X ∈ ̃ ℘ ( κ η ) , let ̃ ι ( X ) ∶ = (( a m ) m < ω , ( σ m , i m ) m < ω ) if (( a m ) m < ω , ( σ m , i m ) m < ω ) ∈ M β with X ∈ ℘ ( κ η ) ∩ V [ ∏ m G ∗ ( a m ) × ∏ m G σ m i m ] , and ψ (( a m ) m < ω , ( σ m , i m ) m < ω ) is least with this property.Now, we use the Axiom of Choice in V [( G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω G η m i m ↾ [ κ η , κ η m )] , and choose for all (( a m ) m < ω , ( σ m , i m ) m < ω ) ∈ M β an injection χ (( a m ) m < ω , ( σ m ,i m ) m < ω ) ∶ ( ℘ ( κ η ) ∩ V [ ∏ m G ∗ ( a m ) × ∏ m G σ m i m ] ) ↪ ( κ + η ) V . ι ∶ ̃ ℘ ( κ η ) ↪ ( κ + η ) V ⋅ ∣ β ∣ V as follows: For X ∈ ̃ ℘ ( κ η ) , let ι ( X ) ∶ = ( χ ̃ ι ( X ) ( X ) , ψ (̃ ι ( X )) ) . Since ψ and the maps χ (( a m ) m < ω , ( σ m ,i m ) m < ω ) for (( a m ) m < ω , ( σ m , i m ) m < ω ) ∈ M β areinjective, it follows that also ι is injective; which finishes our construction in thecase that (∣ β ∣ + ) V = α η .If ∣ β ∣ + < α η in V , we can take an injection ψ ∶ M β ↪ (∣ β ∣ + ) V , and construct an injec-tive function ι ∶ ̃ ℘ ( κ η ) ↪ ( κ + η ) V ⋅ (∣ β ∣ + ) V in V [( G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω G η m i m ↾ [ κ η , κ η m )] similarly as before.This gives the following proposition: Proposition 42. If (∣ β ∣ + ) V = α η , then there is in V [( G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω G η m i m ↾ [ κ η , κ η m )] an injection ι ∶ ̃ ℘ ( κ η ) ↪ ∣ β ∣ V , where ̃ ℘ ( κ η ) ∶ = ⋃ { ℘ ( κ η ) ∩ V [ ∏ m G ∗ ( a m ) × ∏ m G σ m i m ] ∣ (( a m ) m < ω , ( σ m , i m ) m < ω ) ∈ M β } . If (∣ β ∣ + ) V < α η , there is in V [( G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω G η m i m ↾ [ κ η , κ η m )] aninjection ι ∶ ̃ ℘ ( κ η ) ↪ (∣ β ∣ + ) V . This leads to our desired contradiction: We assumed that f β ∶ dom f β → α η was surjective. By Chapter 6.2 B), Definition 37 and Proposition 38, it followsthat f β ∈ V [( G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m G η m i m ↾ [ κ η , κ η m )] ; where ( G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m G η m i m ↾ [ κ η , κ η m ) is a V -generic filter on the forcing notion (( P β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m P η m ↾ [ κ η , κ η m ) , which preserves cardinals ≥ α η byChapter 6.2 C), Proposition 41.However, since dom f β ⊆ ̃ ℘ ( κ η ) and ∣ β ∣ V < α η , it follows that f β together withthe map ι from Proposition 42 above, collapses the cardinal α η in V [( G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m G η m i m ↾ [ κ η , κ η m )] . Contradiction.Thus, we have shown that our initial assumption that f β ∶ dom f β → α η was sur-jective, was wrong.Hence, there must be α < α η with α ∉ rg f β . E) We use an isomorphism argument and obtain a contradiction.
We fix an ordinal α < α η with α ∉ rg f β . By surjectivity of f , there must be X ⊆ κ η , X ∈ N , with f ( X ) = α . Hence, there is an η -good pair ̺ = (( a m ) m < ω , ( σ m , i m ) m < ω ) with X ∈ V [ ∏ m G ∗ ( a m ) × ∏ m G σ m i m ] ; but since X ∉ dom f β , there must be at leastone index m < ω with i m ≥ β . Let S denote the set of all ( σ m , i m ) with i m < β ,and let S be the set of all ( σ m , i m ) with i m ≥ β . Then ∣ S ∣ ≥ ( λ m , k m ) ∶ = ( σ m , i m ) in the case that m ∈ S . We denote our η -good pair ̺ by ̺ = (( a m ) m < ω , (( σ m , i m ) m ∈ S , ( λ m , k m ) m ∈ S )) . Then X = ˙ X ∏ m G ∗ ( a m ) × ∏ m ∈ S G σmim × ∏ m ∈ S G λmkm for some ˙ X ∈ N ame (( P η ) ω × ∏ m ∈ S P σ m × ∏ m ∈ S P λ m ) , such that the followingholds: • ( a m ∣ m < ω ) is a sequence of pairwise disjoint κ η -subsets, such that for all m < ω and κ ν, < κ η , it follows that ∣ a m ∩ [ κ ν, , κ ν, + )∣ = • S ⊆ ω , and for all m ∈ S , we have σ m ∈ Succ with σ m ≤ η , i m < min { α σ m , β } , • if m , m ′ ∈ S with m ≠ m ′ , then ( σ m , i m ) ≠ ( σ m ′ , i m ′ ) , • ∅ ≠ S ⊆ ω , and for all m ∈ S , we have λ m ∈ Succ with λ m ≤ η , k m ∈ [ β, α λ m ) , • if m , m ′ ∈ S with m ≠ m ′ , then ( λ m , k m ) ≠ ( λ m ′ , k m ′ ) .Since ( X, α ) ∈ f , take p ∈ G with p ⊩ s ( τ ̺ ( ˙ X ) , α ) ∈ ˙ f . Since we are using countable support, we can asssume w.l.o.g. that σ m ∈ supp p , i m ∈ dom x p ( σ m ) for all m ∈ S ; and λ m ∈ supp p , k m ∈ dom x p ( λ m ) for all m ∈ S .The idea can roughly be explained as follows: Recall that we have β = ̃ β + κ + η (addition of ordinals), where the ordinal ̃ β is large enough for ( A ˙ f ) . In particular, κ + η < ̃ β < β < α η . We will now extend p and obtain a condition q ∈ G , q ≤ p , suchthat there is a sequence ( l m ∣ m ∈ S ) with l m ∈ (̃ β, β ) for all m ∈ S , such that q λ m k m = q λ m l m for all m ∈ S . Then we construct an isomorphism π ∈ A that swaps any ( λ m , k m ) -coordinate with the according ( λ m , l m ) -coordinate.Then πq = q ; and we will see that π ∈ ⋂ m F ix ( η m , i m ) ∩ ⋂ m H λ m k m , since ̃ β is largeenough for ( A ˙ f ) . Hence, πf D π = f D π ; so from q ⊩ s ( τ ̺ ( ˙ X ) , α ) ∈ ˙ f , we obtain that q ⊩ s ( πτ ̺ ( ˙ X ) D π , α ) ∈ f D π . Setting Y ∶ = ( πτ ̺ ( ˙ X ) D π ) G , it follows that ( Y, α ) ∈ f. Y = ˙ X ∏ m G ∗ ( a m ) × ∏ m ∈ S G σmim × ∏ m ∈ S G λmlm ; where i m < β for all m ∈ S , but also l m < β for all m ∈ S . But then, the η -good pair ̺ ′ = (( a m ) m < ω , (( σ m , i m ) m ∈ S , ( λ m , l m ) m ∈ S )) is an element of M β , and it follows that ( Y, α ) = ( ˙ X ∏ m G ∗ ( a m ) × ∏ m ∈ S G σmim × ∏ m ∈ S G λmlm , α ) ∈ f β . But this would be a contradiction towards α ∉ rg f β .We start our proof with the following lemma: Lemma 43.
