Inner Models from Extended Logics: Part 1
aa r X i v : . [ m a t h . L O ] J u l Inner Models from Extended Logics: Part 1 ∗ Juliette Kennedy † Helsinki Menachem Magidor ‡ JerusalemJouko V¨a¨an¨anen § Helsinki and AmsterdamJuly 22, 2020
Abstract
If we replace first order logic by second order logic in the original def-inition of G ¨odel’s inner model L , we obtain HOD ([32]). In this paper weconsider inner models that arise if we replace first order logic by a logic thathas some, but not all, of the strength of second order logic. Typical examplesare the extensions of first order logic by generalized quantifiers, such as theMagidor-Malitz quantifier ([23]), the cofinality quantifier ([34]), or station-ary logic ([6]). Our first set of results show that both L and HOD manifestsome amount of formalism freeness in the sense that they are not very sen-sitive to the choice of the underlying logic. Our second set of results showsthat the cofinality quantifier gives rise to a new robust inner model between L and HOD. We show, among other things, that assuming a proper class ofWoodin cardinals the regular cardinals > ℵ of V are weakly compact inthe inner model arising from the cofinality quantifier and the theory of thatmodel is (set) forcing absolute and independent of the cofinality in question. ∗ The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for itshospitality during the programme Mathematical, Foundational and Computational Aspects of theHigher Infinite supported by EPSRC Grant Number EP/K032208/1. The authors are grateful toJohn Steel, Philip Welch and Hugh Woodin for comments on the results presented here. † Research partially supported by grant 322488 of the Academy of Finland. ‡ Research supported by the Simons Foundation and the Israel Science Foundation grant817/11. § Research supported by the Simons Foundation and grant 322795 of the Academy of Finland. e do not know whether this model satisfies the Continuum Hypothesis, as-suming large cardinals, but we can show, assuming three Woodin cardinalsand a measurable above them, that if the construction is relativized to a real,then on a cone of reals the Continuum Hypothesis is true in the relativizedmodel. Inner models, together with the forcing method, are the basic building blocks usedby set theorists to prove relative consistency results on the one hand and to try tochart the “true” universe of set theory V on the other hand.The first and best known, also the smallest of the inner models is G¨odel’s L ,the universe of constructible sets. An important landmark among the largest innermodels is the universe of hereditarily ordinal definable sets HOD, also introducedby G¨odel . In between these two extremes there is a variety of inner models arisingfrom enhancing G¨odel’s L by normal ultrafilters on measurable cardinals, or in amore general case extenders, something that L certainly does not have itself.We propose a construction of inner models which arise not from adding normalultrafilters, or extenders, to L , but by changing the underlying construction of L .We show that the new inner models have similar forcing absoluteness propertiesas L ( R ) , but at the same time they satisfy the Axiom of Choice.G¨odel’s hierarchy of constructible sets is defined by reference to first order G¨odel introduced HOD in his 1946
Remarks before the Princeton Bicenntenial conference onproblems in mathematics [12]. The lecture was given during a session on computability organizedby Alfred Tarski, and in it G¨odel asks whether notions of definability and provability can beisolated in the set-theoretic formalism, which admit a form of robustness similar to that exhibitedby the notion of general recursiveness: “Tarski has stressed in his lecture the great importance(and I think justly) of the concept of general recursiveness (or Turing computability). It seemsto me that this importance is largely due to the fact that with this concept one has succeeded ingiving an absolute definition of an interesting epistemological notion, i.e. one not depending onthe formalism chosen. In all other cases treated previously, such as definability or demonstrability,one has been able to define them only relative to a given language, and for each individual languageit is not clear that the one thus obtained is not the one looked for. For the concept of computabilityhowever. . . the situation is different. . . This, I think, should encourage one to expect the same thingto be possible also in other cases (such as demonstrability or definability).”G¨odel contemplates the idea that constructibility might be a suitable analog of the notion ofgeneral recursiveness. G¨odel also considers the same for HOD, and predicts the consistency of theaxiom V = HOD + 2 ℵ > ℵ (proved later by McAloon [28]). See [14] for a development ofG¨odel’s proposal in a “formalism free” direction. L enjoys strong forcing absoluteness: truth in L cannot be changed by forcing, in fact not by any method of extending the universewithout adding new ordinals. Accordingly, it is usually possible to settle in L , oneway or other, any set theoretical question which is otherwise independent of ZFC.However, the problem with L is that it cannot have large cardinals on the levelof the Erd˝os cardinal κ ( ω ) or higher. To remedy this, a variety of inner models,most notably the smallest inner model L µ with a measurable cardinal, have beenintroduced (see e.g. [37]).We investigate the question to what extent is it essential that first order defin-ability is used in the construction of G¨odel’s L . In particular, what would be theeffect of changing first order logic to a stronger logic? In fact there are two prece-dents: Scott and Myhill [32] showed that if first order definability is replaced bysecond order definability the all-encompassing class HOD of hereditarily ordinaldefinable sets is obtained. The inner model L is thus certainly sensitive to thedefinability concept used in its construction. The inner model HOD has consis-tently even supercompact cardinals [29]. However, HOD does not solve any ofthe central independent statements of set theory; in particular, it does not solvethe Continuum Hypothesis or the Souslin Hypothesis [28].A second precedent is provided by Chang [7] in which first order definabilitywas replaced by definability in the infinitary language L ω ω , obtaining what cameto be known as the Chang model . Kunen [17] showed that the Chang model failsto satisfy the Axiom of Choice, if the existence of uncountably many measurablecardinals is assumed. We remark that the inner model L ( R ) arises in the sameway if L ω ω is used instead of L ω ω . Either way, the resulting inner model failsto satisfy the Axiom of Choice if enough large cardinals are assumed. This putsthese inner models in a different category. On the other hand, the importanceof both the Chang model and L ( R ) is accentuated by the result of Woodin [41]that under large cardinal assumptions the first order theory of the Chang model, aswell as of L ( R ) , is absolute under set forcing. So there would be reasons to expectthat these inner models would solve several independent statements of set theory,e.g. the CH. However, the failure of the Axiom of Choice in these inner modelsdims the light such “solutions” would shed on CH. For example, assuming largecardinals, the model L ( R ) satisfies the statement “Every uncountable set of realscontains a perfect subset”, which under AC would be equivalent to CH . On theother hand, large cardinals imply that there is in L ( R ) a surjection from R onto ω , which under AC would imply ¬ CH .In this paper we define analogs of the constructible hierarchy by replacing first3rder logic in G¨odel’s construction by any one of a number of logics. The innermodels HOD, L ( R ) and the Chang model are special cases, obtained by replacingfirst order definability by definability in L , L ω ω and L ω ω , respectively. Ourmain focus is on extensions of first order logic by generalized quantifiers in thesense of Mostowski [30] and Lindstr¨om [21]. We obtain new inner models whichare L -like in that they are models of ZFC and their theory is absolute under setforcing, but at the same time these inner models contain large cardinals, or innermodels with large cardinals.The resulting inner models enable us to make distinctions in set theory thatwere previously unknown. However, we also think of the arising inner models as atool to learn more about extended logics. As it turns out, for many non-equivalentlogics the inner model is the same. In particular for many non-elementary logicsthe inner model is the same as for first order logic. We may think that such logicshave some albeit distant similarity to first order logic. On the other hand, someother logics give rise to the inner model HOD. We may say that they bear someresemblance to second order logic.Our main results can be summarized as follows: (A) For the logics L ( Q α ) we obtain just L , for any choice of α . If exists thesame is true of the Magidor-Malitz logics L ( Q MM α ) . (B) If ♯ exists the cofinality quantifier logic L ( Q cf ω ) yields a proper extension C ∗ of L . But C ∗ = HOD if there are uncountably many measurable cardinals. (C)
If there is a proper class of Woodin cardinals, then regular cardinals > ℵ areMahlo and indiscernible in C ∗ , and the theory of C ∗ is invariant under (set)forcing. (D) The Dodd-Jensen Core Model is contained in C ∗ . If there is an inner modelwith a measurable cardinal, then such an inner model is also contained in C ∗ . (E) If there is a Woodin cardinal and a measurable cardinal above it, then CH istrue in the version C ∗ ( x ) of C ∗ , obtained by allowing a real parameter x ,for a cone of reals x . 4 Basic concepts
We define an analogue of the constructible hierarchy of G¨odel by replacing firstorder logic in the construction by an arbitrary logic L ∗ . We think of logics inthe sense of Lindstr¨om [22], Mostowski [31], Barwise [4], and the collection [2].What is essential is that a logic L ∗ has two components i.e. L ∗ = ( S ∗ , T ∗ ) , where S ∗ is the class of sentences of L ∗ and T ∗ is the truth predicate of L ∗ . We usuallywrite ϕ ∈ L ∗ for ϕ ∈ S ∗ and M | = ϕ for T ∗ ( M , ϕ ) . We can talk about formulas with free variables by introducing new constant symbols and letting the constantsymbols play the role of free variables. The classes S ∗ and T ∗ may be defined withparameters, as in the case of L κλ , where κ and λ can be treated as parameters. Alogic L ∗ is a sublogic of another logic L + , L ∗ ≤ L + , if for every ϕ ∈ S ∗ there is ϕ + ∈ S + such that for all M : M | = ϕ ⇐⇒ M | = ϕ + . We assume that ourlogics have first order order logic as sublogic. Example 2.1. First order logic L ωω (or FO) is the logic ( S ∗ , T ∗ ) , where S ∗ is the set of first order sentences and T ∗ is the usual truth definition for firstorder sentences.2. Infinitary logic L κλ , where κ and λ ≤ κ are regular cardinals, is the logic ( S ∗ , T ∗ ) , where S ∗ consists of the sentences built inductively from conjunc-tions and disjunctions of length < κ of sentences of L κλ , and homogeneousstrings of existential and universal quantifiers of length < λ in front of for-mulas of L κλ . The class T ∗ is defined in the obvious way. We allow alsothe case that κ or λ is ∞ . We use L ωκλ to denote that class of formulae of L κλ with only finitely many free variables.3. The logic L ( Q ) with a generalized quantifier Q is the logic ( S ∗ , T ∗ ) , where S ∗ is obtained by adding the new quantifier Q to first order logic. The exactsyntax depends on the type of Q , (see our examples below). The class T ∗ isdefined by first fixing the defining model class K Q of Q and then defining T ∗ by induction on formulas: M | = Qx , . . . , x n ϕ ( x , . . . , x n ,~b ) ⇐⇒ ( M, { ( a , . . . , a n ) ∈ M n : M | = ϕ ( a , . . . , a n ,~b ) } ) ∈ K Q . Thought of in this way, the defining model class of the existential quantifieris the class K ∃ = { ( M, A ) : ∅ 6 = A ⊆ M } , and the defining model class ofthe universal quantifier is the class K ∀ = { ( M, A ) : A = M } . Noting that5he generalisations of ∃ with defining class { ( M, A ) : A ⊆ M, | A | ≥ n } ,where n is fixed, are definable in first order logic, Mostowski [30] intro-duced the generalisations Q α of ∃ with defining class K Q α = { ( M, A ) : A ⊆ M, | A | ≥ ℵ α } . Many other generalized quantifiers are known today in the literature and wewill introduce some important ones later.4.
Second order logic L is the logic ( S ∗ , T ∗ ) , where S ∗ is obtained from firstorder logic by adding variables for n -ary relations for all n and allowingexistential and universal quantification over the new variables. The class T ∗ is defined by the obvious induction. In this inductive definition of T ∗ thesecond order variables range over all relations of the domain (and not onlye.g. over definable relations).We now define the main new concept of this paper: Definition 2.2.
Suppose L ∗ is a logic. If M is a set, let Def L ∗ ( M ) denote the setof all sets of the form X = { a ∈ M : ( M, ∈ ) | = ϕ ( a,~b ) } , where ϕ ( x, ~y ) is anarbitrary formula of the logic L ∗ and ~b ∈ M . We define a hierarchy ( L ′ α ) of setsconstructible using L ∗ as follows: L ′ = ∅ L ′ α +1 = Def L ∗ ( L ′ α ) L ′ ν = S α<ν L ′ α for limit ν We use C ( L ∗ ) to denote the class S α L ′ α .Thus a typical set in L ′ α +1 has the form X = { a ∈ L ′ α : ( L ′ α , ∈ ) | = ϕ ( a,~b ) } (1)where ϕ ( x, ~y ) is a formula of L ∗ and ~b ∈ L ′ α . It is important to note that ϕ ( x, ~y ) is a formula of L ∗ in the sense of V , not in the sense of C ( L ∗ ) , i.e. we assume S ∗ ( ϕ ( x, ~y )) is true rather than being true in ( L ′ α , ∈ ) . In extensions of first orderlogic of the form L ( Q ) this is not a problem because being a formula is absoluteto high degree. For example, in a countable vocabulary we have G¨odel-numberingfor the set of formulas of L ( Q ) which renders the set of G¨odel-numbers of formu-las primitive recursive. On the other hand, the set of formulas of L ω ω is highly6on-absolute, because an infinite conjunction may be uncountable in L ′ α but count-able in V . Also, note that ( L ′ α , ∈ ) | = ϕ ( a,~b ) refers to T ∗ in the sense of V , notin the sense of C ( L ∗ ) . This is a serious point. For example, if ϕ ( a,~b ) comparescardinalities or cofinalities of a,~b to each other, the witnessing mappings do nothave to be in L ′ α .By definition, C ( L ωω ) = L . Myhill-Scott [32] showed that C ( L ) = HOD(See Theorem 7.1 below). Chang [7] considered C ( L ω ω ) and pointed out thatthis is the smallest transitive model of ZFC containing all ordinals and closedunder countable sequences. Kunen [17] showed that C ( L ω ω ) fails to satisfy theAxiom of Choice, if we assume the existence of uncountably many measurablecardinals (see Theorem 5.10 below). Sureson [38, 39] investigated a CoveringLemma for C ( L ω ω ) . Proposition 2.3.
For any L ∗ the class C ( L ∗ ) is a transitive model of ZF contain-ing all the ordinals.Proof. As in the usual proof of ZF in L . Let us prove the Comprehension Schemaas an example. Suppose A,~b are in C ( L ∗ ) , ϕ ( x, ~y ) is a first order formula of settheory and X = { a ∈ A : C ( L ∗ ) | = ϕ ( a,~b ) } . Let α be an ordinal such that A ∈ L ′ α and ϕ ( x, y ) is absolute for L ′ α , C ( L ∗ ) (seee.g. [19, IV.7.5]). Now X = { a ∈ L ′ α : L ′ α | = a ∈ A ∧ ϕ ( a,~b ) } . Hence X ∈ C ( L ∗ ) .We cannot continue and follow the usual proof of AC in L , because the syntax of L ∗ may introduce sets into C ( L ∗ ) without introducing a well-ordering for them(See Theorem 2.11). Also, formulas of L ∗ , such as W n x = y n if in L ω ω , maycontain infinitely many free variables and that could make C ( L ∗ ) closed under ω -sequences, rendering it vulnerable to the failure of AC. To overcome this difficulty,we introduce the following concept, limiting ourselves to logics in which everyformula has only finitely many free variables: Definition 2.4.
A logic L ∗ is adequate to truth in itself if for all finite vocabu-laries K there is function ϕ p ϕ q from all formulas ϕ ( x , . . . , x n ) ∈ L ∗ in thevocabulary K into ω , and a formula Sat L ∗ ( x, y, z ) in L ∗ such that: This is a special case of a concept with the same name in [10].
7. The function ϕ p ϕ q is one to one and has a recursive range.2. For all admissible sets M , formulas ϕ of L ∗ in the vocabulary K , structures N ∈ M in the vocabulary K , and a , . . . , a n ∈ N the following conditionsare equivalent:(a) M | = Sat L ∗ ( N , p ϕ q , h a , . . . , a n i ) (b) N | = ϕ ( a , . . . , a n ) . We may admit ordinal parameters in this definition.Most logics that one encounters in textbooks and research articles of logic areadequate to truth in themselves. To find counter examples one has to consider e.g.logics with infinitely many generalized quantifiers.
Example 2.5.
First order logic L ωω and the logic L ( Q α ) are adequate to truth inthemselves. Also second order logic is adequate to truth in itself in the slightlyweaker sense that M has to be of size ≥ | N | because we also have second ordervariables. Infinitary logics are for obvious reasons (cannot use natural numbers forG¨odel-numbering) not adequate to truth in themselves, but there is a more generalnotion which applies to them (see [10, 40]). In infinitary logic what accounts asa formula depends on set theory. For example, in the case of L ω ω the formulasessentially code in their syntax all reals.The following proposition is instrumental in showing that C ( L ∗ ) , for certain L ∗ , satisfies the Axiom of Choice: Proposition 2.6. If L ∗ is adequate to truth in itself, there are formulas Φ L ∗ ( x ) and Ψ L ∗ ( x, y ) of L ∗ in the vocabulary {∈} such that if M is an admissible set and α = M ∩ On , then:1. { a ∈ M : ( M, ∈ ) | = Φ L ∗ ( a ) } = L ′ α ∩ M. { ( a, b ) ∈ M × M : ( M, ∈ ) | = Ψ L ∗ ( a, b ) } is a well-order < ′ α the field ofwhich is L ′ α ∩ M . I.e. transitive models of the Kripke-Platek axioms KP of set theory. The only reason why weneed admissibility is that admissible sets are closed under inductive definitions of the simple kindthat are used in the syntax and semantics of many logics. For more on admissibility we refer to[5].