Let D be the set of all q ∈ P for which there exists a sequence ofpairwise distinct ordinals ( l m ∣ m ∈ S ) with l m ∈ (̃ β, β ) ∖ { i m ∣ m ∈ S } for all m ∈ S , such that q λ m k m = q λ m l m holds for all m ∈ S . Then D is dense below p .Proof. Consider q ∈ P with q ≤ p . We have to construct q ≤ q with q ∈ D . Theidea is that for every m ∈ S , we enlarge dom x q ( λ m ) by some suitable k m , and set q ( λ m )( k m , ζ ) ∶ = q ( λ m )( l m , ζ ) = q ( λ m )( l m , ζ ) for all ζ ∈ dom y q ( λ m ) = dom y q ( λ m ) .Note that for every m ∈ S , we have λ m ∈ supp q with ∣ dom x q ( λ m )∣ < κ λ m ≤ κ η ,since λ m ≤ η . Hence, it follows that ∣ ⋃ m ∈ S dom x q ( λ m )∣ ≤ κ η < κ + η ; and similarly, ∣ ⋃ m ∈ S dom x q ( σ m )∣ ≤ κ η < κ + η . Thus, the set∆ ∶ = (̃ β, β ) ∖ ( ⋃ m ∈ S dom x q ( λ m ) ∪ ⋃ m ∈ S dom x q ( σ m ) ) has cardinality κ + η .Recall that for every m ∈ S , we have assumed that i m ∈ dom x p ( σ m ) ⊆ dom x q ( σ m ) ;hence i m ∉ ∆.For m ∈ S , we have k m ∈ [ β, α λ m ) ; hence, β < α λ m and ∆ ⊆ (̃ β, β ) ⊆ α λ m follows.We take a sequence of pairwise distinct ordinals ( l m ∣ m ∈ S ) in ∆ (then { l m ∣ m ∈ S } ⊆ (̃ β, β ) ∖ { i m ∣ m ∈ S } ), and define the extension q ≤ q as follows:Set q ∶ = q , and supp q = supp q . (From q ≤ p it follows that λ m ∈ supp q forall m ∈ S .) For σ ∈ supp q with σ ∉ { λ m ∣ m ∈ S } , we set q ( σ ) ∶ = q ( σ ) . For σ ∈ { λ m ∣ m ∈ S } , we proceed as follows: Let S ( σ ) ∶ = { m ∈ S ∣ σ = λ m } . We setdom y q ( σ ) ∶ = dom y q ( σ ) , and dom x q ( σ ) ∶ = dom x q ( σ ) ∪ { l m ∣ m ∈ S ( σ )} . Notethat by construction of ∆ this union is disjoint, since l m ∉ dom x q ( σ ) = dom x q ( λ m ) for all m ∈ S ( σ ) .Note that for every m ∈ S ( σ ) , we have k m ∈ dom x p ( σ ) ⊆ dom x q ( σ ) ⊆ dom x q ( σ ) .We let q ( σ )( i, ζ ) ∶ = q ( σ )( i, ζ ) whenever ( i, ζ ) ∈ dom x q ( σ ) × dom y q ( σ ) . If ( i, ζ ) ∈ dom q ( σ ) ∖ dom q ( σ ) , then ζ ∈ dom y q ( σ ) and i = l n for some n ∈ S ( σ ) , i.e. n ∈ S with σ = λ n . In this case, we set q ( σ )( i, ζ ) = q ( λ n )( l n , ζ ) ∶ = q ( λ n )( k n , ζ ) = q ( σ )( k n , ζ ) . 71his defines q ≤ q with the property that q λ m k m = q λ m l m holds for all m ∈ S .Thus, it follows that D is dense below p .Since p ∈ G , we can now take q ∈ G , q ≤ p with q ∈ D . Take ( l m ∣ m ∈ S ) as inthe definition of D , with l m ∈ (̃ β, β ) ∖ { i m ∣ m ∈ S } and q λ m k m = q λ m l m for all m ∈ S .Then the sets {( λ m , l m ) ∣ m ∈ S } and {( σ m , i m ) ∣ m ∈ S } are disjoint.Since q ≤ p , we have q ⊩ s ( τ ̺ ( ˙ X ) , α ) ∈ ˙ f . The next step is to construct an isomorphism π that swaps every ( λ m , k m ) -coordinate with the according ( λ m , l m ) -coordinate for m ∈ S , and does nothingelse. Definition 44.
We define an isomorphism π ∈ A as follows: • The map π is the identity on D π = P . • We set supp π ∶ = supp q , and for every σ ∈ supp q , we let dom π ( σ ) ∶ = dom q ( σ ) .Then for all m ∈ S , it follows that λ m ∈ supp p ⊆ supp q = supp π ;and k m ∈ dom x p ( λ m ) ⊆ dom x q ( λ m ) = dom x π ( λ m ) , l m ∈ dom x q ( λ m ) = dom x π ( λ m ) . • Consider σ ∈ supp π with κ σ = κ σ + . In the case that σ ∉ { λ m ∣ m ∈ S } , weset supp π ( σ ) ∶ = ∅ , and let π ( σ )( i, ζ ) ∶ → be the identity map for all ( i, ζ ) ∈ α σ × [ κ σ , κ σ ) . • For σ ∈ { λ m ∣ m ∈ S } , consider the set S ( σ ) ∶ = { m ∈ S ∣ σ = λ m } , and let supp π ( σ ) ∶ = { k m ∣ m ∈ S ( σ )} ∪ { l m ∣ m ∈ S ( σ )} . Then supp π ( σ ) is asubset of dom x π ( σ ) .The map f π ( σ ) ∶ supp π ( σ ) → supp π ( σ ) is defined as follows: Let f π ( σ )( k m ) = l m , and f π ( σ )( l m ) = k m for all m ∈ S ( σ ) .Then f π ( σ ) is well-defined and bijective, since k m ≥ β for all m ∈ S , and l m < β for all m ∈ S .It remains to define the maps π ( ζ ) ∶ supp π ( σ ) → supp π ( σ ) for ζ ∈ dom y π ( σ ) :Let π ( ζ )( ǫ i ∣ i ∈ supp π ( σ )) ∶ = (̃ ǫ i ∣ i ∈ supp π ( σ )) , where ̃ ǫ k m ∶ = ǫ l m , ̃ ǫ l m ∶ = ǫ k m for all m ∈ S ( σ ) .Finally, for every ( i, ζ ) ∈ α σ × [ κ σ , κ σ ) , we let π ( σ )( i, ζ ) ∶ → be theidentity. π ∈ A . Lemma 45.
For Y ∶ = ( πτ ̺ ( ˙ X ) D π ) G , it follows that ( Y, α ) ∈ f .Proof. By construction of π it follows that whenever r is a condition in D π with r ′ ∶ = πr , then the following holds: Firstly, for all m ∈ S , we have ( r ′ ) λ m k m = r λ m l m and ( r ′ ) λ m l m = r λ m k m . Secondly, whenever σ ∈ supp r , i ∈ dom x r ( σ ) with ( σ, i ) ∉ {( λ m , k m ) ∣ m ∈ S } ∪ {( λ m , l m ) ∣ m ∈ S } , then ( r ′ ) σi = r σi .In particular, ( r ′ ) σ m ′ i m ′ = r σ m ′ i m ′ holds for all m ′ ∈ S :On the one hand, we have ( σ m ′ , i m ′ ) ∉ {( λ m , k m ) ∣ m ∈ S } for all m ′ ∈ S , since i m ′ < β ; but k m ≥ β for all m ∈ S . On the other hand, ( σ m ′ , i m ′ ) ∉ {( λ m , l m ) ∣ m ∈ S } for all m ′ ∈ S follows by construction of the set D .In other words: The map π swaps for all m ∈ S the ( λ m , k m ) -coordinate with theaccording ( λ m , l m ) -coordinate, and does nothing else.Hence, it follows that πq = q ; since q λ m k m = q λ m l m for all m ∈ S .Next, we want to show that π ∈ ⋂ m F ix ( η m , i m ) ∩ ⋂ m H λ m k m . Then πf D π = f D π follows. Regarding π ∈ ⋂ m F ix ( η m , i m ) , it suffices to make sure that for all m < ω ,we have ( η m , i m ) ∉ {( λ m ′ , k m ′ ) ∣ m ′ ∈ S } ∪ {( λ m ′ , l m ′ ) ∣ m ′ ∈ S } . But this followsfrom the fact that λ m ′ ≤ η and k m ′ ≥ β > ̃ β , l m ′ > ̃ β for all m ′ ∈ S ; but ̃ β is largeenough for ( A ˙ f ) , so for any η m with η m ≤ η , it follows that i m < ̃ β . This implies ( η m , i m ) ∉ {( λ m ′ , k m ′ ) ∣ m ′ ∈ S } ∪ {( λ m ′ , l m ′ ) ∣ m ′ ∈ S } for all m < ω as desired.Hence, π ∈ ⋂ m F ix ( η m , i m ) .Regarding π ∈ ⋂ m H λ m k m , we have to make sure that whenever λ m = λ m ′ for some m < ω and m ′ ∈ S , then supp π ( λ m ) = supp π ( λ m ′ ) ⊆ ( k m , α λ m ) holds; i.e. k m ′ > k m and l m ′ > k m . Again, this follows from the fact that λ m ′ ≤ η and k m ′ ≥ β > ̃ β , l m ′ > ̃ β for all m ′ ∈ S ; and ̃ β is large enough for ( A ˙ f ) , so whenever λ m ≤ η , then k m < ̃ β follows. Hence, π ∈ ⋂ m H λ m k m .Thus, it follows that πf D π = f D π .Now, from q ⊩ s ( τ ̺ ( ˙ X ) , α ) ∈ ˙ f , we obtain πq ⊩ s ( πτ ̺ ( ˙ X ) D π , α ) ∈ πf D π ; hence, q ⊩ s ( πτ ̺ ( ˙ X ) D π , α ) ∈ f D π . With Y ∶ = ( πτ ̺ ( ˙ X ) D π ) G , it follows from q ∈ G that ( Y, α ) ∈ f as desired.We will now show that ( Y, α ) ∈ f implies that also ( Y, α ) ∈ f β must hold. Thisfinally gives our desired contradiction, since α ∉ rg f β .73ndeed, we will prove that Y = ˙ X ∏ m G ∗ ( a m ) × ∏ m ∈ S G σmim × ∏ m ∈ S G λmlm . Since i m < β for all m ∈ S and l m < β for all m ∈ S , it follows that the η -goodpair ̺ ′ ∶ = (( a m ) m < ω , (( σ m , i m ) m ∈ S , ( λ m , l m ) m ∈ S )) is an element of M β . Hence, ( Y, α ) ∈ f would then imply that also ( Y, α ) ∈ f β must hold, and we are done.Recall that ̺ ∶ = (( a m ) m < ω , (( σ m , i m ) m ∈ S , ( λ m , k m ) m ∈ S )) , ˙ X ∈ Name (( P η ) ω × ∏ m ∈ S P σ m × ∏ m ∈ S P λ m ) , and τ ̺ ( ˙ X ) is the canonical exten-sion of ˙ X to a name for P (see Definition 34).We will show recursively: Lemma 46.
For every ˙ Y ∈ Name (( P η ) ω × ∏ m ∈ S P σ m × ∏ m ∈ S P λ m ) , it followsthat π τ ̺ ( ˙ Y ) D π = τ ̺ ′ ( ˙ Y ) D π . Proof.
Consider ˙ Y ∈ Name α + (( P η ) ω × ∏ m ∈ S P σ m × ∏ m ∈ S P λ m ) , and assumerecursively that the claim was true for all ˙ Z ∈ Name α (( P η ) ω × ∏ m ∈ S P σ m × ∏ m ∈ S P λ m ) .First, τ ̺ ( ˙ Y ) D π = { ( τ ̺ ( ˙ Z ) D π , r ) ∣ r ∈ D π , ˙ Z ∈ dom ˙ Y , r ⊩ s τ ̺ ( ˙ Z ) ∈ τ ̺ ( ˙ Y ) } , and πτ ̺ ( ˙ Y ) D π = { ( πτ ̺ ( ˙ Z ) D π , πr ) ∣ r ∈ D π , ˙ Z ∈ dom ˙ Y , r ⊩ s τ ̺ ( ˙ Z ) ∈ τ ̺ ( ˙ Y ) } . Now, for any H a V -generic filter on P and ˙ Z ∈ Name (( P η ) ω × ∏ m ∈ S P σ m × ∏ m ∈ S P λ m ) , it follows by construction of the map π that ( τ ̺ ( ˙ Z )) H = ˙ Z ∏ m < ω H ∗ ( a m ) × ∏ m ∈ S H σmim × ∏ m ∈ S H λmkm = ˙ Z ∏ m < ω ( πH ) ∗ ( a m ) × ∏ m ∈ S ( πH ) σmim × ∏ m ∈ S ( πH ) λmlm = ( τ ̺ ′ ( ˙ Z )) πH , since π swaps any ( λ m , k m ) -coordinate with the according ( λ m , l m )-coordinate,and does nothing else. 74ence, whenever r ∈ D π , then r ⊩ s τ ̺ ( ˙ Z ) ∈ τ ̺ ( ˙ Y ) if and only if πr ⊩ s τ ̺ ′ ( ˙ Z ) ∈ τ ̺ ′ ( ˙ Y ) . Thus, by our recursive assumption, πτ ̺ ( ˙ Y ) D π = { ( πτ ̺ ( ˙ Z ) D π , πr ) ∣ πr ∈ D π , ˙ Z ∈ dom ˙ Y , πr ⊩ s τ ̺ ′ ( ˙ Z ) ∈ τ ̺ ′ ( ˙ Y ) } = { ( τ ̺ ′ ( ˙ Z ) D π , r ) ∣ r ∈ D π , ˙ Z ∈ dom ˙ Y , r ⊩ s τ ̺ ′ ( ˙ Z ) ∈ τ ̺ ′ ( ˙ Y ) } = τ ̺ ′ ( ˙ Y ) D π . Hence, Y = ( πτ ̺ ( ˙ X ) D π ) G = ( τ ̺ ′ ( ˙ X ) D π ) G = ( τ ̺ ′ ( ˙ X )) G == ˙ X ∏ m G ∗ ( a m ) × ∏ m ∈ S G σmim × ∏ m ∈ S G λmlm . Hence, by Lemma 45 above, it follows that ( ˙ X ∏ m G ∗ ( a m ) × ∏ m ∈ S G σmim × ∏ m ∈ S G λmlm , α ) ∈ f. But i m < β for all m ∈ S and l m < β for all m ∈ S ; hence, ( ˙ X ∏ m G ∗ ( a m ) × ∏ m ∈ S G σmim × ∏ m ∈ S G λmlm , α ) ∈ f β . But this contradicts our choice of α ∉ rg f β .Thus, in either case our assumption of a surjective function f ∶ ℘ N ( κ η ) → α η in N has lead to a contradiction, and it follows that indeed, θ N ( κ η ) ≤ α η .Recall that we have assumed throughout our proof that κ η + > κ + η . In the nextChapter 6.3, we will treat the case that κ η + = κ + η , and discuss where the argumentsfrom Chapter 6.2 have to be modified. ∀ η ( κ η + = κ + η Ð→ θ N ( κ η ) ≤ α η ) . If κ η + = κ + η , we need the notion of an η -almost good pair (cf. Definition 26 andProposition 27): For any X ∈ N , X ⊆ κ η , there exists an η -almost good pair (( a m ) m < ω , ( σ m , i m ) m < ω ) such that X ∈ V [ ∏ m G ∗ ( a m ) × ∏ m G σ m i m × G η + ] .Throughout this Chapter 6.3, we assume that κ η + = κ + η . As before in Chapter 6.2, we assume towards a contradiction that there was asurjective function f ∶ ℘ N ( κ η ) → α η in N with πf D π = f D π for all π ∈ A with [ π ] contained in the intersection ⋂ m < ω F ix ( η m , i m ) ∩ ⋂ m < ω H λ m k m ( A ˙ f ) .
75e take ̃ β large enough for ( A ˙ f ) as in Chapter 6.2, Definition 29, and set β ∶ = ̃ β + κ + η (addition of ordinals).Now, we can adapt our definition of f β to η -almost good pairs, and obtain: f β ∶ = { ( X, α ) ∈ f ∣ ∃ (( a m ) m < ω , ( σ m , i m ) m < ω ) η -almost good pair ∶ ( ∀ m i m < β ) ∧∃ ˙ X ∈ Name (( P η ) ω × ∏ m P σ m × P η + ) X = ˙ X ∏ m G ∗ ( a m ) × ∏ m G σmim × G η + } . First, we assume towards a contradiction that f β ∶ dom f β → α η is surjective . A) Constructing ̃ P β ↾ ( η + ) . As before, we only treat the case that β < α ̃ η or Lim ∩ ( η, γ ) ≠ ∅ , where ̃ η ∶ = sup { σ < η ∣ σ ∈ Lim } , i.e. we presume that there exist ( σ, i ) with σ ∈ Lim and i ≥ β or σ > η .This times, we construct a forcing notion ̃ P β ↾ ( η + ) instead of P β ↾ ( η + ) ; whichshould be like P β ↾ ( η + ) , except that firstly, we use restrictions p ∗ ↾ κ η + insteadof p ∗ ↾ κ η , and secondly, we include P η + . Definition 47.
For p ∈ P , let ̃ p β ↾ ( η + ) ∶ = ( p ∗ ↾ κ η + , ( p σi , a σi ) σ ≤ η,i < β , ( p σ ↾ ( β × dom y p σ )) σ ≤ η , p η + , X p ) , and denote by ̃ P β ↾ ( η + ) the collection of all ̃ p β ↾ ( η + ) such that p ∈ P (i.e. p ∈ P with ∣{( σ, i ) ∈ supp p ∣ σ > η ∨ i ≥ β }∣ = ℵ ); together with the maximalelement ̃ βη + . The order relation ̃ ≤ βη + is defined as in Definition 31 . Like in Chapter 6.2 A), one can write down a projection of forcing posets ̃ ρ βη + ∶ P → ̃ P β ↾ ( η + ) and conclude that ̃ G β ↾ ( η + ) ∶ = { p ∈ ̃ P β ↾ ( η + ) ∣ ∃ q ∈ G ∩ P ̃ q β ↾ ( η + ) ̃ ≤ βη + p } is a V -generic filter on ̃ P β ↾ ( η + ) . B) Capturing f β . We define a forcing notion (̃ P β ↾ ( η + )) ( η m ,i m ) m < ω , which will be obtained from ̃ P β ↾ ( η + ) by using ̃ X p instead of X p (cf. Chapter 6.2 B) ), and including for η m ∈ Lim, η m > η the verticals p η m i m ↾ κ η + , and also a η m i m ∩ κ η + , the accordinglinking ordinals up to κ η + .The restriction (̃ p β ↾ ( η + )) ( η m ,i m ) m < ω for p ∈ P is defined as follows: (̃ p β ↾ ( η + )) ( η m ,i m ) m < ω ∶ = ( p ∗ ↾ κ η + , ( p σi , a σi ) σ ≤ η,i < β , ( p η m i m ↾ κ η + , a η m i m ∩ κ η + ) m < ω , η m > η , p σ ↾ ( β × dom y p σ )) σ ≤ η , ̃ X p , p η + ) . Roughly speaking, the difference with the restrictions ( p β ↾ ( η + )) ( η m ,i m ) m < ω intro-duced in Chapter 6.2 B) is, that we are now reaching up to κ η + = κ + η instead of κ η .We denote by (̃ P β ↾ ( η + )) ( η m ,i m ) m < ω the collection of all (̃ p β ↾ ( η + )) ( η m ,i m ) m < ω for p ∈ P together with the maximal element (̃ βη + ) ( η m ,i m ) m < ω . The order relation“ ≤ ” is defined like in Definition 35.Finally, we include the verticals P η m ↾ [ κ η + , κ η m ) for η m > η +
1, which gives theproduct (̃ P β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω P η m ↾ [ κ η + , κ η m ) . Let ( ̃ G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω G η m i m ↾ [ κ η + , κ η m ) denote the collection of all ((̃ p β ↾ ( η + )) ( η m ,i m ) m < ω , ( p η m i m ↾ [ κ η + , κ η m ) m < ω )) such that there exists q ∈ G ∩ P with (̃ q β ↾ ( η + )) ( η m ,i m ) m < ω ≤ (̃ p β ↾ ( η + )) ( η m ,i m ) m < ω and q η m i m ↾ [ κ η + , κ η m ) ⊇ p η m i m ↾ [ κ η + , κ η m ) for all m < ω .As in Proposition 36, one can construct a projection of forcing posets (̃ ρ β ) ( η m ,i m ) m < ω ∶ P → (̃ P β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω P η m ↾ [ κ η + , κ η m ) , and it follows that ( ̃ G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω G η m i m ↾ [ κ η + , κ η m ) is a V -genericfilter on (̃ P β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω P η m ↾ [ κ η + , κ η m ) .Like in Chapter 6.2 B), we want to define a map ( f β ) ′ contained in V [ ̃ G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω G η m i m ↾ [ κ η + , κ η m )] , and then use an isomorphism ar-gument to show that f β = ( f β ) ′ .Before that, we have to modify our transformations of names τ ̺ (where ̺ is an η -good pair), and define transformations ̃ τ ̺ (where ̺ is an η -almost good pair) with ̃ τ ̺ ∶ Name (( P η + ) ω × ∏ m < ω P σ m × P η + ) → Name ( P ) as follows (cf. Definition 34): Definition 48.