8t is important to note that the formulas Φ L ∗ ( x ) and Ψ L ∗ ( x, y ) are in the ex-tended logic L ∗ , not necessarily in first order logic.Recall that we have defined the logic L ∗ as a pair ( S ∗ , T ∗ ) . We can use theset-theoretical predicates S ∗ and T ∗ to write “( M, ∈ ) | = Φ L ∗ ( a )” and “( M, ∈ ) | =Ψ L ∗ ( x, y )” of Proposition 2.6 as formulas ˜Φ L ∗ ( M, x ) and ˜Ψ L ∗ ( M, x, y ) of the firstorder language of set theory, such that for all M with α = M ∩ On and a, b ∈ M :1. ˜Φ L ∗ ( M, a ) ↔ [( M, ∈ ) | = Φ L ∗ ( a )] ↔ a ∈ L ′ α .2. ˜Ψ L ∗ ( M, a, b ) ↔ [( M, ∈ ) | = Ψ L ∗ ( a, b )] ↔ a < ′ α b . Proposition 2.7. If L ∗ is adequate to truth in itself, then C ( L ∗ ) satisfies the Axiomof Choice.Proof. Let us fix α and show that there is a well-order of L ′ α in C ( L ∗ ) . Let κ = | α | + . Then Ψ L ∗ ( x, y ) defines on L ′ κ a well-order < ′ κ of L ′ κ . The relation < ′ κ is in L ′ κ +1 ⊆ C ( L ∗ ) by the definition of C ( L ∗ ) .There need not be a first order definable well-order of the class C ( L ∗ ) (see theproof of Theorem 6.6 for an example) although there always is in V a definablerelation which well-orders C ( L ∗ ) . Of course, in this case V = C ( L ∗ ) . Proposi-tion 2.7 holds also for second order logic, even though it is only adequate to truthin itself in a slight weaker sense.Note that trivially L ∗ ≤ L + implies C ( L ∗ ) ⊆ C ( L + ) . Thus varying the logic L ∗ we get a whole hierarchy of inner models C ( L ∗ ) . Manyquestions can be asked about these inner models. For example we can ask: (1) canall the known inner models be obtained in this way, (2) under which conditionsdo these inner models satisfy GCH, (3) do inner models obtained in this way haveother characterisations (such as L , HOD and C ( L ω ω ) have), etc. Definition 2.8.
A set a is ordinal definable if there is a formula ϕ ( x, y , . . . , y n ) and ordinals α , . . . , α n such that ∀ x ( x ∈ a ⇐⇒ ϕ ( x, α , . . . , α n )) . (2)A set a is hereditarily ordinal definable if a itself and also every element of TC ( a ) is ordinal definable. 9hen we look at the construction of C ( L ∗ ) we can observe that sets in C ( L ∗ ) are always hereditarily ordinal definable when the formulas of L ∗ are finite (moregenerally, the formulas may be hereditarily ordinal definable): Proposition 2.9. If L ∗ is any logic such that the formulas S ∗ and T ∗ do not con-tain parameters (except hereditarily ordinal definable ones) and in addition everyformula of L ∗ (i.e. element of the class S ∗ ) is a finite string of symbols (or moregenerally hereditarily ordinal definable, with only finitely many free variables),then every set in C ( L ∗ ) is hereditarily ordinal definable.Proof. Recall the construction of the successor stage of C ( L ∗ ) : X ∈ L ′ α +1 if andonly if for some ϕ ( x, ~y ) ∈ L ∗ and some ~b ∈ L ′ α X = { x ∈ L ′ α : ( L ′ α , ∈ ) | = ϕ ( x,~b ) } . Now we can note that X = { x ∈ L ′ α : T ∗ (( L ′ α , ∈ ) , ϕ ( x,~b )) } . Thus if L ′ α is ordinal definable, then so is X . Moreover, ∀ z ( z ∈ L ′ α +1 ⇐⇒ ∃ ϕ ( x, ~u )( S ∗ ( ϕ ( x, ~u )) ∧∀ y ( y ∈ z ⇐⇒ y ∈ L ′ α ∧ T ∗ (( L ′ α , ∈ ) , ϕ ( x, ~u ))) , or in short ∀ z ( z ∈ L ′ α +1 ⇐⇒ ψ ( z, L ′ α )) , where ψ ( z, w ) is a first order formula in the language of set theory. When wecompare this with (2) we see that if L ′ α is ordinal definable and if the (first order)set-theoretical formulas S ∗ and T ∗ have no parameters, then also L ′ α +1 is ordinaldefinable. It follows that the class h L ′ α : a ∈ On i is ordinal definable, whence h L ′ α : α < ν i , and thereby also L ′ ν , is in HOD for all limit ν .Thus, unless the formulas of the logic L ∗ are syntactically complex (as hap-pens in the case of infinitary logics like L ω ω and L ω ω , where a formula can codean arbitrary real), the hereditarily ordinal definable sets form a firm ceiling for theinner models C ( L ∗ ) . Theorem 2.10. C ( L ∞ ω ) = V. roof. Let ( L ′ α ) α be the hierarchy behind C ( L ∞ ω ) , as in Definition 2.2. We show V α ⊆ C ( L ∞ ω ) by induction on α . For any set a let the formulas θ a ( x ) of settheory be defined by the following transfinite recursion: θ a ( x ) = ^ b ∈ a ∃ y ( yEx ∧ θ b ( y )) ∧ ∀ y ( yEx → _ b ∈ a θ b ( y )) . Note that in any transitive set M containing a : ( M, ∈ ) | = ∀ x ( θ a ( x ) ⇐⇒ x = a ) . Let us assume V α ⊆ C ( L ∞ ω ) , or more exactly, V α ∈ L ′ β . Let X ⊆ V α . Then X = { a ∈ L ′ β : L ′ β | = a ∈ V α ∧ _ b ∈ X θ b ( a ) } ∈ L ′ β +1 . Note that the proof actually shows C ( L ω ∞ ω ) = V . Theorem 2.11. C ( L ωω ω ) = L ( R ) . Proof.
Let ( L ′ α ) α be the hierarchy behind C ( L ωω ω ) . We first show L ( R ) ⊆ C ( L ωω ω ) .Since C ( L ωω ω ) is clearly a transitive model of ZF it suffices to show that R ⊆ C ( L ωω ω ) . Let X ⊆ ω . Let ϕ n ( x ) be a formula of set theory which defines thenatural number n in the obvious way. Then X = { a ∈ L ′ α : L ′ α | = a ∈ ω ∧ _ n ∈ X ϕ n ( a ) } ∈ L ′ α +1 . Next we show C ( L ωω ω ) ⊆ L ( R ) . We prove by induction on α that L ′ α ⊆ L ( R ) .Suppose this has been proved for α and L ′ α ∈ L β ( R ) . Suppose X ∈ L ′ α +1 . Thismeans that there is a formula ϕ ( x, ~y ) of L ωω ω and a finite sequence ~b ∈ L ′ α suchthat X = { a ∈ L ′ α : L ′ α | = ϕ ( a,~b ) } . It is possible (see e.g. [5, page 83]) to write a first order formula Φ of set theorysuch that X = { a ∈ L β ( R ) : L β ( R ) | = Φ( a, L ′ α , ϕ,~b ) } . Since there is a canonical coding of formulas of L ωω ω by reals we can consider ϕ as a real parameter. Thus X ∈ L β +1 ( R ) .11 heorem 2.12. C ( L ω ω ) = C ( L ω ω ) (= Chang model ) . Proof.
The model C ( L ω ω ) is closed under countable sequences, for if a n ∈ C ( L ω ω ) for n < ω , then the L ω ω -formula ∀ y ( y ∈ x ↔ _ n y = h n, a n i ) . defines the sequence h a n : n < ω i . Since the Chang model is the smallest transi-tive model of ZF closed under countable sequences, the claim follows.We already known that several familiar inner models ( L itself, L ( R ) , Changmodel) can be recovered in the form C ( L ∗ ) . We can also recover the inner model L µ of one measurable cardinal as a model of the form C ( L ∗ ) in the followingsomewhat artificial way: Definition 2.13.
Suppose U is a normal ultrafilter on κ . We define a generalisedquantifier Q κU as follows: M | = Q Uκ wxyvθ ( w ) ϕ ( x, y ) ψ ( v ) ⇐⇒∃ π : ( S, R ) ∼ = ( κ, < ) ∧ π ′′ A ∈ U, where S = { a ∈ M : M | = θ ( a ) } R = { ( a, b ) ∈ M : M | = ϕ ( a, b ) } A = { a ∈ M : M | = ψ ( a ) } Theorem 2.14. C ( Q Uκ ) = L U .Proof. Let ( L ′ α ) be the hierarchy that defines C ( Q Uκ ) . We prove for all a : L ′ α = L Uα . We use induction on α . Suppose the claim is true up to α . Suppose X ∈ L ′ α +1 ,e.g. X = { a ∈ L ′ α : ( L ′ a , ∈ ) | = ϕ ( a,~b ) } , where ϕ ( x, ~y ) ∈ F O ( Q Uκ ) and ~b ∈ L ′ α . We show X ∈ L Uα . To prove this we useinduction on ϕ ( x, ~y ) . Suppose X = { a ∈ L ′ α : ( L ′ α , ∈ ) | = Q Uκ wxyvθ ( z, a,~b ) ϕ ( x, y, a,~b ) ψ ( v, a,~b ) } θ , ϕ and ψ . Let Y a = { c ∈ L ′ α : ( L ′ α , ∈ ) | = θ ( c, a,~b ) } ,R a = { ( c, d ) ∈ L ′ α : ( L ′ α , ∈ ) | = ϕ ( c, d, a,~b ) } , and S a = { c ∈ L ′ α : ( L ′ α , ∈ ) | = ψ ( c, a,~b ) } . Thus X = { a ∈ L ′ α : ∃ π : ( Y a , R a ) ∼ = ( κ, < ) ∧ π ′′ S a ∈ U } . But now X = { a ∈ L Uα : ∃ π : ( Y a , R a ) ∼ = ( κ, < ) ∧ π ′′ S a ∈ U ∩ L U } , so X ∈ L U . Claim 2:
For all a : L Uα ∈ C ( Q Uκ ) . We use induction on α . It suffices to provefor all α : U ∩ L Uα ∈ C ( Q Uκ ) . Suppose the claim is true up to α . We show U ∩ L Uα +1 ∈ C ( Q Uκ ) . Now U ∩ L Uα +1 = U ∩ Def ( L Uα , ∈ , U ∩ L Uα )= { X ⊆ L Uα : X ∈ U ∧ X ∈ Def ( L Uα , ∈ , U ∩ L Uα ) } The concept of an absolute logic attempts to capture the first-order content of L ωω . Is it possible that logics that are “first order” in the way L ωω is turn out to besubstitutable with L ωω in the definition of the constructible hierarchy?Barwise writes in [3, pp. 311-312]:“Imagine a logician k using T as his metatheory for defining the basicnotions of a particular logic L ∗ . When is it reasonable for us, asoutsiders looking on, to call L ∗ a “first order” logic? If the words“first order” have any intuitive content it is that the truth or falsity of M | = ∗ ϕ should depend only on ϕ and M , not on what subsets of M may or may not exist in k ’s model of his set theory T . In other13ords, the relation | = ∗ should be absolute for models of T . Whatabout the predicate ϕ ∈ L ∗ of ϕ ? To keep from ruling out L ω ω (thepredicate ϕ ∈ L ω ω is not absolute since the notion of countable is notabsolute) we demand only that the notion of L ∗ -sentence be persistentfor models of T : i.e. that if ϕ ∈ L ∗ holds in k ’s model of T then itshould hold in any end extension of it.”Using absoluteness as a guideline, Barwise [3] introduced the concept of an abso-lute logic: Definition 3.1.
Suppose A is any class and T is any theory in the language ofset theory. A logic L ∗ is T -absolute if there are a Σ -predicate S ( x ) , a Σ -predicate T ( x, y ) , and a Π -predicate T ′ ( x, y ) such that ϕ ∈ L ∗ ⇐⇒ S ( ϕ ) , M | = ϕ ⇐⇒ T ( M, ϕ ) and T ⊢ ∀ x ∀ y ( S ( x ) → ( T ( x, y ) ↔ T ′ ( x, y ))) . Ifparameters from a class A are allowed, we say that L ∗ is absolute with parametersfrom A .Note that the stronger T is, the weaker the notion of T -absoluteness is. Bar-wise [3] calls KP -absolute logics strictly absolute.As Theorems 2.10 and . demonstrate, absolute logics (such as L ωω ω ) maybe very strong from the point of view of the inner model construction. However,this is so only because of the potentially complex syntax of the absolute logics, asis the case with L ω ω . Accordingly we introduce the following notion: Definition 3.2.
An absolute logic L ∗ has T -absolute syntax if its sentences are(coded as) natural numbers and there is a Π -predicate S ′ ( x ) such that T ⊢∀ x ( S ( x ) ↔ S ′ ( x )) . We may allow parameters, as in Definition 3.1.In other words, to say that a logic L ∗ has “absolute syntax” means that theclass of L ∗ -formulas has a ∆ T -definition. Obviously, L ω ω does not satisfy thiscondition. On the other hand, many absolute logics, such as L ωω , L ( Q ) , weaksecond order logic, L HYP , etc have absolute syntax.The original definition of absolute logics does not allow parameters. Still thereare many logics that are absolute apart from dependence on a parameter. In ourcontext it turns out that we can and should allow parameters.The cardinality quantifier Q α is defined as follows: M | = Q α xϕ ( x,~b ) ⇐⇒ |{ a ∈ M : M | = ϕ ( a,~b ) }| ≥ ℵ α . Kripke-Platek set theory.
14 slightly stronger quantifier is
M | = Q Eα x, yϕ ( x, y, ~c ) ⇐⇒ { ( a, b ) ∈ M : M | = ϕ ( a, b, ~c ) } is anequivalence relation with ≥ ℵ α classes. Example 3.3. L ∞ ω is KP-absolute [3].2. L ( Q α ) is ZFC-absolute with ω α as parameter.3. L ( Q Eα ) is ZFC-absolute with ω α as parameter. Theorem 3.4.
Suppose L ∗ is ZFC+V=L-absolute with parameters from L , andthe syntax of L ∗ is (ZFC+V=L)-absolute with parameters from L . Then C ( L ∗ ) = L .Proof. We use induction on α to prove that L ′ α ⊆ L . We suppose L ′ α ⊆ L andthat ZFC n is a finite part of ZFC so that L ∗ is ZFC n + V = L -absolute. Then L ′ α ∈ L γ for some γ such that L γ | = ZFC n . We show that L ′ α +1 ⊆ L γ +1 . Suppose X ∈ L ′ α +1 . Then X is of the form X = { a ∈ L ′ α : ( L ′ α , ∈ ) | = ϕ ( a,~b ) } , where ϕ ( x, ~y ) ∈ L ∗ and ~b ∈ L ′ α . W.l.o.g., ϕ ( x, ~y ) ∈ L γ . By the definition ofabsoluteness, X = { a ∈ L γ : ( L γ , ∈ ) | = a ∈ L ′ α ∧ S ( ϕ ( x, ~y )) ∧ T ( L ′ α , ϕ ( a,~b )) } . Hence X ∈ L γ +1 . This also shows that h L ′ α : α < ν i ∈ L , and thereby L ′ ν ∈ L ,for limit ordinals ν .A consequence of the Theorem 3.4 is the following: Conclusion:
The constructible hierarchy L is unaffected if first order logic isexchanged in the construction of L for any of the following, simultaneously orseparately: • Recursive infinite conjunctions V ∞ n =0 ϕ n and disjunctions W ∞ n =0 ϕ n . • Cardinality quantifiers Q α , α ∈ On . • Equivalence quantifiers Q Eα , α ∈ On .15 Well-ordering quantifier M | = W x, yϕ ( x, y ) ⇐⇒{ ( a, b ) ∈ M : M | = ϕ ( a, b ) } is a well-ordering . • Recursive game quantifiers ∀ x ∃ y ∀ x ∃ y . . . ∞ ^ n =0 ϕ n ( x , y , . . . , x n , y n ) , ∀ x ∃ y ∀ x ∃ y . . . ∞ _ n =0 ϕ n ( x , y , . . . , x n , y n ) . • Magidor-Malitz quantifiers at ℵ M | = Q MM ,n x , . . . , x n ϕ ( x , . . . , x n ) ⇐⇒∃ X ⊆ M ( | X | ≥ ℵ ∧ ∀ a , . . . , a n ∈ X : M | = ϕ ( a , . . . , a n )) . Thus G¨odel’s L = C ( L ωω ) exhibits some robustness with respect to the choice ofthe logic. The Magidor-Malitz quantifier at ℵ [23] extends Q by allowing us to say thatthere is an uncountable set such that, not only every element of the set satisfiesa given formula ϕ ( x ) , but even any pair of elements from the set satisfy a givenformula ψ ( x, y ) . Much more is expressible with the Magidor-Malitz quantifierthan with Q , e.g. the existence of a long branch or of a long antichain in a tree,but this quantifier is still axiomatizable if one assumes ♦ . On the other hand, theprice we pay for the increased expressive power is that it is consistent, relative tothe consistency of ZF, that Magidor-Malitz logic is very badly incompact [1]. We This quantifier is absolute because the well-foundedness of a linear order < is equivalent tothe existence of a function from the tree of strictly < -decreasing sequences into the ordinals suchthat a strictly longer sequence is always mapped to a strictly smaller ordinal. This quantifier is absolute because the existence of an infinite set X as above is equivalentto the non-well-foundedness of the tree of strictly ⊂ -increasing sequences ( s , . . . , s m ) of finitesubsets of the model with the property that M | = ϕ ( a , . . . , a n ) holds for all a , . . . , a n ∈ s m . L , if we assume ♯ , the innermodel collapses to L . This is a bit surprising, because the existence of ♯ impliesthat L is very “slim”, in the sense that it is not something that an a priori biggerinner model would collapse to. The key to this riddle is that under ♯ the Magidor-Malitz logic itself loses its “sharpness” and becomes in a sense absolute between V and L . Definition 4.1.