For an η -almost good pair ̺ = (( a m ) m < ω , ( σ m , i m ) m < ω ) , definerecursively for ˙ Y ∈ Name (( P η + ) ω × ∏ m < ω P σ m × P η + ) : ̃ τ ̺ ( ˙ Y ) ∶ = { (̃ τ ̺ ( ˙ Z ) , q ) ∣ q ∈ P , ∃ ( ˙ Z , (( p ∗ ( a m )) m < ω , ( p σ m i m ) m < ω , p η + )) ∈ ˙ Y ∶∀ m ( q ∗ ( a m ) ⊇ p ∗ ( a m ) , q σ m i m ⊇ p σ m i m ) , q η + ⊇ p η + } . Y ∏ m < ω G ∗ ( a m ) × ∏ m < ω G σmim × G η + = (̃ τ ̺ ( ˙ Y )) G holds for all ˙ Y ∈ Name (( P η + ) ω × ∏ m < ω P σ m × P η + ) . Definition 49.
Let ( f β ) ′ denote the set of all ( X, α ) for which there exists an η - almost good pair ̺ = (( a m ) m < ω , ( σ m , i m ) m < ω ) with i m < β for all m < ω , suchthat X = ˙ X ∏ m G ∗ ( a m ) × ∏ m G σmim × G η + , and there is a condition p ∈ P with the following properties: • ∣{( σ, i ) ∈ supp p ∣ σ > η or i ≥ β }∣ = ℵ , • p ⊩ s (̃ τ ̺ ( ˙ X ) , α ) ∈ ˙ f , • ( (̃ p β ↾ ( η + )) ( η m ,i m ) m < ω , ( p η m i m ↾ [ κ η + , κ η m )) m < ω ) ∈ ( ̃ G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω G η m i m ↾ [ κ η + , κ η m ) , • ∀ η m ∈ Lim ∶ ( η m , i m ) ∈ supp p with a η m i m = g η m i m . Then ( f β ) ′ ∈ V [( ̃ G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω G η m i m ↾ [ κ η + , κ η m )] . Proposition 50. f β = ( f β ) ′ .Proof. We briefly outline, where the isomorphism argument form Proposition 38has to be modified. We start with ( X, α ) ∈ ( f β ) ′ ∖ f β , X = ˙ X ∏ m G ∗ ( a m ) × ∏ m G σmim × G η + ,for an η -almost good pair ̺ = (( a m ) m < ω , (( σ m , i m ) m < ω )) . Take p as in the defini-tion of ( f β ) ′ with p ⊩ s (̃ τ ̺ ( ˙ X ) , α ) ∈ ˙ f , and p ′ ∈ G with p ′ ⊩ s (̃ τ ̺ ( ˙ X ) , α ) ∉ ˙ f .The first step is the construction of extensions p ≤ p , p ′ ≤ p ′ , such that p and p ′ have “the same shape”, agree on ̃ P β ↾ ( η + ) ; and p η m i m = ( p ′ ) η m i m holds for all m < ω ,and a η m i m = ( a ′ ) η m i m holds for all m < ω with η m ∈ Lim .We proceed as in Proposition 38, with the following modifications: • The construction of p ∗ , p ′∗ that we used in the Proposition 38 for intervals [ κ ν,j , κ ν,j + ) ⊆ κ η , has to be applied to all intervals [ κ ν,j , κ ν,j + ) ⊆ κ η + now,since we need p ∗ and p ′∗ agree on κ η + . • Analogously, the construction of p σi , ( p ′ ) σi for σ ∈ Lim , i < α σ for intervals [ κ ν,j , κ ν,j + ) ⊆ κ η , has to be applied to all intervals [ κ ν,j , κ ν,j + ) ⊆ κ η + now,in the case that σ > η + • Additionally, we have to make sure that p η + = ( p ′ ) η + .The next step is the construction of an isomorphism π such that πp = p ′ , πf D π = f D π , and π ̃ τ ̺ ( ˙ X ) D π = ̃ τ ̺ ( ˙ X ) D π . Again, we take for π a standard isomorphism for πp = p ′ ; but this time, we set G π ( ν, j ) ∶ = F π ( ν, j ) for all intervals [ κ ν,j , κ ν,j + ) ⊆ η + (instead of only intervals [ κ ν,j , κ ν,j + ) ⊆ κ η ), and G π ( ν, j ) = id for all κ ν,j ≥ κ η + (instead of all κ ν,j ≥ κ η ). Then as before, it follows that π ∈ ⋂ m F ix ( η m , i m ) ∩ ⋂ m H λ m k m .For verifying π ̃ τ ̺ ( ˙ X ) D π = ̃ τ ̺ ( ˙ X ) D π , we now additionally have to make sure that π is the identity on P η + . But since we have arranged p η + = ( p ′ ) η + , this is clearby construction of π .Now, it follows from p ⊩ s (̃ τ ̺ ( ˙ X ) , α ) ∈ ˙ f that πp ⊩ s ( π ̃ τ ̺ ( ˙ X ) D π , α ) ∈ πf D π . Hence, p ′ ⊩ s (̃ τ ̺ ( ˙ X ) D π , α ) ∈ f D π , which is a contradiction towards p ′ ⊩ s (̃ τ ̺ ( ˙ X ) , α ) ∉ ˙ f .Thus, f β = ( f β ) ′ ∈ V [( ̃ G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m G η m i m ↾ [ κ η + , κ η m )] as desired. C) (̃ P β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m P η m ↾ [ κ η + , κ η m ) preserves cardinals ≥ α η . Now, we will show that cardinals ≥ α η are absolute between V and V [( ̃ G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω G η m i m ↾ [ κ η + , κ η m )] .As in Chapter 6.2 C), we are using that GCH holds in our ground model V , andwhen we argue that a particular forcing notion preserves cardinals, we mean thatit preserves cardinals under GCH , if not stated differently.
Lemma 51. If ∣ β ∣ + < α η , then (̃ P β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m P η m ↾ [ κ η + , κ η m ) preserves cardinals ≥ α η .Proof. We closely follow the proof of Lemma 39 and Corollary 40.The forcing notion (̃ P β ↾ ( η + )) ( η m ,i m ) m < ω is the set of all ( p ∗ ↾ κ η + , ( p σi , a σi ) σ ≤ η,i < β , ( p η m i m ↾ κ η + , a η m i m ∩ κ η + ) m < ω,η m > η , ( p σ ↾ ( β × dom y p σ )) σ ≤ η , ̃ X p , p η + ) , where p ∈ P with ∣{( σ, i ) ∈ supp p ∣ σ > η ∨ i ≥ β }∣ = ℵ .Since κ η + = κ + η , it follows that the p ∗ ↾ κ η + , as well as ( p η m i m ↾ κ η + , a η m i m ∩ κ η + ) for m < ω are bounded below κ η + ; which gives only ≤ ( κ η + ⋅ κ η ) ω = κ η + = κ + η ≤ ∣ β ∣ -many possibilities.Since ̃ X p ⊆ κ η , there are only ≤ κ + η ≤ ∣ β ∣ -many possibilities for ̃ X p , as well. Regard-ing ( p σi , a σi ) σ ≤ η , i < β and ( p σ ↾ ( dom x p σ × β )) σ ≤ η , it follows as in Lemma 39 thatthere are only ≤ ∣ β ∣ + ⋅ κ + η = ∣ β ∣ + -many possibilities.We denote by ((̃ P β ↾ ( η + )) ( η m ,i m ) m < ω ) ′ the forcing notion that is obtained from (̃ P β ↾ ( η + )) ( η m ,i m ) m < ω by excluding P η + . Then (̃ P β ↾ ( η + )) ( η m ,i m ) m < ω is iso-morphic to the product ((̃ P β ↾ ( η + )) ( η m ,i m ) m < ω ) ′ × P η + .79y what we have just argued, it follows that the forcing notion ((̃ P β ↾ ( η + )) ( η m ,i m ) m < ω ) ′ has cardinality ≤ ∣ β ∣ + ; and the remaining product P η + × ∏ m < ω P η m ↾ [ κ η + , κ η m ) preserves all cardinals by similar arguments as in Proposition 41.Hence, it follows that ((̃ P β ↾ ( η + )) ( η m ,i m ) m < ω ) ′ × P η + × ∏ m < ω P η m ↾ [ κ η + , κ η m ) preserves all cardinals ≥ ∣ β ∣ ++ . Proposition 52.
The forcing (̃ P β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m P η m ↾ [ κ η + , κ η m ) preserves cardinals ≥ α η .Proof. We only have to treat the case that α η = ∣ β ∣ + . Then cf β > ω , and GCH gives ∣ β ∣ ℵ = ∣ β ∣ . The proof is similar as for Proposition 41: We distinguish severalcases, and construct ((̃ P β ↾ ( η + )) ( η m ,i m ) m < ω ) ′′ from (̃ P β ↾ ( η + )) ( η m ,i m ) m < ω bysplitting up P η + , and also one or two factors P σ ↾ ( β × [ κ σ , κ σ )) for σ ∈ Succ with σ = η , or σ = η in the case that η is a successor ordinal with η = η +
1. Thenas in the proof of Proposition 41, it follows that ((̃ P β ↾ ( η + )) ( η m ,i m ) m < ω ) ′′ hascardinality ≤ ∣ β ∣ < α η , and the product of the remaining P σ ↾ ( β × [ κ σ , κ σ )) , P η + and ∏ m P η m ↾ [ κ η + , κ η m ) preserves all cardinals. D) A set ̃ ℘ ( κ η ) ⊇ dom f β with an injection ι ∶ ̃ ℘ ( κ η ) ↪ ∣ β ∣ ℵ . For an η -almost good pair ̺ = (( a m ) m < ω , ( σ m , i m ) m < ω ) , it follows that ∏ m G ∗ ( a m ) × ∏ m G η m i m × G η + is a V -generic filter on ∏ m ( P η ) ω × ∏ m P σ m × P η + , and ( α ) ∏ m G ∗ ( a m ) × ∏ m G σmim × G η + = ( α + ) V holds for all α ≤ κ η by the same proof as for Lemma 2, since P η + ist ≤ κ η -closed.Thus, there is an injection χ ∶ ℘ ( κ η ) ↪ ( κ + η ) V in V [ ∏ m G ∗ ( a m ) × ∏ m G σ m i m × G η + ] .Let ̃ M β denote the set of all η -almost good pairs ̺ = (( a m ) m < ω , ( σ m , i m ) m < ω ) in V with the property that i m < β for all m < ω . Then ̃ M β has cardinality ≤ κ η + ⋅ ∣ η ∣ ℵ ⋅ ∣ β ∣ ℵ = ∣ β ∣ ℵ .Moreover, dom f β is a subset of ̃ ℘ ( κ η ) ∶ = ⋃ { ℘ ( κ η ) ∩ V [ ∏ m G ∗ ( a m ) × ∏ m G σ m i m × G η + ] ∣ (( a m ) m < ω , ( σ m , i m ) m < ω ) ∈ ̃ M β } . Now, we can proceed as in Chapter 6.2 D) and construct in V [( ̃ G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m G η m i m ↾ [ κ η + , κ η m )] an injection ι ∶ ̃ ℘ ( κ η ) ↪ ∣ β ∣ V in the case that α η = (∣ β ∣ + ) V ,and an injection ι ∶ ̃ ℘ ( κ η ) ↪ (∣ β ∣ + ) V in the case that α η > (∣ β ∣ + ) V . Together withChapter 6.3 B) and 6.3 C), this gives the desired contradiction.Thus, we have shown that the map f β ∶ dom f β → α η must not be surjective.80 ) We use an isomorphism argument and obtain a contradiction. The arguments for this part are the very same as in the case that κ η + > κ + η , exceptthat we are now working with η -almost good pairs ̺ = (( a m ) m < ω , ( σ m , i m ) m < ω ) in-stead of η -good pairs.Thus, also in the case that κ η + = κ + η , it follows that θ N ( κ η ) = α η . λ ∈ ( κ η , κ η + ) . So far, we have shown that θ N ( κ η ) = α η holds for all 0 < η < γ . Recall that inthe very beginning (see Chapter 2), we started by “thinning out” our sequence ( κ η ∣ < η < γ ) and assuming w.l.o.g. that ( α η ∣ < η < γ ) is strictly increasing.Thus, it remains make sure that for all cardinals λ ∈ ( κ η , κ η + ) in the “gaps” , θ N ( λ ) gets the smallest possible value, i.e. θ N ( λ ) = max { α η , λ ++ } . This will beour aim for this Chapter 6.4.After that, in Chapter 6.5, we make sure that also for all cardinals λ ≥ κ γ , thevalue θ N ( λ ) will be the smallest possible.We consider a cardinal λ in a “gap” λ ∈ ( κ η , κ η + ) (then κ η + > κ + η ), and set α ( λ ) ∶ = max { λ ++ , α η } . Then θ N ( λ ) ≥ α ( λ ) is clear, and it remains to make surethat there is no surjective function f ∶ ℘ N ( λ ) → α ( λ ) in N .First, we want to describe the intermediate generic extensions where the λ -subsets X ∈ ℘ N ( λ ) are located.Let λ ∈ [ κ η, , κ η, + ) , where < cf κ η + in the case that η + ∈ Lim , and = λ ∈ ( κ η, , κ η, ) = ( κ η , κ η + ) in the case that η + ∈ Succ .We will modify our definition of an η -good pair and obtain the notion of an η -goodpair for λ , which will be used to describe the intermediate generic extensions wherethe sets X ∈ ℘ N ( λ ) are located: Definition 53.
We say that (( a m ) m < ω , ( σ m , i m ) m < ω ) is an η -good pair for λ , ifthe following hold: • ( a m ∣ m < ω ) is a sequence of pairwise disjoint subsets of κ η, , such that forall κ ν,l < κ η, , it follows that ∣ a m ∩ [ κ ν,l , κ ν,l + )∣ = , • for all m < ω , we have σ m ∈ Succ with σ m ≤ η , and i m < α σ m , • if m ≠ m ′ , then ( σ m , i m ) ≠ ( σ m ′ , i m ′ ) . Similarly as in Proposition 25, we obtain:
Proposition 54.