The Magidor-Malitz quantifier in dimension n is the following: M | = Q MM ,nα x , . . . , x n ϕ ( x , . . . , x n ) ⇐⇒∃ X ⊆ M ( | X | ≥ ℵ α ∧ ∀ a , . . . , a n ∈ X : M | = ϕ ( a , . . . , a n )) . The original Magidor-Malitz quantifier had dimension and α = 1 : M | = Q MM x , x ϕ ( x , x ) ⇐⇒∃ X ⊆ M ( | X | ≥ ℵ ∧ ∀ a , a ∈ X : M | = ϕ ( a , a )) . The logics L ( Q MM ,<ωκ ) and L ( Q MM ,nκ ) are adequate to truth in themselves (recallDefinition 2.4), with κ as a parameter.Note that putting n = 1 gives us Q : M | = Q xϕ ( x ) ⇐⇒∃ X ⊆ M ( | X | ≥ ℵ ∧ ∀ a ∈ X : M | = ϕ ( a )) . We have already noted in Footnote 6 that for α = 0 this quantifier is absolute. Theorem 4.2. If ♯ exists, then C ( Q MM ,<ωα ) = L .Proof. We treat only the case n = 2 , α = 1 . The general case is treated similarly,using induction on n . The proof hinges on the following lemma: Lemma 4.3.
Suppose ♯ exists and A ∈ L , A ⊆ [ η ] . If there is an uncountable B such that [ B ] ⊆ A , then there is such a set B in L .Proof. Let us first see how the Lemma helps us to prove the theorem. We will useinduction on α to prove that L ′ α ⊆ L . We suppose L ′ α ⊆ L , and hence L ′ α ∈ L γ forsome canonical indiscernible γ . We show that L ′ α +1 ⊆ L γ +1 . Suppose X ∈ L ′ α +1 .Then X is of the form X = { a ∈ L ′ α : ( L ′ α , ∈ ) | = ϕ ( a,~b ) } , ϕ ( x, ~y ) ∈ L ( Q MM ) and ~b ∈ L ′ α . For simplicity we suppress the mention of ~b . Since we can use induction on ϕ , the only interesting case is X = { a ∈ L ′ α : ∃ Y ( | Y | ≥ ℵ ∧ ∀ x, y ∈ Y : ( L ′ α , ∈ ) | = ψ ( x, y, a )) } , where we already have for each a ∈ L ′ α A = {{ c, d } ∈ [ L ′ α ] : ( L ′ α , ∈ ) | = ψ ( c, d, a ) } ∈ L. Now the Lemma implies X = { a ∈ L ′ α : ∃ Y ∈ L ( | Y | ≥ ℵ ∧ ∀ x, y ∈ Y : ( L ′ α , ∈ ) | = ψ ( x, y, a )) } . Since L γ ≺ L , we have X = { a ∈ L ′ α : ∃ Y ∈ L γ ( | Y | ≥ ℵ ∧ ∀ x, y ∈ Y : ( L ′ α , ∈ ) | = ψ ( x, y, a )) } . Finally, X = { a ∈ L γ : ( L γ , ∈ ) | = a ∈ L ′ α ∧∃ z (“ ∃ f : ( ℵ ) V − → z ” ∧ ∀ x, y ∈ zψ ( x, y, a ) ( L ′ α , ∈ ) ) } . Now we prove the Lemma. W.l.o.g. the set B of the lemma satisfies | B | = ℵ ,say B = { δ i : i < ω } in increasing order. Let I be the canonical closed un-bounded class of indiscernibles for L . Let δ i = τ i ( α i , . . . , α ik i ) , where α i , . . . , α ik ∈ I . W.l.o.g., τ i is a fixed term τ . Thus also k i is a fixed number k . By the ∆ -lemma,by thinning I if necessary, we may assume that the finite sets { α i , . . . , α ik } , i <ω , form a ∆ -system with a root { α , . . . , α n } and leaves { β i , . . . , β ik } , i < ω .W.l.o.g. the mapping i β i is strictly increasing in i . Let γ = sup { β i : i < ω } .W.l.o.g., the mapping i β i is also strictly increasing in i . Let γ ′ = sup { β i : i < ω } . It may happen that γ = γ . Then we continue to β i , β i , etc until we get γ ′ k = sup { β ik : i < ω } > γ . Then we let γ = γ ′ k . We continue in this wayuntil we have γ < ... < γ k s − , all limit points of I .Recall that whenever γ is a limit point of the set I there is a natural L -ultrafilter U γ ⊆ L on γ , namely A ∈ U γ ⇐⇒ ∃ δ < γ (( I \ δ ) ∩ γ ⊆ A ) . Recall also thefollowing property of the L -ultrafilters U γ : • Rowbottom Property : Suppose γ < ... < γ n are limits of indiscerniblesand U γ ,. . . , U γ n are the corresponding L -ultrafilters. Suppose C ⊆ [ γ ] n × ... × [ γ l ] n l , where C ∈ L . Then there are B ∈ U γ , . . . , B l ∈ U γ l such that [ B ] n × ... × [ B l ] n l ⊆ C or [ B ] n × ... × [ B l ] n l ∩ C = ∅ . (3)18e apply this to the ordinals γ , . . . , γ k s − and to a set C of sequences ( ζ , . . . , ζ k − , η k s − , . . . , η k − , . . . , ζ sk s − , . . . , ζ sk s − , η sk s − , . . . , η sk s − ) (4)such that { τ ( α , . . . , α n , ζ , . . . , ζ k − , . . . , ζ sk s − , . . . , ζ sk s − ) ,τ ( α , . . . , α n , η , . . . , η k − , . . . , η sk s − , . . . , η sk s − ) } ∈ A (5)Since A ∈ L , also C ∈ L . Note that C ⊆ [ γ ] k × ... × [ γ s ] k s By the Rowbottom Property there are B ∈ U γ , . . . , B s ∈ U γ s such that [ B ] k × ... × [ B s ] k s ⊆ C or [ B ] k × ... × [ B s ] k s ∩ C = ∅ . (6) Claim: [ B ] k × ... × [ B s ] k s ⊆ C .To prove the claim suppose [ B ] k × ... × [ B s ] k s ∩ C = ∅ . Since B j ∈ U γ j ,there is ξ j < γ j such that ( I \ ξ j ) ∩ γ j ⊆ B j . We can now find i , i < ω suchthat in the sequence β i l , . . . , β i l k − , . . . , β i l k s − , . . . , β i l k s − , l ∈ { , } , where β i l , . . . , β i l k − < γ and β i l k j − , . . . , β i l k j − < γ j for all j, we actually have ξ < β i l , . . . , β i l k − < γ and for all j : ξ j < β i l k j − , . . . , β ik j − < γ j , l ∈ { , } . Then since τ ( α , . . . , α n , β i l , . . . , β i l k − , . . . , β i l k s − , . . . , β i l k s − ) ∈ B, and [ B ] ⊆ A , we have { τ ( α , . . . , α n , β i , . . . , β i k − , . . . , β i k s − , . . . , β i k s − ) ,τ ( α , . . . , α n , β i , . . . , β i k − , . . . , β i k s − , . . . , β i k s − ) } ∈ A ( β l , . . . , β l k − , β l , . . . , β l k − , . . . , β l k s − , . . . , β l k s − , β l k s − , . . . , β l k s − ) ∈ C (7)contrary to the assumption [ B ] k × ... × [ B s ] k s ∩ C = ∅ . We have proved theclaim.Now we define B ∗ = { τ ( α , . . . , α n , ζ , . . . , ζ k − , . . . , ζ sk s − , . . . , ζ sk s − ) :( ζ , . . . , ζ k − ) ∈ B k , . . . , ( ζ sk s − , . . . , ζ sk s − ) ∈ B k s s } . (8)Then B ∗ ∈ L, | B ∗ | = ℵ and [ B ∗ ] ⊆ A .What if we do not assume ♯ ? We show that if we start from L and use forcingwe can obtain a model in which C ( Q MM , ω ) = L . Theorem 4.4.
If Con(ZF), then Con(ZFC+ C ( Q MM , ω ) = L ).Proof. Assume V = L . Jensen and Johnsbr˚aten [13] define a sequence T n ofSouslin trees in L and a CCC forcing notion P which forces the set a of n suchthat ˇ T n is Souslin to be non-constructible. But a ∈ C ( Q MM , ω ) since the trees T n arein C ( Q MM , ω ) and Sousliness of a tree can be expressed in L ( Q MM , ω ) by [23, page223]. So we are done.This result can be strengthened in a number of ways. In [1] an ω -sequenceof Souslin trees is constructed from ♦ giving rise to forcing extensions in which L ( Q MM , ω ) can express some ostensibly second order properties, and C ( Q MM , ω ) isvery different from L .There are several stronger versions of Q MM ,<ωκ , for example Q MR κ x , x , x ψ ( x , x , x ) ⇐⇒∃X ( ∀ X , X ∈ X )( ∀ x , x ∈ X )( ∀ x ∈ X ) ψ ( x , x , x , ~y ) , where X , X range over sets of size κ and X ranges over families of size κ ofsets of size κ ([24]). The above is actually just one of the various forms of similarquantifiers that L ( Q MR κ ) has. The logic L ( Q MR ℵ ) is still countably compact assuming ♦ . We do not know whether ♯ implies C ( Q MR κ ) = L .20 The Cofinality Quantifier
The cofinality quantifier of Shelah [34] says that a given linear order has cofi-nality κ . Its main importance lies in the fact that it satisfies the compactnesstheorem irrespective of the cardinality of the vocabulary. Such logics are called fully compact . This logic has also a natural complete axiomatization, provably inZFC. This makes the cofinality quantifier particularly appealing in this project,even though we do not have a clear picture yet of the connection between modeltheoretic properties of logics L ∗ and set theoretic properties of C ( L ∗ ) .The cofinality quantifier Q cf κ for a regular κ is defined as follows: M | = Q cf κ xyϕ ( x, y, ~a ) ⇐⇒ { ( c, d ) : M | = ϕ ( c, d, ~a ) } is a linear order of cofinality κ .We will denote by C ∗ κ the inner model C ( Q cf κ ) . Note that C ∗ κ need not computecofinality κ correctly, it just knows which ordinals have cofinality κ in V . Themodel knows this as if the model had an oracle for exactly this but nothing else.Thus while many more ordinals may have cofinality κ in V than in C ∗ κ , still theproperty of an ordinal having cofinality κ in V is recognised in C ∗ κ in the sensethat for all β and A, R ∈ C ∗ κ : • { α < β : cf V ( α ) = κ } ∈ C ∗ κ • { α < β : cf V ( α ) = κ } ∈ C ∗ κ • { α < β : cf V ( α ) = κ ⇐⇒ cf C ∗ κ ( α ) = κ } ∈ C ∗ κ • { a ∈ A : { ( b, c ) : ( a, b, c ) ∈ R } is a linear order on A with cofinality (in V )equal to κ } ∈ C ∗ κ .Let On κ be the class of ordinals of cofinality κ . Let L (On κ ) be L defined inthe expanded language {∈ , On κ } . Now L (On κ ) ⊆ C ∗ κ because we can use theequivalence of On κ ( β ) with Q cf κ xy ( x ∈ y ∧ y ∈ β ) . Conversely, C ∗ κ ⊆ L (On κ ) because if E is a club of β such that for every linear order R ∈ L β (On κ ) there isan ordinal γ < β and a function f ∈ L β (On κ ) mapping γ cofinally into R , then L ′ α ⊆ L β (On κ ) whenever α ≤ β ∈ E . We have proved C ∗ κ = L (On κ ) . We use C ∗ to denote C ∗ ω . 21he following related quantifier turns out to be useful, too: M | = Q cf <κ xyϕ ( x, y, ~a ) ⇐⇒ { ( c, d ) : M | = ϕ ( c, d, ~a ) } is a linear order of cofinality < κ .We use C ∗ κ,λ to denote C ( Q cf κ , Q cf λ ) and C ∗ <κ to denote C ( Q cf <κ ) . Respectively, C ∗≤ κ denotes C ( Q cf ≤ κ ) .Our results show that the inner models C ∗ <κ all resemble C ∗ in many ways (seee.g. Theorem 5.18), and accordingly we indeed focus mostly on C ∗ .The logics C ∗ κ,λ and C ∗ <κ are adequate to truth in themselves (recall Defini-tion 2.4), with κ, λ as parameters, whence these inner models satisfy AC.We can translate the formulas Φ L ( Q cf κ ) ( x ) and Ψ L ( Q cf κ ) ( x, y ) , introduced inProposition 2.6, into ˆΦ L ( Q cf κ ) ( x, κ ) and ˆΨ L ( Q cf κ ) ( x, y, κ ) in the first order languageof set theory by systematically replacing Q cf κ xyϕ ( x, y, ~a ) by the canonical set-theoretic formula saying the same thing. Then for all M with α = M ∩ On and a, b ∈ M :1. ˆΦ L ( Q cf κ ) ( a, κ ) ↔ [( M, ∈ ) | = Φ L ( Q cf κ ) ( a )] ↔ a ∈ C ∗ κ .2. ˆΨ L ( Q cf κ ) ( a, b, κ ) ↔ [( M, ∈ ) | = Ψ L ( Q cf κ ) ( a, b )] ↔ a < ′ α b . Lemma 5.1. If M and M are two transitive models of ZFC such that for all α : M | = cf( α ) = κ ⇐⇒ M | = cf( α ) = κ, then ( C ∗ κ ) M = ( C ∗ κ ) M . Proof.
Let ( L ′ α ) be the hierarchy defining ( C ∗ κ ) M and ( L ′′ α ) be the hierarchy defin-ing ( C ∗ κ ) M . By induction, L ′ α = L ′′ α for all α .By letting M = V in Proposition 5.1 we get Corollary.
Suppose M is a transitive model of ZFC such that for all α : cf( α ) = κ ⇐⇒ M | = cf( α ) = κ, then ( C ∗ κ ) M = C ∗ κ . ( C ∗ κ ) M = C ∗ κ is a perfectly possible sit-uation: In Theorem 6.3 below we construct a model M in which CH is false in C ∗ . So ( C ∗ ) M = L . Thus in M it is true that ( C ∗ ) L = C ∗ . ( C ∗ κ ) M = C ∗ κ also if κ = ω , V = L µ and M = C ∗ (see the below Theorem 5.16). In this respect C ∗ κ resembles HOD. There are other respects in which C ∗ κ resembles L . Lemma 5.2.
Suppose ( L ′ α ) is the hierarchy forming C ∗ κ . Then for α < κ we have L ′ α = L α . We can relativize C ∗ to a set X of ordinals as follows. Let us define a newgeneralized quantifier as follows: M | = Q X xyϕ ( x, y, ~a ) ⇐⇒ { ( c, d ) : M | = ϕ ( c, d, ~a ) } is a well-order of type ∈ X .We define C ∗ ( X ) as C ( Q cf ω , Q X ) . Of course, C ∗ ( X ) = L (On ω , X ) .We will prove a stronger form of the next Proposition in the next Theorem, butwe include this here for completeness: Proposition 5.3. If ♯ exists, then ♯ ∈ C ( Q cf κ ) .Proof. Let I be the canonical set of indiscernibles obtained from ♯ . Let us firstprove that ordinals ξ which are regular cardinals in L and have cofinality > ω in V are in I . Suppose ξ / ∈ I . Note that ξ > min( I ) . Let δ be the largest element of I ∩ ξ . Let λ < λ < ... be an infinite sequence of elements of I above ξ . Let τ n ( x , . . . , x k n ) , n < ω, be a list of all the Skolem terms of the language of set theory relative to the theoryZFC + V = L . If α < ξ , then α = τ n α ( γ , . . . , γ m n , λ , . . . , λ l n ) for some γ , . . . , γ m n ∈ I ∩ δ and some l n < ω . Let us fix n for a moment andconsider the set A n = { τ n α ( β , . . . , β m n , λ , . . . , λ l n ) : β , . . . , β n < δ } . Note that A n ∈ L and | A n | L ≤ | δ | L < ξ , because ξ is a cardinal in L . Let η n = sup( A n ) . Since ξ is regular in L , η n < ξ . Since ξ has cofinality > ω in23 , η = sup n η n < ξ . But we have now proved that every α < ξ is below η , acontradiction. So we may conclude that necessarily ξ ∈ I .Suppose now κ = ω . Let X = { ξ ∈ L ′ℵ ω : ( L ′ℵ ω , ∈ ) | = “ ξ is regular in L ” ∧ ¬ Q cfκ xy ( x ∈ y ∧ y ∈ ξ ) } Now X is an infinite subset of I and X ∈ C ( Q cf κ ) . Hence ♯ ∈ C ( Q cf κ ) : ♯ = { p ϕ ( x , . . . , x n ) q : ( L ℵ ω , ∈ ) | = ϕ ( γ , . . . , γ n ) for some γ < ... < γ n in X } . If κ = ℵ α > ω , then we use X = { ξ ∈ L ′ℵ α + ω : ( L ′ℵ α + ω , ∈ ) | = “ ξ is regular in L ” ∧ Q cfκ xy ( x ∈ y ∧ y ∈ ξ ) } and argue as above that ♯ ∈ C ( Q cf κ ) .More generally, the above argument shows that x ♯ ∈ C ∗ ( x ) for any x such that x ♯ exists. Hence C ∗ = L ( x ) whenever x is a set of ordinals such that x ♯ exists in V (see Theorem 5.4). Theorem 5.4.
Exactly one of the following always holds:1. C ∗ is closed under sharps, (equivalently, x ♯ exists for all x ⊂ On such that x ∈ C ∗ ).2. C ∗ is not closed under sharps and moreover C ∗ = L ( x ) for some set x ⊂ On . (Equivalently, there is x ⊂ On such that x ∈ C ∗ but x ♯ does not exist.)Proof. Suppose (1) does not hold. Suppose a ⊆ λ , λ > ω , such that a ∈ C ∗ but a ♯ does not exist. Let S = { α < λ + : cf V ( α ) = ω } . We show that C ∗ = L ( a, S ) .Trivially, C ∗ ⊇ L ( a, S ) . For C ∗ ⊆ L ( a, S ) it is enough to show that one candetect in L ( a, S ) whether a given δ ∈ On has cofinality ω (in V ) or not. If cf( δ ) = ω , and c ⊆ δ is a cofinal ω -sequence in δ , then the Covering Theorem for L ( a ) gives a set b ∈ L ( a ) such that c ⊆ b ⊂ λ , sup( c ) = sup( b ) and | b | = λ . Theorder type of b is in S . Hence whether δ has cofinality ω or not can be detected in L ( a, S ) . Corollary. If x ♯ does not exist for some x ∈ C ∗ , then there is λ such that C ∗ | =2 κ = κ + for all κ ≥ λ . Theorem 5.5.
The Dodd-Jensen Core model is contained in C ∗ . roof. Let K be the Dodd-Jensen Core model of C ∗ . We show that K is the coremodel of V . Assume otherwise and let M be the minimal Dodd-Jensen mousemissing from K . (Minimality here means in the canonical pre-well ordering ofmice.) Let κ be the cardinal of M on which M has the M -normal measure.Denote this normal measure by U . Note that M = J U α for some α . J α [ U ] isthe Jensen J -hierarchy of constructibility from U , where J α [ U ] = S β<ωα S U β ,where S U β is the finer S -hierarchy.Let ξ be ( κ +0 ) M . (If ( κ +0 ) M does not exist in M put ξ = M ∩ ON .). Let δ = cf V ( ξ ) .For an ordinal β let M β be the β ’th iterated ultrapower of M where for β ≤ γ let j β,γ : M β → M γ be the canonical ultrapower embedding. j β,γ is a Σ -embedding. Let κ β = j β ( κ ) , U β = j β ( U ) , ξ β = j β ( ξ ) . (In case ( κ +0 ) M doesnot exist we put ξ β = M β ∩ ON .). κ β is the critical point of j βγ for β < γ . For alimit β and A ∈ M β , A ⊆ κ β A ∈ U β iff κ γ ∈ A for large enough γ < β . Claim.
1. For every β we have ξ β = sup j ” β ( ξ ) . Hence cf V ( ξ β ) = δ . Proof.
Every η < ξ β is of the form j β ( f )( κ γ . . . , κ γ n − ) for some γ < γ . . . <γ n − < β and for some f ∈ M , f : κ n → ξ . By definition of ξ there is ρ < ξ such that f ( α , . . . α n − ) < ρ for every h α , . . . α n − i ∈ κ n . Hence it followsthat every value of j β ( f ) is bounded by j β ( ρ ) . So η < j β ( ρ ) , which proves theclaim.The usual proof of GCH in L [ U ] shows that κ κ β β ∩ M β ⊆ J U β ξ β and that J U β ξ β isthe increasing union of δ members of M β , each one having cardinality κ β in M β . Claim.
2. Let κ < η < κ β be such that M β | = η is regular, then either there is γ < β such that η = κ γ or cf V ( η ) = δ . Proof.
By induction on β . The claim is vacuously true for β = 0 . For β limit κ β = sup { κ γ | γ < β } . Hence there is α < β such that η < κ α . j αβ ( η ) = η ) so M α | = η is regular. So the claim in this case follows from the inductionassumption.We are left with the case that β = α + 1 . If η ≤ κ α the claim follows from theinductive assumption for α as in the limit case. So we are left with the case κ α <η < κ β . M β is the ultrapower of M α by U α , so η is represented in this ultrapowerby a function f ∈ M α whose domain is κ α . By the assumption η < κ β = j αβ ( κ α ) we can assume f ( ρ ) < κ α for every ρ < κ α . By the assumption κ α < η we canassume that ρ < f ( ρ ) for every ρ < κ α and by the assumption that η is regular25n M β we can assume that f ( ρ ) is regular in M α for every ρ < κ α . In order tosimplify notation put M = M α , κ = κ α , U = U α , and ξ = ξ α .In order to show that cf V ( η ) = δ we shall define (in V ) a sequence h g ν | ν < δ i of functions in κ κ ∩ M such that :1. The sequence is increasing modulo U .2. For every ρ < κ , g ν ( ρ ) < f ( ρ ) .3. The ordinals represented by these functions in the ultrapower of M by U are cofinal in η .By the definition of ξ and the previous claim we can represent κ κ ∩ M as anincreasing union S ψ<δ F ψ where for every ψ < δ , F ψ ∈ M and F ψ has cardinality κ in M . For ψ < δ fix an enumeration in M of h h ψρ | ρ < κ i of the set G ψ = { h ∈ F ψ |∀ ρ < κ ( h ( ρ ) < f ( ρ )) } . Let f ψ ∈ κ κ be defined by f ψ ( ρ ) = sup( { h ψµ ( ρ ) | µ <ρ } ) . Clearly f ψ ∈ M and f ψ bounds all the functions in G ψ modulo U . Alsosince for all ρ < κ and h ∈ G ψ h ( ρ ) < f ( ρ ) we obtain f ψ ( ρ ) < f ( ρ ) . (Recallthat f ( ρ ) > ρ , f ( ρ ) is regular in M and f ψ ( ρ ) is the sup of a set in M whosecardinality in M is ρ . ).Define g ν by induction on ν < δ . By induction we shall also define an increas-ing sequence h ψ ν | ν < δ i such that ψ ν < δ and g ν ∈ G ψ ν . Given h ψ µ | µ < ν i let σ be their sup. Let g ν be f σ and let ψ ν be the minimal member of δ − σ such that f σ ∈ G ψ ν . The induction assumptions on g µ , ψ µ for µ < ν and the properties of f σ yields that g ν and ψ ν also satisfy the required inductive assumption.The fact that the sequence of ordinals represented by h g ν | ν < δ i in the ultra-power of M by U is cofinal in η follows from the fact that every ordinal below η is represented by some function h which is bounded everywhere by f , hence itbelongs to G ψ for some ψ < δ . There is ν such that ψ < ψ ν and then g ν +1 willbound h modulo U .The minimality of M (hence the minimality of the equivalent M β ) impliesthat for every β , P ( κ β ) ∩ K = P ( κ β ) ∩ M β . It follows that ρ ≤ κ β is regular in K iff it is regular in M β . In particular for every β , κ β is regular in K since it isregular in M β . Claim.
3. Let λ be a regular cardinal greater than max( | M | , δ ) . Then there thereis D ∈ C ∗ , D ⊆ E = { κ β | β < λ } which is cofinal in λ . Proof.
Note that λ > | M | implies that the set E = { κ β | β < λ } is a club in λ .Let S λ be the set of ordinals in λ whose cofinality (in V ) is ω . Obviously both26 − S λ and E ∩ S λ are unbounded in λ . Let C be the set of the ordinals of λ − κ which are regular in K . By the definition of C ∗ and K both S λ and C are in C ∗ .Also E ⊆ C since κ β is regular in M β , hence regular in K .If δ = ω then we can take D = C ∩ S λ which by Claim 2 is a subset of E which is unbounded in λ . If δ = ω then similarly we can take D = C − S λ . Inboth cases D ∈ C ∗ .Pick λ, E as in the Claim above and let D ⊆ λ be the witness to the claim. Itis well known that for every X ∈ M λ X ∈ U λ iff X ⊆ λ and X contains a finalsegment of E . Since U λ is an ultrafilter on λ in M λ we get that for X ∈ M λ , X ⊆ λ X ∈ U β iff X contains a final segment of D . Let F D be the filter on λ generatedby final segments of D . D ∈ C ∗ implies that L ( F D ) ⊆ C ∗ . M λ = J U λ α for someordinal α . But since U λ = F D ∩ M λ we get that M λ = J F D α . Now this impliesthat M λ ∈ C ∗ . This is because C ∗ contains an iterate of the mouse M and thenby standard Dodd-Jensen Core model techniques M ∈ C ∗ , which is clearly acontradiction. Theorem 5.6.
Suppose an inner model with a measurable cardinal exists. Then C ∗ contains some inner model L ν for a measurable cardinal.Proof. This is as the proof of Theorem 5.5. Suppose L µ exists, but does not existin C ∗ . Let κ be the cardinal of M = L µ on which L µ has the normal measure.Denote this normal measure by U . Let ξ be ( κ +0 ) M and let δ = cf V ( ξ ) .For an ordinal β let M β be the β ’th iterated ultrapower of M and for β ≤ γ let j β,γ : M β → M γ be the canonical ultrapower embedding. j βγ is a Σ -embedding.Let κ β = j β ( κ ) , U β = j β ( U ) , ξ β = j β ( ξ ) . κ β is the critical point of j βγ for β < γ . For a limit β and A ∈ M β , A ⊆ κ β , A ∈ U β iff κ γ ∈ A for large enough γ < β . Claim.
1. For every β , ξ β = sup j β ”( ξ ) . Hence cf V ( ξ β ) = δ . Proof.
Every η < ξ β is of the form j β ( f )( κ γ . . . , κ γ n − ) for some γ < γ . . . <γ n − < β and for some f ∈ M , f : κ n → ξ . By definition of ξ there is ρ < ξ such that f ( α , . . . α n − ) < ρ for every h α , . . . α n − i ∈ κ n . Hence it followsthat every value of j β ( f ) is bounded by j β ( ρ ) . So η < j β ( ρ ) , which proves theclaim.The usual proof of GCH in L [ U ] shows that κ κ β β ∩ M β ⊆ J U β ξ β and that J U β ξ β isthe increasing union of δ members of M β , each one having cardinality κ β in M β .27 laim.
2. Let κ < η < κ β be such that M β | = η is regular, then either there is γ < β such that η = κ γ or cf V ( η ) = δ . Proof.
By induction on β . The claim is vacuously true for β = 0 . For β limit κ β = sup { κ γ | γ < β } . Hence there is α < β such that η < κ α . j αβ ( η ) = η so M α | = η is regular. So the claim in this case follows from the inductionassumption.We are left with the case that β = α + 1 . If η ≤ κ α the claim follows from theinductive assumption for α as in the limit case. So we are left with the case κ α <η < κ β . M β is the ultrapower of M α by U α , so η is represented in this ultrapowerby a function f ∈ M α whose domain is κ α . By the assumption η < κ β = j αβ ( κ α ) we can assume that f ( ρ ) < κ α for every ρ < κ α . By the assumption κ α < η wecan assume that ρ < f ( ρ ) for every ρ < κ α and by the assumption that η is regularin M β we can assume that f ( ρ ) is regular in M α for every ρ < κ α . In order tosimplify notation put M = M α , κ = κ α , U = U α , and ξ = ξ α .In order to show that cf V ( η ) = δ we shall define (in V ) a sequence h g ν | ν < δ i of functions in κ κ ∩ M such that :1. The sequence is increasing modulo U .2. For every ρ < κ , g ν ( ρ ) < f ( ρ ) .3. The ordinals represented by these functions in the ultrapower of M by U are cofinal in η .By the definition of ξ and by the previous claim we can represent κ κ ∩ M as an increasing union S ψ<δ F ψ where for every ψ < δ F ψ ∈ M and F ψ hascardinality κ in M . For ψ < δ fix an enumeration in M of h h ψρ | ρ < κ i of the set G ψ = { h ∈ F ψ |∀ ρ < κ ( h ( ρ ) < f ( ρ )) } . Let f ψ ∈ κ κ be defined by f ψ ( ρ ) =sup( { h ψµ ( ρ ) | µ < ρ } ) . Clearly f ψ ∈ M and f ψ bounds all the functions in G ψ modulo U . Also because for all ρ < κ and h ∈ G ψ h ( ρ ) < f ( ρ ) we get that f ψ ( ρ ) < f ( ρ ) . (Recall that f ( ρ ) > ρ , f ( ρ ) is regular in M and f ψ ( ρ ) is the sup ofa set in M whose cardinality in M is ρ .).Define g ν by induction on ν < δ . By induction we shall also define an increas-ing sequence h ψ ν | ν < δ i such that ψ ν < δ and g ν ∈ G ψ ν . Given h ψ µ | µ < ν i let σ be their sup. Let g ν be f σ and let ψ ν be the minimal member of δ − σ such that f σ ∈ G ψ ν . The induction assumptions on g µ , ψ µ for µ < ν and the properties of f σ yields that g ν and ψ ν also satisfy the required inductive assumption.The fact that the sequence of ordinals represented by h g ν | ν < δ i in the ultra-power of M by U is cofinal in η follows from the fact that every ordinal bellow28 is represented by some function h which is bounded everywhere by f , hence itbelongs to G ψ for some ψ < δ . There is ν such that ψ < ψ ν . Then g ν +1 willbound h modulo U .We know already that K ⊆ C ∗ . Since P ( κ β ) ∩ K = P ( κ β ) ∩ M β for every β ,it follows that ρ ≤ κ β is regular in K iff it is regular in M β . In particular for every β , κ β is regular in K since it is regular in M β . Claim.
3. Let λ be a regular cardinal greater than max( | M | , δ ) . Then there thereis D ∈ C ∗ , D ⊆ E = { κ β | β < λ } which is cofinal in λ .Proof of the Claim: Note that λ > | M | implies that the set E = { κ β | β < λ } is a club in λ . Let S λ be the set of ordinals in λ whose cofinality (in V ) is ω .Obviously both E − S λ and E ∩ S λ are unbounded in λ . Let C be the set of theordinals of λ − κ which are regular in K . By definition of C ∗ and K both S λ and C are in C ∗ . Also E ⊆ C since κ β is regular in M β , hence regular in K .If δ = ω then we can take D = C ∩ S λ which by Claim 2 is a subset of E which is unbounded in λ . If δ = ω then similarly we can take D = C − S λ . Inboth cases D ∈ C ∗ . The Claim is proved.Pick λ, E as in the Claim above and let D ⊆ λ be the witness to the claim.It is well known that for every X ∈ M λ X ∈ U λ iff X ⊆ λ and X contains afinal segment of E . Since U λ is an ultrafilter on λ in M λ we get that for X ∈ M λ , X ⊆ λ X ∈ U β iff X contains a final segment of D . Let F D be the filter on λ generated by final segments of D . D ∈ C ∗ implies that L ( F D ) ⊆ C ∗ . M λ = J U λ α for some ordinal α . But since U λ = F D ∩ M λ we get that M λ = J F D α . Thus M λ ∈ C ∗ , i.e. C ∗ contains an iterate of M . Hence C ∗ contains an inner modelwith a measurable cardinal.Below (Theorem 5.16) we will show that if L µ exists, then ( C ∗ ) L µ can beobtained by adding to the ω th iterate of L µ the sequence { κ ω · n : n < ω } .In the presence of large cardinals, even with just uncountably many measur-able cardinals, we can separate C ∗ from both L and HOD. We first observe that inthe special case that V = C ∗ , there cannot exist even a single measurable cardinal.The proof is similar to Scott’s proof that measurable cardinals violate V = L : Theorem 5.7.