For every X ∈ N , X ⊆ λ , there is an η -good pair for λ , denotedby ̺ = (( a m ) m < ω , ( σ m , i m ) m < ω ) , such that X ∈ V [ ∏ m < ω G ∗ ( a m ) × ∏ m < ω G σ m i m ] . roof. As in Proposition 25, it follows by the
Approximation Lemma
21 that any X ∈ N , X ⊆ λ is contained in a generic extension X ∈ V [ ∏ m < ω G ∗ ( g σ m i m ) × ∏ m < ω G σ m i m ] , where (( σ m , i m ) ∣ m < ω ) and (( σ m , i m ) ∣ m < ω ) are sequences of pairwise distinctpairs with σ m ∈ Lim , i m < α σ m , and σ m ∈ Succ , i m < α σ m for all m < ω .The forcing ∏ m < ω P σ m × ∏ m < ω P σ m can be factored as ( ∏ m < ω P σ m ↾ κ η, × ∏ σ m ≤ η P σ m ) × ( ∏ m < ω P σ m ↾ [ κ η, , κ η m ) × ∏ σ m > η P σ m ) . In the case that λ ∈ ( κ η, , κ η, + ) , it follows that the “lower part” has cardinality ≤ κ + η, ≤ λ , and the “upper part” is ≤ λ -closed.If λ = κ η, , then firstly, the “lower part” has cardinality ≤ κ + η, = λ + , and secondly,it follows that > κ η + ∈ Lim , so κ η, + ≥ κ ++ η, by construction. Hence, the“upper part” is ≤ λ + -closed.In either case, we obtain X ∈ V [ ∏ m < ω G ∗ ( g σ m i m ∩ κ η, ) × ∏ σ m ≤ η G σ m i m ] . With a m ∶ = g σ m i m ∩ κ η, for all m < ω , it follows by the independence property that (( a m ) m < ω , ( σ m , i m ) m < ω ) is an η -good pair for λ with X ∈ V [ ∏ m < ω G ∗ ( a m ) × ∏ σ m ≤ η G σ m i m ] . (Note that ∏ m G ∗ ( a m ) × ∏ σ m ≤ η G σ m i m is a V -generic filter on the forcing ( P η + ↾ κ η, ) ω × ∏ σ m ≤ η P σ m ).As before, we assume towards a contradiction that there was a surjective function f ∶ ℘ N ( λ ) → α ( λ ) in N , where πf D π = f D π holds for all π ∈ A with [ π ] containedin the intersection ⋂ m < ω F ix ( η m , i m ) ∩ ⋂ m < ω H λ m k m ( A ˙ f ) . We take ̃ β large enough for the intersection ( A ˙ f ) as in Chapter 6.2, Definition 29,and set β ∶ = ̃ β + κ + η (addition of ordinals).Let f β ∶ = { ( X, α ) ∈ f ∣ ∃ (( a m ) m < ω , ( σ m , i m ) m < ω ) η -good pair for λ ∶ ( ∀ m i m < β ) ∧∃ ˙ X ∈ Name (( P η + ↾ κ η, ) ω × ∏ σ m ≤ η P σ m ) X = ˙ X ∏ m G ∗ ( a m ) × ∏ m G σmim } . First, we assume towards a contradiction that f β ∶ dom f β → α ( λ ) is surjective .82 ) Constructing ̃̃ P β ↾ ( η + ) We proceed as in Chapter 6.2 A) and 6.3 A), except that we have to use P ∗ ↾ κ η, instead of P ∗ ↾ κ η , and do not include P η + :For p ∈ P , we set ̃̃ p β ↾ ( η + ) ∶ = ( p ∗ ↾ κ η, , ( p σi , a σi ) σ ≤ η,i < β , ( p σ ↾ ( β × dom y p σ )) σ ≤ η , X p ) , and denote by ̃̃ P β ↾ ( η + ) the collection of all ̃̃ p β ↾ ( η + ) , where p ∈ P (i.e. p ∈ P with ∣{( σ, i ) ∈ supp p ∣ σ > η ∨ i ≥ β }∣ = ℵ ); together with the maximal element ̃̃ βη + , and the order relation ̃̃ ≤ βη + defined similarly as in Definition 31.We denote by ̃̃ G β ↾ ( η + ) the set of all p ∈ ̃̃ P β ↾ ( η + ) such that there exists q ∈ G ∩ P with ̃̃ q β ↾ ( η + ) ̃̃ ≤ βη + p . Then as in Chapter 6.2 A), Proposition 33, itfollows that ̃̃ G β ↾ ( η + ) is a V -generic filter on ̃̃ P β ↾ ( η + ) . B) Capturing f β For p ∈ P , the restriction (̃̃ p β ↾ ( η + )) ( η m ,i m ) m < ω is defined as follows: (̃̃ p β ↾ ( η + )) ( η m ,i m ) m < ω ∶ = ( p ∗ ↾ κ η, , ( p σi , a σi ) σ ≤ η,i < β , ( p η m i m ↾ κ η, , a η m i m ∩ κ η, ) m < ω , η m > η , ( p σ ↾ ( β × dom y p σ )) σ ≤ η , ̃ X p ) . We define (̃̃ P β ↾ ( η + )) ( η m ,i m ) m < ω and ( ̃̃ G β ↾ ( η + )) ( η m ,i m ) m < ω as in Chapter 6.2B) and 6.3 B). Then ( ̃̃ G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω G η m i m ↾ [ κ η, , κ η m ) is a V -generic filter on (̃̃ P β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω P η m ↾ [ κ η, , κ η m ) . The construction of ( f β ) ′ as well as the proof of f β = ( f β ) ′ are as in Chapter 6.2B) and 6.3 B); except that this time, the isomorphism π from the proof of Propo-sition 38 has to be the identity on P ∗ ↾ κ η, (not only on P ∗ ↾ κ η ). This can beachieved by the following modifications: Firstly, we demand that p ∗ and p ′∗ cohereon P ∗ ↾ κ η, (not only P ∗ ↾ κ η ); secondly, we arrange p ∗ ↾ κ η, = p ′∗ ↾ κ η, (instead ofjust p ∗ ↾ κ η = p ′∗ ↾ κ η ); and thirdly, when constructing the isomorphism π , we set G π ( ν, j ) ∶ = F π ( ν, j ) for all κ ν,j < κ η, now, and G π ( ν, j ) = id whenever κ ν,j ≥ κ η, .It follows that f β = ( f β ) ′ ∈ V [( ̃̃ G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m G η m i m ↾ [ κ η, , κ η m )] . C) (̃̃ P β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m < ω P η m ↾ [ κ η, , κ η m ) preserves cardinals ≥ α ( λ ) = max { λ ++ , α η } . The arguments here are similar as in Chapter 6.2 C) and 6.3 C), since there areonly ≤ ( κ η, ) ℵ = κ + η, ≤ λ + < α ( λ ) -many possibilities for p ∗ ↾ κ η, and ( p η m i m ↾ κ η, , a η m i m ∩ κ η, ) m < ω . 83 ) A set ̃ ℘ ( λ ) ⊇ dom f β with an injection ι ∶ ̃ ℘ ( λ ) ↪ λ + ⋅ ∣ β ∣ ℵ . We proceed as in Chapter 6.2 D) and 6.3 D). Whenever (( a m ) m < ω , ( σ m , i m ) m < ω ) is an η -good pair for λ , it follows that ∏ m G ∗ ( a m ) × ∏ m G σ m i m is a V -generic filteron ( P η + ↾ κ η, ) ω × ∏ m P σ m ; and ( α ) V [∏ m G ∗ ( a m ) × ∏ m G σmim ] = ( α + ) V holds for all cardinals α by the same proof as in Lemma 2. Hence, it follows thatin V [ ∏ m G ∗ ( a m ) × ∏ m G σ m i m ] , there is an injection χ ∶ ℘ ( λ ) ↪ ( λ + ) V .Let ̃̃ M β be the set of all ̺ = (( a m ) m < ω , ( σ m , i m ) m < ω ) in V such that ̺ is an η -goodpair for λ with the property that i m < β for all m < ω . Then ̃̃ M β has cardinality ≤ ( κ + η, ) ℵ ⋅ ∣ β ∣ ℵ ≤ λ + ⋅ ∣ β ∣ ℵ .By construction, it follows that dom f β is a subset of ̃ ℘ ( λ ) ∶ = ⋃ { ℘ ( λ ) ∩ V [ ∏ m G ∗ ( a m ) × ∏ m G σ m i m ] ∣ (( a m ) m < ω , ( σ m , i m ) m < ω ) ∈ ̃̃ M β } . As in Chapter 6.2 D), we can now work in V [( ̃̃ G β ↾ ( η + )) ( η m ,i m ) m < ω × ∏ m G η m i m ↾ [ κ η, , κ η m )] and construct there an injection ι ∶ ̃ ℘ ( λ ) ↪ ( λ + ) V ⋅ ∣ β ∣ V in the casethat α η = (∣ β ∣ + ) V , and an injection ι ∶ ̃ ℘ ( λ ) ↪ ( λ + ) V ⋅ (∣ β ∣ + ) V in the case that α η > (∣ β ∣ + ) V .Together with Chapter 6.4 B) and 6.4 C), this gives the desired contradiction.Hence, it follows that there must be α < α ( λ ) with α ∉ rg f β . E) We use an isomorphism argument and obtain a contradiction.
The arguments for this part are the same as in Chapter 6.2 E); except that we areworking with η -good pairs for λ now (instead of η -good pairs).Thus, we have shown that for all cardinals λ ∈ ( κ η , κ η + ) in a “gap” , the value θ N ( λ ) is the smallest possible: θ N ( λ ) = α ( λ ) = max { λ ++ , α η } .It remains to consider the cardinals λ ≥ κ γ ∶ = sup { κ η ∣ < η < γ } . We prove thatfor all λ ≥ κ γ , again, θ N ( λ ) takes the smallest possible value.This will be the aim of the next Chapter 6.5. λ ≥ κ γ ∶= sup { κ η ∣ < η < γ } . Let α γ ∶ = sup { α η ∣ < η < γ } , and consider a cardinal λ ≥ κ γ . We want to showthat θ N ( λ ) takes the smallest possible value α ( λ ) , defined as follows: • In the case that cf α γ = ω , we set α ( λ ) = max { α ++ γ , λ ++ } .84 In the case that α γ = α + for some α with cf α = ω , we set α ( λ ) = max { α + γ , λ ++ } . • In other cases, we set α ( λ ) ∶ = max { α γ , λ ++ } .Then by our remarks from Chapter 2, it follows that indeed, θ N ( λ ) ≥ α ( λ ) holdsfor all λ ≥ κ γ .First, we assume that α ( λ ) > α γ . It remains to prove that there is no surjective function f ∶ ℘ N ( λ ) → α ( λ ) in N .We start with the following observation (again, we use that V ⊧ GCH ): Lemma 55.