If there is a measurable cardinal κ , then V = C ∗ λ for all λ < κ .Proof. Suppose V = C ∗ λ but κ > λ is a measurable cardinal. Let i : V → M withcritical point κ and M κ ⊆ M . Now ( C ∗ λ ) M = ( C ∗ λ ) V = V , whence M = V . Thiscontradicts Kunen’s result [16] that there cannot be a non-trivial i : V → V .29e can strengthen this as follows, at least for λ = ω . Recall that covering is said to hold for a inner model M if for every set X of ordinals there is a set Y ⊇ X of ordinals such that Y ∈ M and | Y | ≤ | X | + ℵ . We can show that ifthere is a measurable cardinal, then not only V = C ∗ , but we do not even havecovering for C ∗ : Theorem 5.8.
If there is a measurable cardinal then covering fails for C ∗ .Proof. Let i : V → M with critical point κ and M κ ⊆ M . As above, i is anembedding of C ∗ into C ∗ . Let κ n be i n ( κ ) and κ ω = sup n κ n . Clearly i ( κ ω ) = κ ω and there are no fixed points of i on the interval [ κ, κ ω ) . We prove that coveringfails for C ∗ by showing that the singular cardinal κ ω is regular in C ∗ . Assumeotherwise. Then the cofinality α of κ ω in C ∗ is, by elementarity, a fixed point of i . Hence α < κ . Let Z be a be a subset of κ ω in C ∗ witnessing the fact that thecofinality of κ ω in C ∗ is α . W.l.o.g., Z is the minimal such set in the canonicalwellordering of C ∗ . Hence i ( Z ) = Z . Let δ = sup( Z ∩ κ ) . Since κ is regular, δ < κ . Hence i ( δ ) = δ . Let δ ∗ be the minimal member of Z above δ . Then δ ∗ ≥ κ and i ( δ ∗ ) = δ ∗ . But there are no fixed points of i on the interval [ κ, κ ω ) . We havereached a contradiction.On the other hand we will now use known results to show that we cannot failcovering for C ∗ without an inner model for a measurable cardinal. It is curiousthat covering for C ∗ is in this way entangled with measurable cardinals. Theorem 5.9.
If there is no inner model with a measurable cardinal then coveringholds for C ∗ .Proof. By Theorem 5.5, K ⊆ C ∗ . If there is no inner model with a measurablecardinal, then K satisfies covering by [9]. Hence all the more we have coveringfor C ∗ .Kunen [17] proved that if there are uncountably many measurable cardinals,then AC fails in Chang’s model C ( L ω ω ) . Recall that Chang’s model contains C ∗ and C ∗ does satisfy AC. Theorem 5.10. If h κ n : n < ω i is any sequence of measurable cardinals (in V ) > λ , then h κ n : n < ω i / ∈ C ∗ λ and C ∗ λ = HOD .Proof.
We proceed as in Kunen’s proof ([17]) that AC fails in the Chang modelif there are uncountably many measurable cardinals, except that we only use in-finitely many measurable cardinals. Suppose κ n , n < ω , are measurable > λ . Let30 = sup n κ n . Let ≺ be the first well-order of µ ω in C ∗ λ in the canonical well-orderof C ∗ λ . Suppose h κ n : n < ω i ∈ C ∗ λ . Then for some η it is the η th element in thewell-order ≺ . By [17, Lemma 2] there are only finitely many measurable cardi-nals ξ such that η is moved by the ultrapower embedding of a normal ultrafilteron ξ . Let n be such that the ultrapower embedding j : V → M by the normalultrafilter on κ n does not move η . Since κ n > λ , ( C ∗ λ ) M = C ∗ λ . Since µ is a stronglimit cardinal > λ , j ( µ ) = µ . Since the construction of C ∗ λ proceeds in M exactlyas it does in V , j ( ≺ ) is also in M the first well-ordering of µ ω that appears in C ∗ λ .Hence j ( ≺ ) = ≺ . Since j ( η ) = η , the sequence h κ n : n < ω i is fixed by j . Butthis contradicts the fact that j moves κ n .If the κ n are the first ω measurable cardinals above λ , then the sequence h κ n : n < ω i is in HOD and hence C ∗ λ = HOD.
Definition 5.11.
The weak Chang model is the model C ωω = C ( L ωω ω ) .We can make the following observations about the relationship between theweak Chang model and the (full) Chang model. The weak Chang model clearlycontains C ∗ and L ( R ) , as it contains C ( L ωω ω ) . It is a potentially interesting inter-mediate model between L ( R ) and the (full) Chang model. If there is a measurableWoodin cardinal, then the Chang model satisfies AD, whence the weak modelcannot satisfy AC, as the even bigger (full) Chang model cannot contain a well-ordering of all the reals. Theorem 5.12.
1. If V = L µ , then C ωω = L ( R ) .2. If V is the inner model for ω measurable cardinals, then C ωω = Changmodel.Proof.
For (1), suppose V = L µ , where µ is a normal measure on κ . Let us firstnote that all the reals are in C ∗ , because under the assumption V = L µ all thereals are in the Dodd-Jensen core model, which by our Theorem 5.5 is containedin C ∗ . Thus all the reals are in C ωω . Also under the same assumption we havein L ( R ) a Σ well ordering of the reals of order type ω . Hence L ( R ) = L ( A ) for some A ⊆ ω . Suppose now C ωω = L ( R ) . Then there is an A ⊆ ω suchthat C ωω = L ( A ) . By Theorem 5.6 there is in C ∗ , hence in C ωω , an inner model L ν with a measurable cardinal δ . But κ is in L µ the smallest ordinal which ismeasurable in an inner model. Hence κ ≤ δ and A ⊆ δ . But by [33] there cannotbe an inner model with a measurable cardinal δ in L ( A ) , where A ⊆ δ . Thereforewe must have C ωω = L ( R ) . 31or (2), we commence by noting that in the inner model for ω measurablecardinals there is a Σ -well-order of R [36]. By means of this well-order we canwell-order the formulas of L ωω ω in the Chang model. In this way we can definea well-order of C ωω in the Chang model. However, since we assume uncountablymany measurable cardinals, the Chang model does not satisfy AC [17] (see alsoTheorem 5.10). Hence it must be that C ωω = Chang model.If there is a Woodin cardinal, then C ∗ = V in the strong sense that ℵ is a largecardinal in C ∗ . So not only are there countable sequences of measurable cardinalswhich are not in C ∗ but there are even reals which are not in C ∗ : Theorem 5.13.
If there is a Woodin cardinal, then ω is (strongly) Mahlo in C ∗ .Proof. To prove that ω is strongly inaccessible in C ∗ suppose α < ℵ and f : ω → (2 α ) C ∗ is 1-1. Let λ be Woodin, Q <λ the countable stationary tower forcing and G genericfor this forcing. In V [ G ] there is j : V → M such that V [ G ] | = M ω ⊂ M and j ( ω ) = λ . Thus j ( f ) : λ → ((2 α ) C ∗ ) M . Let a = j ( f )( ω V ) . If a ∈ V , then j ( a ) = a , whence, as a i.e. j ( a ) is in the rangeof j ( f ) , a = f ( δ ) for some δ < ω . But then a = j ( a ) = j ( f )( j ( δ )) = j ( f )( δ ) , contradicting the fact that a = j ( f )( ω ) . Hence a / ∈ V . However, ( C ∗ ) M = ( C ∗ <λ ) V , since by general properties of this forcing, an ordinal has cofinality ω in M iff ithas cofinality < λ in V . Hence a ∈ C ∗ <λ ⊆ V , a contradiction.To see that ω is Mahlo in C ∗ , suppose D is a club on ω V , D ∈ C ∗ . Let j and M be as above. Then j ( D ) is a club on λ in ( C ∗ ) M . Since ω V is the critical pointof j , j ( D ) ∩ ω V = D . Since j ( D ) is closed, ω V ∈ j ( D ) . Remark.
In the previous theorem we can replace the assumption of a Woodincardinal by MM ++ .For cardinals > ω we have an even better result:32 heorem 5.14. Suppose there is a Woodin cardinal λ . Then every regular cardi-nal κ such that ω < κ < λ is weakly compact in C ∗ .Proof. Suppose λ is a Woodin cardinal, κ > ω is regular and < λ . To prove that κ is strongly inaccessible in C ∗ we use the “ ≤ ω -closed” stationary tower forc-ing from [11, Section 1]. With this forcing, cofinality ω is not changed, whence ( C ∗ ) M = C ∗ , so the proof of Theorem 5.13 can be repeated mutatis mutandis.Thus we need only prove the tree property. Let the forcing, j and M be as above,in Theorem 5.13, with j ( κ ) = λ . Suppose T is a κ -tree in C ∗ . Then j ( T ) is a λ -tree in ( C ∗ ) M = C ∗ . We may assume j ( T ↾ κ ) = T ↾ κ . Let t ∈ j ( T ) be ofheight κ and b = { u ∈ j ( T ) : u < t } = { u ∈ T : u < t } . Now b is a κ -branch of T in C ∗ .As a further application of ω -closed stationary tower forcing we extend theabove result as follows: Theorem 5.15.
If there is a proper class of Woodin cardinals, then the regularcardinals ≥ ℵ are indiscernible in C ∗ .Proof. We use the ω -closed stationary tower forcing of [11]. Let us first prove anauxiliary claim: Claim 1: If λ < . . . < λ k and ¯ λ < . . . < ¯ λ k are Woodin cardinals, and β , . . . , β l < min( λ , ¯ λ ) , then C ∗ | = Φ( β , . . . , β l , λ , . . . , λ k ) ↔ Φ( β , . . . , β l , ¯ λ , . . . , ¯ λ k ) for all formulas Φ( x , . . . , x l , y , . . . , y k ) of set theory.To prove Claim 1, assume w.l.o.g. ¯ λ > λ . The proof proceeds by inductionon k . The case k = 0 is clear. Let us then assume the claim for k − . Let G be generic for the ≤ ω -closed stationary tower forcing of [11, Section 1] with thegeneric embedding j : V → M, M ω ⊆ M, j ( λ ) = ¯ λ , j (¯ λ i ) = ¯ λ i for i > . A special feature of the ω -closed stationary tower forcing of [11] is that it doesnot introduce new ordinals of cofinality ω . Thus C ∗ V = C ∗ V [ G ] = C ∗ M . The cardinals are indiscernible even if the quantifier Q cf ω is added to the language of set theory. C ∗ | = Φ( β , . . . , β l , λ , λ , . . . , λ k ) . By the induction hypothesis, in M , applied to λ , . . . , λ k and ¯ λ , . . . , ¯ λ k , C ∗ | = Φ( β , . . . , β l , λ , ¯ λ , . . . , ¯ λ k ) . Since j is an elementary embedding, C ∗ | = Φ( β , . . . , β l , ¯ λ , ¯ λ , . . . , ¯ λ k ) . Claim 1 is proved.
Claim 2: If λ < . . . < λ k are Woodin cardinals, κ < . . . < κ k are regularcardinals > ℵ , λ > max( κ , . . . , κ k ) , and β , . . . , β l < κ , then C ∗ | = Φ( β , . . . , β l , κ , . . . , κ k ) ↔ Φ( β , . . . , β l , λ , . . . , λ k ) for all formulas Φ( x , . . . , x l , y , . . . , y k ) of set theory.We use induction on k to prove the claim. The case k = 0 is clear. Let usassume the claim for k − . Using ω -stationary tower forcing we can find j : V → M, M ω ⊆ M, j ( κ ) = λ , j ( λ i ) = λ i for i > . Now we use the Claim to prove the theorem. Suppose now C ∗ | = Φ( β , . . . , β l , κ , κ , . . . , κ k ) . By the induction hypothesis applied to κ , . . . , κ k and λ , . . . , λ k , C ∗ | = Φ( β , . . . , β l , κ , λ , . . . , λ k ) . Since j is an elementary embedding, C ∗ | = Φ( β , . . . , β l , λ , λ , . . . , λ k ) . Claim 2 is proved.The theorem follows now immediately from Claim 2.Note that we cannot extend Theorem 5.15 to ℵ , for ℵ has the followingproperty, recognizable in C ∗ , which no other uncountable cardinal has: it is hasuncountable cofinality but all of its (limit) elements have countable cofinality.34 heorem 5.16. If V = L µ , then C ∗ is exactly the inner model M ω [ E ] , where M ω is the ω th iterate of V and E = { κ ω · n : n < ω } .Proof. In order to prove the theorem, we have to show that in M ω we can recog-nize which ordinals have cofinality ω in V . The following lemma gives a generalanalysis about the relation between the cofinality of the ordinal in a universe andits cofinality in an iterated ultrapower of it. Lemma 5.17.
Let M be a transitive model of ZFC+GCH with a measurable car-dinal κ which is iterable. (Namely the iterated ultrapowers by a normal ultrafilteron κ are all well founded.) For β ∈ On let M β be the β -th iterate of M . Then forevery ordinal δ ∈ M β if cf M ( δ ) < κ then either cf M β ( δ ) = cf M ( δ ) or there is alimit γ ≤ β such that cf M β ( δ ) = cf M ( γ ) .Proof. Let j β,γ be the canonical embedding j βγ : M β → M γ and let κ β = j β ( κ ) .Let ξ β = ( κ + β ) M β and let η = cf M ( ξ ) . As in claim 1 of the proof of Theorem 5.5we can show by induction that cf M ( ξ β ) = η . Claim. cf M ( κ β +1 ) = η . Proof. M β +1 is the ultrapower of M β by a normal ultrafilter on κ β . Since M β | = GCH , it is well known that cf M β ( j β,β +1 ( κ β )) = cf M β ( ξ β ) but since we have cf M ( ξ β ) = η we get cf M ( κ β +1 ) = η .Without loss of generality we can assume that δ (in the formulation of thelemma) is regular in M β . We distinguish several cases : δ ≤ κ We know that in this case the iterated ultrapower does not change thatcofinality of δ . Hence cf M ( δ ) = cf M β ( δ ) . κ < δ ≤ κ β An argument like in the proof of claim 2 of the proof of Theorem5.5 will show that either cf M ( δ ) = η or there is γ ≤ β such that δ = κ γ .The first case cannot occur since we assumed that cf M ( δ ) < κ < η . In thesecond case, if γ is successor or again we get by the previous claim that cf M ( δ ) ≥ κ , contradicting again the assumption. If γ is limit, the lemma isverified. κ β < δ For simplifying notation let j = j β . Every ordinal in M β is of the form j ( F )( κ γ . . . κ γ k ) for some γ < . . . γ k < β and F ∈ M an ordinal valuedfunction defined on κ k +1 . In particular for every ordinal ν in M β there isa function F ∈ M , F : κ <ω → On , such that ν ∈ j ( F )” j ( κ ) <ω . Since35 f M ( δ ) < κ there is in M an ordinal µ < κ and a sequence h F η | η < µ i suchthat for η < µ F η is a function from κ <ω such that the union of the ranges of h j ( F η ) ∩ δ | η < µ i is cofinal in δ . But h j ( F η ) ∩ δ | η < µ i = j ( h F η | η < µ i ) ∈ M β . But the union of the ranges is the union of µ sets each of cardinality ≤ j ( κ ) = κ β . In M β δ is a regular cardinal above j ( κ ) , hence this union isbounded in δ . A contradiction. Corollary. If V | = GCH , κ measurable, then an ordinal has cofinality ω in V iffits cofinality in M ω is either ω or of the form κ γ for some limit γ ≤ ω . From the point of M ω E is a Prikry generic sequence with respect to theimage of µ . Hence the only cardinal of M ω that changes its cofinality is κ ω . Soin M ω it is still true that ordinal has cofinality ω in V iff its cofinality in M ω [ E ] is in { ω } ∪ E ∪ { sup( E ) } . It follows that C ∗ ⊆ M ω [ E ] .For the other direction, let µ ′ be the image of µ in M ω . By Theorem 5.5 weknow that the Dodd-Jensen Core model, K DJ is the same as the Dodd-Jensencore model of C ∗ . V = L µ . Hence by claim 2 of the proof of theorem 5.6, for η regular in K DJ , κ < η ≤ κ ω , cf( η ) = ω iff η = κ γ for some γ ≤ ω . Bythe above lemma we know that for successor γ the ordinal κ γ has cofinality κ + .Hence E is exactly the set of ordinals η which are regular in K DJ , κ ≤ η ≤ κ ω and cf( η ) = ω . This shows that E ∈ C ∗ .It is well known that if we define the filter F on κ ω generated by final segmentof E then M ω = L µ ′ = L [ F ] [15]. Therefore M ω is a definable class in C ∗ . Weconclude M ω ⊆ C ∗ and hence finally, M ω [ E ] ⊆ C ∗ .The situation is similar with the inner model for two measurable cardinals: Toget C ∗ we first iterate the first measurable ω times, then the second ω times, andin the end take two Prikry sequences.We now prove the important property of C ∗ that its truth is invariant under(set) forcing. We have to assume large cardinals because conceivably C ∗ couldsatisfy V = L but in a (set) forcing extension C ∗ would violate V = L (seeSection 6 below). Theorem 5.18.
Suppose there is a proper class of Woodin cardinals. Suppose P is a forcing notion and G ⊆ P is generic. ThenTh (( C ∗ ) V ) = Th (( C ∗ ) V [ G ] ) . oreover, the theory Th ( C ∗ ) is independent of the cofinality used , and forcingdoes not change the reals of these models.Proof. Let G be P -generic. Let us choose a Woodin cardinal λ > | P | . Let H be generic for the countable stationary tower forcing Q <λ . In V [ H ] there is ageneric embedding j : V → M such that V [ H ] | = M ω ⊆ M and j ( ω ) = λ .Hence ( C ∗ ) V [ H ] = ( C ∗ ) M and j : ( C ∗ ) V → ( C ∗ ) M = ( C ∗ ) V [ H ] = ( C ∗ <λ ) V . The last equality uses the fact that an ordinal has cofinality ω in V [ H ] iff it hascofinality < λ in V . Now by elementarity Th (( C ∗ ) V ) = Th (( C ∗ <λ ) V ) .Since | P | < λ , λ is still Woodin in V [ G ] . Let H be generic for the countablestationary tower forcing Q <λ over V [ G ] . Let j : V [ G ] → M be the genericembedding. Now V [ G, H ] | = M ω ⊆ M and j ( ω ) = λ . Hence j : ( C ∗ ) V [ G ] → ( C ∗ ) M = ( C ∗ ) V [ G,H ] = ( C ∗ <λ ) V [ G ] = ( C ∗ <λ ) V . and therefore by elementarity ( C ∗ ) V ≡ ( C ∗ <λ ) V ≡ ( C ∗ ) V [ G ] .We know (Theorem 5.13) that under the existence of a Woodin cardinal theset of reals of C ∗ is countable (in V ). Hence when we define j : V → M thereare no new reals added to the C ∗ of the corresponding models. Hence the realsof ( C ∗ ) V are the same as the reals of ( C ∗ ) M . We argued that the last model isexactly ( C ∗ <λ ) V . The same is true in V [ G ] . But C ∗ <λ does not change when wemove from V to V [ G ] . So ( C ∗ ) V and ( C ∗ ) V [ G ] have the same reals.The argument for the elementary equivalence of C ∗ and C ∗ <κ for κ regular,proceeds in a similar manner. We use stationary tower forcing which produces anelementary embedding j with critical point κ such that j ( κ ) = λ , where λ is aWoodin cardinal above κ . Then we argue that j ( C ∗ <κ ) is ( C ∗ <λ ) V .We may ask, for which λ and µ is C ∗ <λ = C ∗ <µ ? Observations: • It is possible that C ∗ = C ∗ <ω . Let us use the ≤ ω -closed stationary towerforcing of [11, Section 1] to map ω to λ . In this model V the inner model C ∗ is preserved. It is easy to see that in the extension the set A of ordinalsbelow λ of cofinality ω is not in V . If C ∗ = C ∗ <ω , then A is in C ∗ <ω . Weare done. I.e. Th ( C ∗ ) = Th ( C ∗ <κ ) for all regular κ . It is possible that C ∗ changes. Extend the previous model V to V by col-lapsing ω to ω . Then ( C ∗ ) V = ( C ∗ <ω ) V = ( C ∗ ) V . So C ∗ has changed. • Question: Does a Woodin cardinal imply C ∗ <ω = C ∗ ω,ω ?We do not know whether the CH is true or false in C ∗ . Forcing absolutenessof the theory of C ∗ under the hypothesis of large cardinals implies, however, thatlarge cardinals decide the CH in C ∗ in forcing extensions. This would seem togive strong encouragement to try to solve the problem of CH in C ∗ . The situationis in sharp contrast to V itself where we know that large cardinals definitely do notdecide CH [20]. We can at the moment only prove that the size of the continuumof C ∗ is at most ω V . In the presence of a Woodin cardinal this tells us absolutelynothing, as then ω V is (strongly) Mahlo in C ∗ (Theorem 5.13), and hence certainlyfar above the continuum of C ∗ . So the below result is mainly interesting becauseit is a provable result of ZFC, independent of whether we assume the existence ofWoodin cardinals. However, we show later that in the presence of large cardinalsthere is a cone of reals x such that the relativized version of C ∗ , C ∗ ( x ) , satisfies CH . In the light of this it is tempting to conjecture that CH is indeed true in C ∗ ,assuming again the existence of sufficiently large cardinals. Theorem 5.19. |P ( ω ) ∩ C ∗ | ≤ ℵ . Proof.
We use the notation of Definition 2.2. Suppose a ⊆ ω and a ∈ L ′ ξ for some ξ . Let µ > ξ be a sufficiently large cardinal. We build an increasing elementarychain ( M α ) α<ω such that1. a ∈ M and M | = a ∈ C ∗ .2. | M α | ≤ ω .3. M α ≺ H ( µ ) .4. M γ = S α<γ M α , if γ = ∪ γ .5. If β ∈ M α and cf V ( β ) = ω , then M α +1 contains an ω -sequence from H ( µ ) ,cofinal in β .6. If β ∈ M α and cf V ( β ) > ω then for unboundedly many γ < ω there is ρ ∈ M γ +1 with sup( [ ξ<γ ( M ξ ∩ β )) < ρ < β. M be S α<ω M α , N the transitive collapse of M , and ζ the ordinal N ∩ On .Note that | N | ≤ ω , whence ζ < ω . By construction, an ordinal in N hascofinality ω in V if and only if it has cofinality ω in N . Thus ( L ′ ξ ) N = L ′ ξ for all ξ < ζ . Since N | = a ∈ C ∗ , we have a ∈ L ′ ζ . The claim follows.The proof of Theorem 5.19 gives the following more general result: Theorem 5.20.
Let κ be a regular cardinal and δ an ordinal. Then |P ( δ ) ∩ C ∗ κ | ≤ ( | δ | · κ + ) + . Corollary. If δ ≥ κ + is a cardinal in C ∗ κ and λ = | δ | + , then C ∗ κ | = 2 δ ≤ λ . Corollary.
Suppose V = C ∗ . Then ℵ α = ℵ α +1 for α ≥ , and ℵ = ℵ or ℵ = ℵ . Theorem 5.21.
Suppose E = { α < ω V : cf V ( α ) = ω V } . Then ♦ ℵ V ( E ) holds in C ∗ .Proof. The proof is as the standard proof of ♦ ℵ ( E ) in L , with a small necessarypatch. We construct a sequence s = { ( S α , D α ) : α < ℵ V } taking always forlimit α the pair ( S α , D α ) to be the least ( S, D ) ∈ L ′ℵ V in the well-order (seeProposition 2.6) R = { ( a, b ) ∈ ( L ′ℵ V ) : L ′ℵ V | = Ψ L ( Q cf ω ) ( a, b ) } such that S ⊆ α , D ⊆ α a club, and S ∩ β = S β for β ∈ D , if any exists, and S α = D α = α otherwise. Note that s ∈ C ∗ . We show that the sequence s is adiamond sequence in C ∗ . Suppose it is not and ( S, D ) ∈ C ∗ is a counter-example, S ⊆ ℵ V and D ⊆ ℵ V club such that S ∩ β = S β for all β ∈ D . As in the proof ofTheorem 5.19 we can construct M ≺ H ( µ ) such that | M | = ℵ V , the order-type of M ∩ ℵ V is in E , { s, ( S, D ) } ⊂ M , and if N is the transitive collapse of M , withordinal δ ∈ E , then { s ↾ δ, ( S ∩ δ, D ∩ δ ) } ⊂ N and ( L ′ ξ ) N = L ′ ξ for all ξ < δ .Because of the way M is constructed, the well-order R restricted to L ′ δ is definedin M on L ′ℵ V by the same formula Ψ L ( Q cf κ ) ( x, y ) as R is defined on L ′ℵ V in H ( µ ) .Since S ∩ δ ∈ L ′ℵ V and S ∩ β = S β for β ∈ D ∩ δ , we may assume, w.l.o.g., that ( S, D ) ∈ L ′ℵ V . Furthermore, we may assume, w.l.o.g., that ( S, D ) is the R -leastcounter-example to s being a diamond sequence. Thus the pair ( S ∩ δ, D ∩ δ ) isthe R -least ( S ′ , D ′ ) such that S ′ ⊆ δ , D ′ ⊆ δ a club, and S ′ ∩ β = S ′ β for β ∈ D ′ .It follows that ( S ′ , D ′ ) = ( S δ , D δ ) and, since δ ∈ D , a contradiction.39 problem in using condensation type arguments, such as we used in theproofs of Theorem 5.19 and Theorem 5.21 above, is the non-absoluteness of C ∗ .There is no reason to believe that ( C ∗ ) C ∗ = C ∗ in general (see Theorem 6.3).Moreover, we prove in Theorem 6.7 the consistency of C ∗ failing to satisfy CH,relative to the consistency of an inaccessible cardinal.We now prove that CH holds in C ∗ ( y ) for a cone of y . By C ∗ ( y ) we meanthe extension of C ∗ in which the real y is allowed as a parameter throughout theconstruction.Suppose N is a well-founded model of ZFC − and N thinks that λ ∈ M is aWoodin cardinal. We say that N is iterable , if all countable iterations of forminggeneric ultrapowers of N by stationary tower forcing at λ are well-founded. If N ≺ H ( θ ) for large enough θ and N contains a measurable cardinal (of V ) abovethe Woodin cardinal, then it is iterable for the following reason: Suppose α < ω .Suppose N ∗ is an iteration of N at the measurable cardinal until α ≤ N ∗ ∩ On .This is well-founded because it can be embedded into a long enough iterationof V at the measurable cardinal. It is well known that an iteration of length α ≤ N ∗ ∩ On of forming generic ultrapowers of N ∗ by stationary tower forc-ing is well founded (see e.g. [42, Lemma 4.5]). Now the iteration of forminggeneric ultrapowers of N by stationary tower forcing can be embedded to the cor-responding iteration of N ∗ . Since the latter iteration is well-founded, so is theformer.We use the notation L ′ α ( y ) for the levels of the construction of C ∗ ( y ) . Lemma 5.22.
Suppose there is Woodin cardinal and a measurable cardinal aboveit. Suppose a ⊆ γ < ω V . Then the following conditions are equivalent: (i) a ∈ C ∗ ( y ) . (ii) There is a countable transitive iterable model N of ZF C − +“there is a Woodincardinal” such that { a, y } ⊂ N , γ < ω N , and N | = “ a ∈ C ∗ ( y )” .Proof. (i) → (ii): Suppose first a ∈ C ∗ ( y ) . Pick a large enough θ and a countable M ≺ H θ such that γ ∪ { γ, a, y } ⊂ M and both the Woodin cardinal and themeasurable above it are in M . Then M is iterable. Let π : M ∼ = N with N transitive. This N is as required in (ii). In particular, π ( a ) = a, π ( y ) = y and N | = “ a ∈ C ∗ ( y )” since M ≺ H θ .(ii) → (i): Suppose N is as in (ii). Since N | = “ a ∈ C ∗ ( y )” , there is ¯ β < ω N such that N | = a ∈ L ′ ¯ β ( y ) . We form an iteration sequence { N α : α < ω V } withelementary embeddings { π αβ : α < β < ω } . Let N = N . Let N γ +1 be the40ransitive collapse of a generic ultrapower of the stationary tower on the image of λ in N γ . Let π αα +1 be the canonical embedding N α → N α +1 . For limit α ≤ ω V ,the model N α is the transitive collapse of the direct limit of the models N β , β < α ,under the mappings π βγ , β < γ < α . By the iterability condition each N α is well-founded, so the transitive collapse exists. Since π γ ( ω N ) is extended in each stepof this iteration of length ω of countable models, π ω ( ω N ) = ω V . Moreover, π ω ( y ) = y and π ω ( a ) = a , as a ⊆ γ and γ < ω N . Now by elementarity, N ω | = “ a ∈ L ′ β ( y )” , where β = π ω ( ¯ β ) .We now show ( L ′ β ( y )) N ω = ( L ′ β ( y )) V . (9)This is proved level by level. If N ω | = “ cf( δ ) = ω ” , then of course cf( δ ) = ω .Suppose then N ω | = “ cf( δ ) > ω ” , where δ < β . Let ¯ δ < ¯ β such that π ω (¯ δ ) = δ . Then ¯ δ < ω N . Thus N | = cf(¯ δ ) = ω . By elementarity, N ω | = cf( δ ) = ω .But ω N ω = ω V . Hence cf( δ ) = ω V . Equation (9) is proved.Now we can prove (i): Since a ∈ ( L ′ β ( y )) N ω , equation (9) implies a ∈ L ′ β ( y ) ⊂ C ∗ ( y ) .Note that condition (ii) above is a Σ -condition. Thus, if there is a Woodincardinal and a measurable above, then the set of reals of C ∗ is a countable Σ -setwith a Σ -well-ordering. Lemma 5.23.
Suppose there is a Woodin cardinal and a measurable cardinalabove it. Then the following conditions are equivalent: (i) C ∗ ( y ) | = CH . (ii) There is a countable transitive iterable model M of ZF C − plus “there is aWoodin cardinal” such that y ∈ M , M | = “ C ∗ ( y ) | = CH ” , and ( P ( ω ) ∩ C ∗ ( y )) M = P ( ω ) ∩ C ∗ ( y ) .Proof. (i) → (ii): Since we assume the existence of a Woodin cardinal, there areonly countably many reals in C ∗ ( y ) . Let γ = ω C ∗ ( y )1 . Thus γ < ω V . By (i) wemay find a subset a ∈ C ∗ ( y ) of γ that codes an enumeration of P ( ω ) ∩ C ∗ ( y ) in order-type γ together with a well-ordering of ω of each order-type < γ . By41emma 5.22 there is a countable transitive iterable model N of ZF C − +“ there isa Woodin cardinal ” such that { a, y } ⊂ N , γ < ω N and N | = “ a ∈ C ∗ ( y )” .We show N | = “ C ∗ ( y ) | = CH ” . Suppose b ∈ N is real such that N | = “ b ∈ C ∗ ( y )” . By Lemma 5.22, b ∈ C ∗ ( y ) . Hence b is coded by a . The length of thesequence a is γ , so we only have to show that N | = “ γ ≤ ω C ∗ ( y )1 ” . Suppose N | = “ γ > ω C ∗ ( y )1 ” . In such a case, by assumption, a codes a well-ordering R of ω of order-type ( ω C ∗ ( y )1 ) N . But N | = “ a ∈ C ∗ ( y )” , whence N | = “ R ∈ C ∗ ( y )” , acontradiction. The proof that ( P ( ω ) ∩ C ∗ ( y )) N = P ( ω ) ∩ C ∗ ( y ) is similar.Assume then (ii). Let N be as in (ii). Let γ = ( ω C ∗ ( y )1 ) N . Since N | =“ C ∗ ( y ) | = CH ” , we can let some a ⊆ γ code ( P ( ω ) ∩ C ∗ ( y )) N as a sequence oforder-type γ . By lemma 5.22, a ∈ C ∗ ( y ) . Since ( P ( ω ) ∩ C ∗ ( y )) N = P ( ω ) ∩ C ∗ ( y ) , a is an enumeration of all the reals in C ∗ ( y ) and γ = ω C ∗ ( y )1 . Hence C ∗ ( y ) | = CH .Note that condition (ii) above is a Σ -condition. Also, forgetting y , “ C ∗ | = CH ” itself is a Σ -sentence of set theory.Using the above two Lemmas, we now prove a result which seems to lendsupport to the idea that C ∗ satisfies CH, at least assuming large cardinals. Let ≤ T be the Turing-reducibility relation between reals. The cone of a real x is the setof all reals y with x ≤ T y . A set of reals is called a cone if it is the cone of somereal. Suppose A is a projective set of reals closed under Turing-equivalence. If weassume PD, then by a result of D. Martin [26] there is a cone which is included in A or is disjoint from A . Theorem 5.24.