Let λ ≥ κ γ with α ( λ ) > α γ . Then P preserves cardinals ≥ α ( λ ) .Proof. For every p ∈ P , p = ( p ∗ , ( p σi , a σi ) σ < γ,i < α σ , ( p σ ) σ < γ ) , there are • ≤ κ + γ -many possibilities for p ∗ , • ≤ α ℵ γ -many possibilities for the countable support of ( p σi , a σi ) σ < γ,i < α σ , • ≤ κ + γ -many possibilities for ( p σi , a σi ) σ < γ,i < α σ when the support is given.In the case that γ is a limit ordinal , it follows by the strict monotonicity ofthe sequence ( α σ ∣ < σ < γ ) that α σ < α γ holds for all 0 < σ < γ . Hence, forany σ ∈ Succ , the forcing notion P σ has cardinality ≤ α + σ ≤ α γ ; and it follows bycountable support that we have ≤ ∣ γ ∣ ℵ ⋅ α ℵ γ = α ℵ γ -many possibilities for ( p σ ) σ < γ .Hence, the forcing P has cardinality ≤ κ + γ ⋅ α ℵ γ ≤ λ + ⋅ α ℵ γ . If cf α γ > ω , GCH gives ∣ P ∣ ≤ λ + ⋅ α γ , and α ( λ ) = max { λ ++ , α + γ } . Hence, P preserves cardinals ≥ α ( λ ) asdesired. If cf α γ = ω , then α ( λ ) = max { λ ++ , α ++ γ } ≥ ∣ P ∣ + ; and again, it follows that P preserves cardinals ≥ α ( λ ) .It remains to consider the case that γ = γ + . Then oursequences ( κ σ ∣ < σ < γ ) = ( κ σ ∣ < σ ≤ γ ) and ( α σ ∣ < σ < γ ) = ( α σ ∣ < σ ≤ γ ) have a maximal element, and κ γ = κ γ , α γ = α γ .If γ ∈ Lim , i.e. κ γ is a limit cardinal, it follows that for any σ ∈ Succ , we have σ < γ ; hence, α + σ ≤ α γ = α γ . This gives ∣ P ∣ ≤ κ + γ ⋅ α ℵ γ ≤ λ + ⋅ α ℵ γ as before, and α ( λ ) ≥ ∣ P ∣ + .If γ ∈ Succ , i.e. κ γ is a successor cardinal, then P γ has to be treated separately.We factor P ≅ P ′ × P γ with P ′ ∶ = { ( p ∗ , ( p σi , a σi ) σ ≤ γ , i < α σ , ( p σ ) σ < γ ) ∣ p ∈ P } . Then P γ preserves cardinals, and P ′ has cardinality ≤ ( λ + ) V ⋅ ( α ℵ γ ) V as before (in V , andhence, also in any P γ -generic extension); where α ( λ ) ≥ ∣ P ′ ∣ + . Hence, the forcing P ≅ P ′ × P γ preserves cardinals ≥ α ( λ ) as desired.85ow, we assume towards a contradiction that there was a surjective function f ∶ ℘ N ( λ ) → α ( λ ) in N .By the Approximation Lemma 21 , it follows that any X ∈ N , X ⊆ λ , is contained inan intermediate generic extension V [ ∏ m < ω G σ m i m ] , with a sequence (( σ m , i m ) ∣ m < ω ) of pairwise distinct pairs in V such that 0 < σ m < γ , i m < α σ m for all m < ω .Denote by M the collection of these (( σ m , i m ) ∣ m < ω ) . Then ∣ M ∣ ≤ α ℵ γ in V ; and α ℵ γ < α ( λ ) as argued before.The product ∏ m P σ m preserves cardinals and the GCH . Hence, it follows thatin any generic extension V [ ∏ m G σ m i m ] , there is an injection χ ∶ ℘ ( λ ) ↪ ( λ + ) V .Now, one can argue as in Chapter 6.2 D), and define in V [ G ] a set ̃ ℘ ( λ ) ⊇ ℘ N ( λ ) with an injection ι ∶ ̃ ℘ ( λ ) ↪ ( λ + ) V ⋅ α γ , or ι ∶ ̃ ℘ ( λ ) ↪ ( λ + ) V ⋅ ( α + γ ) V in the casethat α ( λ ) ≥ ( α ++ γ ) V . Together with Lemma 55, this gives the desired contradiction.Thus, we have shown that in the case that α ( λ ) > α γ , there can not be a surjectivefunction f ∶ ℘ ( λ ) → α ( λ ) in N .It remains to consider the case that α ( λ ) = α γ . Then λ + < α γ , cf α γ > ω ; and if α γ = α + for some α , then cf α > ω .Assume towards a contradiction that there was a surjective function f ∶ ℘ N ( λ ) → α ( λ ) in N , f = ˙ f G with πf D π = f D π for all π ∈ A with [ π ] contained in theintersection ⋂ m < ω F ix ( η m , i m ) ∩ ⋂ m < ω H λ m k m ( A ˙ f ) . Similary as before, we take ̃ β < α ( λ ) large enough for the intersection ( A ˙ f ) , i.e. ̃ β > λ + with ̃ β > sup { i m ∣ m < ω } ∪ sup { k m ∣ m < ω } (this is possible, since cf α ( λ ) > ω ) . Let β ∶ = ̃ β + κ + γ (addition of ordinals). Then κ + γ ≤ λ + < α ( λ ) gives λ + < β < α ( λ ) .By the Approximation Lemma
21, it follows as in Proposition 25 that any X ∈ N , X ⊆ λ , is contained in an intermediate generic extension V [ ∏ m G ∗ ( a m ) × ∏ m G σ m i m ] ,where (( a m ) m < ω , ( σ m , i m ) m < ω ) is a good pair for κ γ , i.e. • ( a m ∣ m < ω ) is a sequence of pairwise disjoint subsets of κ γ , such that forall κ ν, < κ γ and m < ω , it follows that ∣ a m ∩ [ κ ν, , κ ν, + )∣ = • for all m < ω , we have σ m ∈ Succ , 0 < σ < γ , and i m < α σ m , • if m ≠ m ′ , then ( σ m , i m ) ≠ ( σ m ′ , i m ′ ) .As before, let f β ∶ = { ( X, α ) ∈ f ∣ ∃ (( a m ) m < ω , ( σ m , i m ) m < ω ) good pair for κ γ ∶ ( ∀ m i m < β ) ∧ ∃ ˙ X ∈ Name (( P γ ) ω × ∏ m P σ m ) X = ˙ X ∏ m G ∗ ( a m ) × ∏ m G σmim } . First, we assume towards a contradiction that f β ∶ dom f β → α ( λ ) is surjective . A) + B) Constructing P β and capturing f β . For a condition p ∈ P , let p β ∶ = ( p ∗ , ( p σi , a σi ) σ ∈ Lim , i < β , ( p σ ↾ ( β × dom x p σ )) σ ∈ Succ , X p ) , where X p ∶ = ⋃ { a σi ∣ σ ∈ Lim , i ≥ β } . We define P β and G β as before.The construction of ( f β ) ′ ∈ V [ G β ] and the isomorphism argument for f β = ( f β ) ′ are as in Chapter 6.2 and 6.3; except that when constructing the isomorphism π ,we now have to set G π ( ν, j ) ∶ = F π ( ν, j ) for all κ ν,j < κ γ . C) P β preserves cardinals ≥ α ( λ ) = α γ = sup { α η ∣ < η < γ } . The arguments here are similar as in Chapter 6.2 C): If α γ > ∣ β ∣ + , it follows asin Lemma 39 that ∣ P β ∣ ≤ κ + γ ⋅ ∣ β ∣ ℵ ≤ λ + ⋅ ∣ β ∣ + < α γ . In the case that α γ = ∣ β ∣ + , itfollows that cf ∣ β ∣ > ω , and as before, we distinguish several cases, whether γ isa limit ordinal or γ = γ +
1, and in the latter case, whether γ ∈ Lim , or γ ∈ Succ with γ = γ + P γ (or P γ , or both), and obtain that P γ (or P γ ,or the product P γ × P γ ) preserves cardinals, while the remaining forcing is nowsufficiently small. D) A set ̃ ℘ ( λ ) ⊇ dom f β with an injection ι ∶ ̃ ℘ ( λ ) ↪ λ + ⋅ ∣ β ∣ ℵ . As in Chapter 6.2 D) and 6.4 D), we construct in V [ G β ] a set ̃ ℘ ( λ ) ⊇ dom f β withan injection ι ∶ ̃ ℘ ( λ ) ↪ ( λ + ) V ⋅ (∣ β ∣ + ) V in the case that (∣ β ∣ + ) V < α ( λ ) , and aninjection ι ∶ ̃ ℘ ( λ ) ↪ ( λ + ) V ⋅ ∣ β ∣ V in the case that (∣ β ∣ + ) V = α ( λ ) .Together with Chapter 6.5 B) and 6.5 C), this gives the desired contradiction.Hence, it follows that there must be some α < α ( λ ) with α ∉ rg f β . E) We use an isomorphism argument and obtain a contradiction.