If there are three Woodin cardinals and a measurable cardinalabove them, then there is a cone of reals x such that C ∗ ( x ) satisfies the ContinuumHypothesis.Proof. We first observe that if two reals x and y are Turing-equivalent, then C ∗ ( x ) = C ∗ ( y ) . Hence the set A = { y ⊆ ω : C ∗ ( y ) | = CH } is closed under Turing-equivalence, and therefore by [27] amenable to the abovementioned result by Martin on cones. We already know from Lemma 5.23 that theset A is projective, in fact Σ . Now we need to show that for every real x there isa real y such that x ≤ T y and y is in the set. Fix x . Let P be the standard forcingwhich, in C ∗ ( x ) , forces a subset B of ω C ∗ ( x )1 , such that B codes, via the canonical42airing function in C ∗ ( x ) , an onto mapping ω C ∗ ( x )1 → P ( ω ) ∩ C ∗ ( x ) . Let B be P -generic over C ∗ ( x ) . Note that P does not add any new reals. Now we code B bya real by means of almost disjoint forcing. Let Z α , α < ω C ∗ ( x )1 , be a sequence in C ∗ ( x ) of almost disjoint subsets of ω . Let Q be the standard CCC-forcing, knownfrom [25], for adding a real y ′ such that for all α < ω C ∗ ( x )1 : | z α ∩ y ′ | ≥ ω ⇐⇒ α ∈ B. Let y = x ⊕ y ′ . Of course, x ≤ T y . Now C ∗ ( x ) ⊆ C ∗ ( x )[ B ] ⊆ C ∗ ( y ) . By the definition of B , C ∗ ( x )[ B ] | = CH . The forcing Q is of cardinality ℵ in C ∗ ( x )[ B ] , hence C ∗ ( y ) | = CH .Assuming large cardinals, the set of reals of C ∗ seems like an interestingcountable Σ -set with a Σ -well-ordering. It might be interesting to have a betterunderstanding of this set. This set is contained in the reals of the so called M ♯ ,the smallest inner model for a Woodin cardinal (M. Magidor and R. Schindler,unpublished).In Part 2 of this paper we will consider the so-called stationary logic [6], astrengthening L ( aa ) of L ( Q cf ω ) , and the arising inner model C ( aa ) , a supermodelof C ∗ . We will show that, assuming a proper class of measurable Woodin cardi-nals, uncountable regular cardinals are measurable in C ( aa ) , and the theory of C ( aa ) is absolute under set forcing. These results remain true if we enhance theexpressive power of L ( aa ) slightly, and then the inner model arising from theenhanced stationary logic satisfies the Continuum Hypothesis, assuming again aproper class of measurable Woodin cardinals. C ∗ We define a version of Namba forcing that we call modified Namba forcing andthen use this to prove consistency results about C ∗ .Suppose S = { λ n : n < ω } is a sequence of regular cardinals > ω such thatevery λ n occurs infinitely many times in the sequence. Let h B n : n < ω i be apartition of ω into infinite sets. 43 efinition 6.1. The forcing P is defined as follows: Conditions are trees T with ω levels, consisting of finite sequences of ordinals, defined as follows: If ( α , . . . , α i ) ∈ T , let Suc T (( α , . . . , α i )) = { β : ( α , . . . , α i , β ) ∈ T } . The forcing P consists of trees, called S -trees, such that if ( α , . . . , α i ) ∈ T and i ∈ B n , then1. | Suc T (( α , . . . , α i − )) | ∈ { , λ n } ,2. For every n there are α i , . . . , α k such that k ∈ B n and | Suc T (( α , . . . , α k )) | = λ n .If | Suc T (( α , . . . , α i − )) | = λ n , we call ( α , . . . , α i − ) a splitting point of T .Otherwise ( α , . . . , α i − ) is a non-splitting point of T . The stem stem ( T ) of T is the maximal (finite) initial segment that consists of non-splitting points. If s = ( α , . . . , α i ) ∈ T , then T s = { ( α , . . . , α i , α i +1 , . . . , α n ) ∈ T : i ≤ n < ω } . A condition T ′ extends another condition T , T ′ ≤ T , if T ′ ⊆ T . If h T n : n < ω i} is a generic sequence of conditions, then the stems of the trees T n form a sequence h α n : n < ω i such that h α i : i ∈ B n i is cofinal in λ n . Thus in the generic extension cf( λ n ) = ω for all n < ω .We shall now prove that no other regular cardinals get cofinality ω . Proposition 6.2.
Suppose κ / ∈ S ∪ { ω } is regular. Then P (cid:13) cf( κ ) = ω .Proof. Let us first prove that if τ is a name for an ordinal, then for all T ∈ P there is T ∗ ≤ T such that stem ( T ∗ ) = stem ( T ) and if T ∗∗ ≤ T ∗ decides whichordinal τ is, and s = stem ( T ∗∗ ) , then T ∗ s decides τ . Suppose T is given and thelength of its stem is l ∈ B n . Let us look at the level l + 1 of T . Let us call a node s on level l + 1 of T good if the claim is true when T is taken to be T s . Supposefirst there are λ n good nodes. For each good s we choose T ∗ ( s ) ≤ T s such thatstem ( T ∗ ( s )) = stem ( T s ) and if some T ∗∗ ≤ T ∗ ( s ) decides which ordinal τ is,and s ′ = stem ( T ∗∗ ) , then already T ∗ ( s ) s ′ decides τ . W.l.o.g. the length of such s ′ is a fixed k . We get the desired T ∗ by taking the fusion. Suppose then there arenot λ n many good nodes. So there must be λ n bad nodes. We repeat this processon the next level. Suppose the process does not end. We get T ′ ≤ T consistingof bad nodes. Since T forces that τ is an ordinal, there is T ′′ ≤ T ′ such that T ′′ τ is. We get a contradiction: the node of the stem of T ′′ ,which is also a node of T ′ , cannot be a bad one.Suppose now h β n : n < ω i is a name for an ω -sequence of ordinals below κ ,and T ∈ P forces this. We construct T ∗ ≤ T and an ordinal δ < κ such that T ∗ forces the sequence h β n : n < ω i to be bounded below κ by δ . For each n wehave a partial function f n defined on s ∈ T of such that if T s decides a value for β n and then the value is defined to be f n ( s ) . Let us call T good for h β n : n < ω i if for all infinite branches B through T and all n there is k such that f n restrictedto the initial segment of B of length k is defined. It follows from the above thatwe can build, step by step a T ∗ ≤ T with the same stem as T such that T ∗ is goodfor h β n : n < ω i .Without loss of generality, T itself is good for h β n : n < ω i . Fix δ < κ .We consider the following game G δ . During the game the players determine aninfinite branch through T . If the game has reached node t on height k with k + 1 ∈ B n we consider two cases:Case 1: κ > λ n . Bad moves by giving an immediate successor of t .Case 2: κ < λ n . First Bad plays a subset A of (not necessarily immediate)successors of t such that | A | < λ n . Then Good moves a successor not in theset.Good player loses this game if at some stage of the game a member of the se-quence β n is forced to go above δ . Note that the game is determined. Main Claim:
There is δ < κ such that Bad does not win G δ (hence Good wins). Proof.
Assume the contrary, i.e. that Bad wins for all δ < κ . Let τ δ be a strategyfor Bad for any given δ < κ . Let θ be a large enough cardinal and M ≺ H θ suchthat T, P , { β n : n < ω } , { λ n : n < ω } , { ( δ, τ δ ) : δ < κ } etc are in M , α ⊆ M whenever α ∈ M ∩ κ and | M | < κ . Let δ = M ∩ κ . We define a play of G δ whereBad uses τ δ but all the individual moves are in M . Suppose we have reached anode t of T such that len( t ) = k and k + 1 ∈ B n . If κ > λ n , λ n ⊆ M , so themove of Bad is in M . Suppose then κ < λ n . The strategy τ δ tells Bad to play aset A of successors (not necessarily immediate) of t such that | A | < λ n . The nextmove of Good has to avoid this set A . Still we want the move of Good to be in M . We look at all the possibilities according to all the strategies τ ν , ν < κ . If theplay according to τ ν has reached t the strategy τ ν gives a set A ν of size < λ n ofsuccessors (not necessarily immediate) of t . Let B be the union of all these sets.Still | B | < λ n , as λ n is regular. By elementarity, B ∈ M , hence Good can playa successor of t staying in M . Since Bad is playing the winning strategy τ δ , he45hould win this play. However, Good can play all the moves inside M withoutlosing. This is a contradiction.Now we return to the main part of the proof. By the Main Claim there is δ suchthat Good wins G δ . Let us look at the subtree of all plays of G δ where Good playsher winning strategy. A subtree T ∗ of T is generated and T ∗ forces the sequence h β n : n < ω i to be bounded by δ .The above modified Namba forcing permits us to carry out the following basicconstruction: Suppose V = L . Let us add a Cohen real r . We can code this realwith the above modified Namba forcing so that in the end for all n < ω : cf V ( ℵ Ln +2 ) = ω ⇐⇒ n ∈ r. Thus in the extension r ∈ C ∗ . Theorem 6.3.
Con ( ZF ) implies Con (( C ∗ ) C ∗ ) = C ∗ ) .Proof. We start with V = L . We add a Cohen real a . In the extension C ∗ = L , for cofinalities have not changed, so to decide whether cf( α ) = ω or not itsuffices to decide this in L . With modified Namba forcing we can change—asabove—the cofinality of ℵ Ln +2 to ω according to whether n ∈ a or n / ∈ a . In theextension C ∗ = L ( a ) , for cofinality ω has only changed from L to the extent thatthe cofinalities of ℵ Ln may have changed, but this we know by looking at a . Thus ( C ∗ ) C ∗ = ( C ∗ ) L ( a ) = L , while C ∗ = L . Thus ( C ∗ ) C ∗ = C ∗ .We now prepare ourselves to iterating this construction in order to code moresets into C ∗ . Definition 6.4 (Shelah) . Suppose S = { λ n : n < ω } is a sequence of regularcardinals > ω . A forcing notion P satisfies the S -condition if player II hasa super strategy (defined below) in the following came G in which the playerscontribute a tree of finite sequences of ordinals:1. There are two players I and II and ω moves.2. In the start of the game player I plays a tree T of finite height and a function f : T → P such that for all t, t ′ ∈ T : t < T t ′ ⇒ f ( t ′ ) < P f ( t ) .3. Then II decides what the successors of the top nodes of T are and extends f . 46. Player I extends the tree with non-splitting nodes of finite height and extends f .5. Then II decides what the successors of the top nodes are and extends f .6. etc, etcPlayer II wins if the resulting tree T is an S -tree (see Definition 6.1), and for every S -subtree T ∗ of T there is a condition B ∗ ∈ P such that B ∗ (cid:13) “ The f -image of some branch through T ∗ is included in the generic set ” . A super strategy of II is a winning strategy in which the moves depend only onthe predecessors in T of the current node, as well as on their f -images.By [35, Theorem 3.6] (see also [8, 2.1]), revised countable support iteration offorcing with the S -condition does not collapse ℵ . Lemma 6.5.
Modified Namba forcing satisfies the S -condition.Proof. Suppose the game has progressed to the following:1. A tree T has been constructed, as well as f : T → P .2. Player I has played a non-splitting end-extension T ′ of T .Suppose η is a maximal node in T ′ . We are in stage n . Now II adds λ n extensionsto η . Let E denote these extensions. Let B be the S -tree f ( η ) . Find a node ρ in B which is a splitting node and splits into λ n nodes. Let g map the elements of E ρ in B . Now we extend f to E by letting the image of e ∈ E be the subtree B η of B consisting of ρ and the predecessors of ρ extended by first g ( e ) and then the subtree of B above g ( e ) .We can easily show that this is a super strategy. We show that II wins. Suppose T is a tree resulting from II playing the above strategy. Let T ∗ be any S -subtreeof T . We construct an S -tree B ∗ ∈ P as follows. Let B ∗ be the union of all thestems of the trees B η , where η is a splitting point of T ∗ . Clearly, B ∗ is an S -tree.To see that B ∗ (cid:13) “ The f -image of some branch through T ∗ is included in the generic set ” , let G be a generic containing B ∗ . This generic is a branch γ through B ∗ . In viewof the definition of B ∗ , there is a branch β through T ∗ such that f “ β = γ .47 heorem 6.6. Suppose V = L and κ is a cardinal of cofinality > ω . There is aforcing notion P which forces C ∗ | = 2 ω = κ and preserves cardinals between L and C ∗ .Proof. Suppose V = L . Let us add κ Cohen reals { r α : α < κ } . We code thesereals with revised countable support (see [35]) iterated modified Namba forcingso that in the end we have a forcing extension in which for α < κ and n < ω : cf V ( ℵ Lω · α + n +2 ) = ω ⇐⇒ n ∈ r α . Thus in the extension r α ∈ C ∗ for all α < κ . We can now note that in theextension C ∗ = L [ { r α : α < κ } ] . First of all, each r α is in C ∗ . This gives “ ⊇ ” .For the other direction, we note that whether an ordinal has cofinality ω in V canbe completely computed from the set { r α : α < κ } .Note that the above theorem gives a model in which, e.g. C ∗ | = 2 ω = ℵ , butthen in the extension |ℵ C ∗ | = ℵ , so certainly V = C ∗ . Note also, that the abovetheorem starts with V = L , so whether large cardinals, beyond those consistentwith V = L , decide CH in C ∗ , remains open. Theorem 6.7.
The following conditions are equivalent: (i)
ZF+“there is an inaccessible cardinal” is consistent. (ii)
ZFC+“ V = C ∗ and ℵ = ℵ ” is consistent.Proof. (i) → (ii): We start with an inaccessible κ and V = L . We iterate over κ with revised countable support forcing adding Cohen reals and coding genericsets using modified Namba forcing. Suppose we are at a stage α and we need tocode a real r . We choose ω uncountable cardinals below κ and code the real r bychanging the cofinality of some of these cardinals to ω . We do this only if at stage α we already have enough reals in order to code the new ω -sequences by reals. Inthe end all the reals are coded by changing cofinalities to ω , and at the same timethe ω -sequences witnessing the cofinalities are coded by reals. In consequencewe have in the end V = C ∗ . The iteration satisfies the S -condition, hence ℵ ispreserved, but the cardinals used for coding the reals all collapse to ℵ . Hence κ is the new ℵ . In the extension ℵ = ℵ and V = C ∗ .(ii) → (i): Suppose V = C ∗ and C ∗ | = “2 ω ≥ ω V ” but ω V is not inaccessible in L . Then ω V = ( λ + ) L for some L -cardinal λ . Let A ⊆ ω V such that A codes the48ountability of all ordinals < ω V and also codes a well-ordering of ω of order-type λ . Now L [ A ] | = ω V = ω ∧ ω V = ω . We show now that ( C ∗ ) L [ A ] = C ∗ .For this to hold it suffices to show that L [ A ] agrees with V about cofinality ω .If α has cofinality ω in L [ A ] , then trivially it has cofinality ω in V . Supposethen L [ A ] | = cf( α ) > ω . If α < ω V , then L [ A ] | = α < ω , whence L [ A ] | =cf( α ) = ω . Since ω L [ A ]1 = ω V , we obtain cf( α ) = ω . Suppose therefore α ≥ ω V , but cf V ( α ) = ω . Note that we can assume ¬ ♯ , because otherwise ω V isinaccessible in L already by the general properties of ♯ . By the Covering Lemma,a consequence of ¬ ♯ , we have L [ A ] | = cf( α ) ≤ ω . Since L [ A ] | = cf( α ) > ω ,we obtain L [ A ] | = cf( α ) = ω , and since ω L [ A ]1 = ω , we have cf V ( α ) = ω . Thisfinishes our proof that ( C ∗ ) L [ A ] = C ∗ . Note that L [ A ] satisfies CH . On the otherhand, we have assumed that there are ℵ V reals in C ∗ . Thus there are ℵ reals in ( C ∗ ) L [ A ] ⊆ L [ A ] , a contradiction. The basic result about higher order logics, proved in [32], is that they give riseto the inner model HOD of hereditarily ordinal definable sets. In this section weshow that this result enjoys some robustness, i.e. ostensibly much weaker logicsthan second order logic still give rise to HOD.
Theorem 7.1 (Myhill-Scott [32]) . C ( L ) = HOD .Proof.