With the same isomorphism argument as in Chapter 6.2 E), it follows that θ N ( λ ) = α ( λ ) as desired.Thus, we have shown that also for all cardinals λ ≥ κ γ , θ N ( λ ) takes the smallestpossible value.This was the last step in the proof of our main theorem.87 Discussion and Remarks
Our result confirms Shelah’s thesis from [She10, p.2] that in ZF + DC + AX itis “better” to look at ( [ κ ] ℵ ∣ κ a cardinal ) rather than ( ℘ ( κ ) ∣ κ a cardinal ) , inthe sense that by what we have shown, the only restrictions that can be imposedon the θ -function on a set of cardinals in ZF + DC + AX , are the obvious ones.From Theorem 1 in [AK10], it follows that increasing the surjective size of [ ℵ ω ] ℵ together with preserving GCH below ℵ ω requires a measurable cardinal, whichagain underlines how differently ℘ ( ℵ ω ) and [ ℵ ω ] ℵ behave without the Axiom ofChoice . In further investigation, one might look at the cardinal arithmetic in ourconstructed model, such as possible (surjective) sizes of ( κ λ ∣ κ a cardinal ) for λ << κ .Another question one might ask is, under what circumstances certain ¬ AC -largecardinal properties are preserved in our symmetric extension N . As an example,we will now briefly look at the question whether an inaccessible cardinal κ in theground model could remain inaccessible in N .The notion of inaccessibility in ZF C reads as follows:
A cardinal κ is inaccessible (or strongly inaccessible ) if κ is regular and λ < κ holds for all cardinals λ < κ . Hence, it can not be transferred directly to the ¬ AC -context, since the power sets ℘ ( λ ) for λ < κ are ususally not well-ordered. In [BDL07, Chapter 2], one can findseveral characterizations how inaccessibility can be defined in ZF : Definition ([BDL07]) . • A regular uncountable cardinal κ is i -inaccessible if for all λ < κ , there is an ordinal α < κ with an injection ι ∶ ℘ ( λ ) ↪ α . • A regular uncountable cardinal κ is v -inaccessible if for all λ < κ , there isno surjection s ∶ V λ → κ . • A regular uncountable cardinal κ is s -inaccessible if for all λ < κ , there isno surjection s ∶ ℘ ( λ ) → κ . Note that i -inaccessibility implies v -inaccessibility, and v -inaccessibility implies s -inaccessibility. It is not difficult to see that a cardinal κ is v -inaccessible if andonly if V κ is a model of second-order ZF (see [BDL07, Chapter 2]).Let now κ be an inaccessible cardinal in the setting of our theorem: V ⊧ ZF C + GCH with sequences ( κ η ∣ < η < γ ) , ( α η ∣ < η < γ ) as before, with the additionalproperty that for all κ η < κ , it follows that also α η < κ . Then by construction,it follows that κ is s -inaccessible in N , while i -inaccessibility of κ is out of reach,since we do not have our power set well-ordered.The question remains whether κ is v -inaccessible in N . By induction over λ , wecould prove (using several isomorphism and factoring arguments, similar to those88n Chapter 6.2): Proposition.
Let V be a ground model of ZF C + GCH with γ ∈ Ord , and se-quences of uncountable cardinals ( κ η ∣ < η < γ ) and ( α η ∣ < η < γ ) with theproperties listed in Chapter . Moreover, let N ⊇ V denote the symmetric extensionconstructed in Chapter , and .If κ is an inaccessible cardinal in V with the property that for all κ η < κ it followsthat α η < κ , then κ is v -inaccessible in N : For any λ < κ , there is no surjectivefunction s ∶ V Nλ → κ in N . In our inductive proof, we show that for any cardinal λ < κ , there exists some κ ν, ( λ ) < κ and a cardinal β λ < κ with an injection ι ∶ V Nλ ↪ β λ in V [ G ↾ κ ν, ( λ )] .Our next remark is about the following requirement that we put on the sequences ( κ η ∣ < η < γ ) , ( α η ∣ < η < γ ) : ∀ η ( α η = α + → cf α > ω ) . We mentioned in Chapter 2 that this condition is necessary if we want ∀ η θ N ( κ η ) = α η under AX .Moreover, we proved in Chapter 2 that whenever we start from a ground model V ⊧ ZF C + GCH , and construct a symmetric extension N ⊇ V with N ⊧ ZF + DC such that V and N have the same cardinals and cofinalities, the following holds: If κ , α ∈ Card with θ N ( κ ) = α + , then cf N ( α ) > ω . One could ask what happens if we drop the requirement that N should extend aground model V ⊧ ZF C + GCH cardinal-preservingly.Can there be any inner model N ⊧ ZF + DC with cardinals κ , α such that θ N ( κ ) = α + and cf N ( α ) = ω ?Let s ∶ κ → α denote a surjective function in N . Then with DC , it follows thatthere is also a surjection s ∶ ( κ ) ω → α ω in N ; and we also have a surjectivefunction s ∶ κ → ( κ ) ω .Recall that in Chapter 2, we then took a surjection ̃ s ∶ ( α ω ) V → ( α + ) V fromour ground model V , which gave a surjection s ∶ ( α ω ) N → ( α + ) N in N . Then s ○ s ○ s ∶ κ → α + was a surjective function in N ; hence, θ N ( κ ) ≥ α ++ .In a more general setting, where we can not refer to a ground model V , we try touse the constructible universe L = L N instead. Under the assumption that 0 ♯ doesnot exist, it follows by Jensen’s Covering Theorem ([DJ75]) that L does not differdrastically from N : In particular, L and N have the same successors of singularcardinals; so if cf N ( α ) = ω , then ( α + ) L = ( α + ) N .This gives the following lemma: 89 emma. Let N be an inner model of ZF + DC with N ⊧ “ ♯ does not exist”, and α ∈ Card N with cf N ( α ) = ω . Then there exists a surjective function s ∶ ( α ω ) N → ( α + ) N in N .Proof. Let ( α i ∣ i < ω ) denote a strictly increasing sequence in N which is cofinalin α . First, we construct in N an injection ι ∶ ( α ) L ↪ ( α ω ) N , ι = ι ○ ι ○ ι , asfollows: • Let ι ∶ ( α ) L → ∏ i < ω ( α i ) L denote the injection that maps any g ∶ α → g ∈ L , to the sequence of its restrictions (( g ↾ α i ) ∣ i < ω ) . • For any i < ω , there is in L an injection γ ∶ ( α i ) L ↪ ( α + i ) L ; so with DC in N , we can choose a sequence of injective maps ( γ i ∣ i < ω ) such that γ i ∶ ( α i ) L ↪ ( α + i ) L for all i < ω . Then we define in N an injection ι ∶ ∏ i < ω ( α i ) L → ∏ i < ω ( α + i ) L by setting ι ( X i ∣ i < ω ) ∶ = ( γ i ( X i ) ∣ i < ω ) . • Finally, since ( α + i ) L ≤ ( α + i ) N < α for all i < ω , it follows that there is in N an injective map ι ∶ ∏ i < ω ( α + i ) L ↪ ( α ω ) N .Thus, ι ∶ = ι ○ ι ○ ι ∶ ( α ) L ↪ ( α ω ) N is an injection in N ; which yields a surjection s ∶ ( α ω ) N → ( α ) L , or s ∶ ( α ω ) N → ( α + ) L .Since we have assumed that N ⊧ “0 ♯ does not exist” and cf N ( α ) = ω , it follows by Jensen’s Covering Lemma in N that ( α + ) L = ( α + ) N .This gives our surjecion s ∶ ( α ω ) N → ( α + ) N in N as desired. Corollary.
Let N be an inner model of ZF + DC with N ⊧ “ ♯ does not exist”,and cardinals κ , α such that θ N ( κ ) = α + . Then cf N ( α ) > ω .Proof. Let s ∶ κ → α denote a surjective function in N , and assume towards acontradiction that cf N ( α ) = ω . As mentioned before, we have surjections s ∶ κ → ( κ ) ω and s ∶ ( κ ) ω → α ω . By the previous lemma, it follows that there is also asurjection s ∶ α ω → α + in N . Setting s ∶ = s ○ s ○ s , we obtain in N a surjectivefunction s ∶ κ → α + . Contradiction.Thus, without large cardinal assumptions, it is not possible to achieve θ N ( κ ) = α + for cardinals κ , α with cf N ( α ) = ω .Finally, we remark that our theorem gives a result about possible behaviors ofthe θ -function on a set of uncountable cardinals. Unfortunately, a straightforwardgeneralization of our forcing notion to ordinal-length sequences ( κ η ∣ η ∈ Ord ) , ( α η ∣ η ∈ Ord ) does not result in a ZF -model:Denote by P the class forcing which canonically generalizes our forcing notion P to sequences ( κ η ∣ η ∈ Ord ) , ( α η ∣ η ∈ Ord ) of ordinal length; denote by G a90 -generic filter on P , and N ∶ = V ( G ) . Then N ⊭ P ower Set : Assume towardsa contradiction that Z ∶ = ℘ N ( ℵ ) ∈ N . Then there would be an ordinal γ and asymmetric name ˙ Z ∈ HS ∩ Name ( P ↾ γ ) with Z = ˙ Z G ↾ γ , where P ↾ γ denotes theinitial part of P up to κ γ . Now, by an isomorphism argument similar as in the Approximation Lemma
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Anne Fernengel, Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universit¨at,Bonn, Germany
E-Mail address: [email protected]
Peter Koepke, Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universit¨at,Bonn, Germany
E-Mail address: [email protected]@math.uni-bonn.de