We give the proof for completeness. We show HOD ⊆ C ( L ) . Let X ∈ HOD. There is a first order ϕ ( x, ~y ) and ordinals ~β such that for all aa ∈ X ⇐⇒ ϕ ( a, ~β ) . By Levy Reflection there is an α such that X ⊆ V α and for all a ∈ V α a ∈ X ⇐⇒ V α | = ϕ ( a, ~β ) . Since we proceed by induction, we may assume X ⊆ C ( L ) . Let γ be such that X ⊆ L ′ γ . We can choose γ so big that | L ′ γ | ≥ | V α | . We show now that X ∈ L ′ γ +1 .We give a second order formula Φ( x, y, ~z ) such that X = { a ∈ L ′ γ : L ′ γ | = Φ( a, α, ~β ) } . We know X = { a ∈ L ′ γ : V α | = ϕ ( a, ~β ) } . X is the set of a ∈ L ′ γ such that in L ′ γ some ( M, E, a ∗ , α ∗ , ~β ∗ ) ∼ =( V α , ∈ , a, α, ~β ) satisfies ϕ ( a ∗ , ~β ∗ ) . Let θ ( x, y, ~z ) be a second order formula of thevocabulary { E } such that for any M , E ⊆ M and a ∗ , α ∗ , ~β ∗ ∈ M : ( M, E ) | = θ ( a ∗ , α ∗ , ~β ∗ ) iff there are an isomorphism π : ( M, E ) ∼ = ( V δ , ∈ ) such that π :( α ∗ , E ) ∼ = ( δ, ∈ ) , and ( V δ , ∈ ) | = ϕ ( π ( a ∗ ) , π ( ~β )) .We conclude X ∈ L ′ γ +1 by proving the: Claim
The following are equivalent for a ∈ L ′ γ : (1) a ∈ X . (2) L ′ γ | = ∃ M, E ( TC ( { a } ) ∪ α + 1 ∪ ~β ∪ { ~β } ⊆ M ∧ ( M, E ) | = θ ( a, α, ~β )) } . (1) → (2) : Suppose a ∈ X . Thus V α | = ϕ ( a, ~β ) . Let M ⊆ L ′ γ and E ⊆ M suchthat α + 1 , TC ( a ) , ~β ∈ M and there is an isomorphism f : ( V α , ∈ , α, a, ~β ) ∼ = ( M, E, α ∗ , a ∗ , ~β ∗ ) . We can assume α ∗ = α , a ∗ = a and ~β ∗ = ~β by doing a partial Mostowski collapsefor ( M, E ) . So then ( M, E ) | = ϕ ( a, ~β ) , whence ( M, E ) | = θ ( a, α, ~β ) . We haveproved (2). (2) → (1) : Suppose M ⊆ L ′ γ and E ⊆ M such that TC ( { a } ) ∪ α +1 ∪ ~β ∪{ ~β } ⊆ M and ( M, E ) | = θ ( a, α, ~β ) . We may assume E ↾ TC ( { a } ) ∪ α +1 ∪ ~β ∪{ ~β } = ∈ ↾ TC ( { a } ) ∪ α + 1 ∪ ~β ∪ { ~β } . There is an isomorphism π : ( M, E ) ∼ = ( V α , ∈ ) suchthat ( V α , ∈ ) | = ϕ ( π ( a ) , π ( ~β )) . But π ( a ) = a and π ( ~β ) = ~β . So in the end ( V α , ∈ ) | = ϕ ( a, ~β ) . We have proved (1).In second order logic L one can quantify over arbitrary subsets of the domain.A more general logic is obtained as follows: Definition 7.2.
Let F be any class function on cardinal numbers. The logic L ,F is like L except that the second order quantifiers range over a domain M oversubsets of M of cardinality ≤ κ whenever F ( κ ) ≤ | M | .Examples of possible functions are F ( κ ) = 0 , κ , κ + , κ , ℵ κ , i κ , etc. Notethat L = L ,F whenever F ( κ ) ≤ κ for all κ . The logic L ,F is weaker thebigger values F ( κ ) takes on. For example, if F ( κ ) = 2 κ , the second ordervariables of L ,F range over “tiny” subsets of the universe. Philosophically second50rder logic is famously marred by the difficulty of imagining how a universallyquantified variable could possibly range over all subsets of an infinite domain. Ifthe universally quantified variable ranges only over “tiny” size subsets, one canconceivably think that there is some coding device which uses the elements of thedomain to code all the “tiny” subsets.Inspection of the proof of Theorem 7.1 reveals that actually the following moregeneral fact holds: Theorem 7.3.
For all F : C ( L ,F ) = HOD . Let L κ denote the modification of L in which the second order variables rangeover subsets (relations, functions, etc) of cardinality at most κ . Theorem 7.4.
Suppose ♯ exists. Then ♯ ∈ C ( L κ ) Proof.
As in the proof of Theorem 5.3.A consequence of Theorem 7.3 is the following:
Conclusion:
The second order constructible hierarchy C ( L ) = HOD is unaf-fected if second order logic is modified in any of the following ways: • Extended in any way to a logic definable with hereditarily ordinal definableparameters. This includes third order logic, fourth order logic, etc. • Weakened by allowing second order quantification in domain M only oversubsets X such that | X | ≤ | M | . • Weakened by allowing second order quantification in domain M only oversubsets X such that | X | ≤ | M | . • Any combination of the above.Thus G¨odel’s HOD = C ( L ) has some robustness as to the choice of the logic L .It is the common feature of the logics that yield HOD that they are able to expressquantification over all subsets of some part of the universe the size of which isnot a priori bounded. We can perhaps say, that this is the essential feature ofsecond order logic that results in C ( L ) being HOD. What is left out are logicsin which one can quantify over, say all countable subsets. Let us call this logic L ℵ . Consistently , C ( L ℵ ) = HOD. Many would call a logic such as L ℵ secondorder. Assume V = L and add a Cohen subset X of ω . Now code X into HOD with countablyclosed forcing using [28]. In the resulting model C ( L ℵ ) = L = HOD. Σ n denote the fragment of second order logic in which the formulas have,if in prenex normal form with second order quantifiers preceding all first orderquantifiers, only n second order quantifier alternations, the first second orderquantifier being existential. Note that trivially C (Σ n ) = C (Π n ) . Let us writeHOD n = df C (Σ n ) . The Myhill-Scott proof shows that HOD n = HOD for n ≥ . What about HOD ?Note that for all β and A ∈ HOD : • { α < β : cf V ( α ) = ω } ∈ HOD • { ( a, b ) ∈ A : | a | V ≤ | b | V } ∈ HOD • { α < β : α cardinal in V } ∈ HOD • { ( α , α ) ∈ β : | α | V ≤ (2 | α | ) V } ∈ HOD • { α < β : (2 | α | ) V = ( | α | + ) V } ∈ HOD These examples show that HOD contains most if not all of the inner modelsconsidered above. In particular we have: Lemma 7.5. C ∗ ⊆ HOD .2. C ( Q MM ,<ω ) ⊆ HOD
3. If ♯ exists, then ♯ ∈ HOD Naturally, HOD = HOD is consistent, since we only need to assume V = L .So we focus on HOD = HOD.
Theorem 7.6.
It is consistent, relative to the consistency of infinitely many weaklycompact cardinals that for some λ : { κ < λ : κ weakly compact (in V ) } / ∈ HOD , and, moreover, HOD = L = HOD . roof. Let us assume V = L . Let κ n , n < ω be a sequence of weakly compactcardinals. Let D δ be the forcing notion for adding a Cohen subset of the regularcardinal δ . Let λ = sup n κ n . We proceed as in [18]. Let η < κ be two regularcardinals. We denote by R η,κ the Easton support iteration of D δ for η ≤ δ ≤ κ .The forcing R κ + n − ,κ n , where for n = 0 we take κ − = ω , we denote by P n . Notethat forcing with P n preserves the weak compactness of κ n . Let D n be the namefor the forcing D κ n defined in V P n . Note that P n ∗ D κ n is forcing equivalent to P n .Let Q be the full support product of P n , n < ω . Let V ∗ = V Q . Claim.
For every n < ω the cardinal κ n is weakly compact in V ∗ .The argument uses the fact that for each n the forcing Q can be decomposedas Q n × P n × Q n where Q n has cardinality κ n − and Q n is κ + n closed. Hence Q n and Q n do not change the weak compactness of κ n , which is preserved by P n .As in [18], we define in V P n a forcing S n to be the canonical forcing whichintroduces a κ n homogeneous Soulin tree. In particular it kills the weak com-pactness of κ n . Let T n be the forcing which introduces a branch through the treeforced by S n . As in [18] we can show that S n ∗ T n is forcing equivalent to D κ n .Therefore if we force with T n over V P n ∗ S n , we regain the weak compactness of κ n . Also a generic object for D κ n introduces a generic object for T n .We are going to describe three models V ⊆ V ⊆ V . Let first V ∗ = L Q .Let G n be the generic filter in P n , introduced by Q . The model V ∗ is the modelone gets from V ∗ by forcing over it with the full support product of D κ n . ( D n is as realized according to G n .). Let H n ⊆ D n be the generic filter introducedby this forcing. Note that V ∗ can also be obtained from L by forcing with Q .In particular both in V and in V ∗ the cardinals κ n are weakly compact for every n < ω . Let V be an extension of V ∗ by adding a Cohen real a ⊆ ω . Let A bethe Cohen forcing on ω . Then define V = V ∗ ( a ) . Both V and V are obtainedby forcing over L with Q × A which is a homogenous forcing notion. Hence HOD V = HOD V = L . Again we did not kill the weak compactness of thecardinals κ n .Now we define V . Each H n introduces a generic filter for the forcing S n (Asdefined according to G n ). Let K n ⊆ S n be this generic filter. We define V = V [ a, h K n | n a i , h H n | n ∈ a i ] . For n < ω we define an auxiliary universe W n as follows: W n = L ( a, h G i | i ≤ n i , h K i | i ≤ n, i a i , h H i | i ≤ n, i ∈ a i ) . n ∈ a then W n is obtained from L by a product of P n ∗ D n and some forcingsof size < κ n . Since P n ∗ D n preserves the weak compactness of κ n , κ n is weaklycompact in W n . If n a then K n generates a tree on κ n which is still Souslin in W n . (Small forcings do not change the Souslinity of a tree.), So κ n is not weaklycompact in W n . We proved: Claim. κ n is weakly compact in W n iff n ∈ a .The following claim follows from the standard arguments analysing the power-set of a cardinal δ under a forcing which is the product of a forcing of size µ < δ ,a forcing of size δ which is µ + -distributive, and a forcing which is δ + -distributive. Claim.
For n < ω P ( κ n ) V = P ( κ n ) W n .From the last two claims it follows that V | = a = { n < ω | κ n is weakly compact } . Therefore a ∈ HOD V .The proof of the Theorem will be finished if we show that HOD V = L . Foran ordinal α let L α , L α , L α be the α -th step of the construction of ( C (Σ )) V , ( C (Σ )) V , ( C (Σ )) V respectively. Lemma 7.7.
For every α L α = L α = L α . The proof of the lemma is by induction on α where the cases α = 0 and α limit are obvious. So given α , by the induction assumption on α we can put M = L α = L α = L α . Note that M ∈ L since M ∈ HOD V = L . Let Φ( ~x ) be a Σ formula and let ~b be a vector of elements of M . Lemma 7.8.
The following are equivalent1. ( M | = Φ( ~b )) V ( M | = Φ( ~b )) V ( M | = Φ( ~b )) V Without loss of generality, Φ( ~x ) has the form ∃ X Ψ( X, ~x ) , where X is a sec-ond order variable and all the quantifiers of Ψ are first order. Both V and V areobtained form L by forcing over L with Q × A . This forcing is homogeneous. M and all the elements of the vector ~b are in L . So (1) is clearly equivalent to (2).Now suppose that ( M | = Φ( ~b )) V . Let Z ⊆ M be the witness for the exis-tential quantifier of Φ . Then ( M | = Ψ( Z,~b )) V . But all the quantifiers of Ψ are54rst order, so ( M | = Ψ( Z,~b )) V . So (3) implies (2), and hence (1). For the otherdirection, if ( M | = Φ( ~b )) V , then we know that ( M | = Φ( ~b )) V . Let Z ∈ V satisfy M | = Ψ( Z,~b ) . So ( M | = Ψ( Z,~b )) V , and therefore ( M | = Φ( ~b )) V .It follows from the lemma that every Σ formula defines the same subset of M in V , V and V . It follows that L α +1 = L α +1 = L α +1 .This proves the lemma and the theorem.The above proof works also with “weakly compact” replaced by other largecardinal properties, e.g. “measurable” or “supercompact”. We can start, for exam-ple, with a ω supercompact cardinals, code each one of them into cardinal expo-nentiation, detectible by means of HOD , above all of them, without losing theirsupercompactness or introducing new supercompact cardinals, and then proceedas in the proof of Theorem 7.6. Note that we can also start with a supercompactcardinal and code, using the method of [29], every set into cardinal exponentia-tion, detectible by means of HOD , without losing the supercompact cardinal. Inthe final model there is a super compact cardinal while V = HOD .We shall now prove an analogue of Theorem 7.6 without assuming any largecardinals. Let C ( κ ) be Cohen forcing for adding a subset for a regular cardinal κ .Let R ( κ ) be the statement that there is a bounded subset A ⊆ κ and a set C ⊆ κ which is C ( κ ) -generic over L [ A ] , such that P ( κ ) ⊆ L [ A, C ] . Theorem 7.9.
It is consistent, relative to the consistency of ZFC that: { n < ω : R ( ℵ n ) } / ∈ HOD , and, moreover, HOD = L = HOD .Proof.
The proof is very much like the proof of Theorem 7.6 so we only indicatethe necessary modifications. Let us assume V = L . As a preliminary forcing C we apply Cohen forcing C n = C ( ℵ n ) for each ℵ n (including ℵ ) adding a Cohensubset C n ⊆ ℵ n \ { , } . W.l.o.g. min( C n +1 ) > ℵ n . Let V denote the extension.Let P be the product forcing in V which adds a non-reflecting stationary set A n to κ n , n / ∈ C , by means of: P n = { p : γ → ℵ n − < γ < ℵ n , ∀ α < γ (cf( α ) > ω →{ β < α : p ( β ) = 0 } is non-stationary in α ) } . Let us note that P n is strategically ℵ n − -closed, for the second player can playsystematically at limits in such a way that during the game a club is left out of { β : p ( β ) = 0 } . Let V denote the extension of V by P . Now V | = C = { n < ω : R ( ℵ n ) } , n ∈ C , then R ( ℵ n ) holds in V by construction, and on the other hand,if n / ∈ C , then R ( ℵ n ) fails in V because one can show with a back-and-forthargument that with P and C as above, we always have V P = V C .Let Q force in V a club into A n , n ∈ C ω , by closed initial segments with a lastelement. The crucial observation now is that P n ⋆ Q n is the same forcing as C n .To see this, it suffices to find a dense ℵ n − -closed subset of P n ⋆ Q n of cardinality ℵ n . Let D consist of pairs ( p, A ) ∈ P n ⋆ C n such that dom( p ) = max( A ) + 1 and ∀ β ∈ A ( p ( β ) = 1) . This set is clearly ℵ n − -closed. Proposition 7.10. If ♯ exists, then ♯ ∈ C (∆ ) , hence C (∆ ) = L .Proof. As Proposition 5.3.
For another kind of application of extended logics in set theory we consider thefollowing concept:
Definition 8.1.
Suppose L ∗ is an abstract logic. We use ZFC ( L ∗ ) to denote theusual ZFC-axioms in the vocabulary {∈} with the modification that the formula ϕ ( x, ~y ) in the Schema of Separation ∀ x ∀ x ... ∀ x n ∃ y ∀ z ( z ∈ y ↔ ( z ∈ x ∧ ϕ ( z, ~x ))) and the formula ψ ( u, z, ~x ) in the Schema of Replacement ∀ x ∀ x ... ∀ x n ( ∀ u ∀ z ∀ z ′ (( u ∈ x ∧ ψ ( u, z, ~x ) ∧ ψ ( u, z ′ , ~x )) → z = z ′ ) → ∃ y ∀ z ( z ∈ y ↔ ∃ u ( u ∈ x ∧ ψ ( u, z, ~x )))) . is allowed to be taken from L ∗ .The concept of a a model ( M, E ) , E ⊆ M × M , satisfying the axiomsZFC ( L ∗ ) is obviously well-defined. Note that ZFC ( L ∗ ) is at least as strong asZFC in the sense that every model of ZFC ( L ∗ ) is, a fortiori, a model of ZFC.The class of (set) models of ZFC is, of course, immensely rich, ZFC being afirst order theory. If ZFC is consistent, we have countable models, uncountablemodels, well-founded models, non-well-founded models etc. We now ask the56uestion, what can we say about the models of ZFC ( L ∗ ) for various logics L ∗ ?Almost by definition, the inner model C ( L ∗ ) is a class model of ZFC ( L ∗ ) : C ( L ∗ ) | = ZFC ( L ∗ ) . But ZFC ( L ∗ ) can very well have other models. Theorem 8.2.
A model of ZFC is a model of ZFC ( L ( Q )) if and only if it is an ω -model.Proof. Suppose first ( M, E ) is an ω -model of ZFC. Then we can eliminate Q in ( M, E ) : Given a first order formula ϕ ( x, ~a ) with some parameters ~a there is, bythe Axiom of Choice, either a one-one function from { b ∈ M : ( M, E ) | = ϕ ( b, ~a ) } (10)onto a natural number of ( M, E ) or onto an ordinal of ( M, E ) which is infinite in ( M, E ) . Since ( M, E ) is an ω -model, these two alternatives correspond exactlyto (10) being finite (in V) or infinite (in V). So Q has, in ( M, E ) , a first orderdefinition. For the converse, suppose ( M, E ) is a model of ZFC ( L ( Q )) but someelement a in ω ( M,E ) has infinitely many predecessors in V . By using the Schemaof Separation, applied to L ( Q ) , we can define the set B ∈ M of elements a in ω ( M,E ) that have infinitely many predecessors in V . Hence we can take thesmallest element of B in ( M, E ) . This is clearly a contradiction.In similar way one can show that a model of ZFC is a model of ZFC ( L ( Q )) if and only if it its set of ordinals is ℵ -like or it has an ℵ -like cardinal. Theorem 8.3.
A model of ZFC is a model of ZFC ( L ( Q MM )) if and only if it iswell-founded.Proof. Suppose first ( M, E ) is a well-founded model of ZFC. Then we can elim-inate Q MM in ( M, E ) because it is absolute: The existence of an infinite set X such that every pair from the set satisfies a given first-order formula can be writ-ten as the non-well-foundedness of a relation in M and non-well-foundednessis an absolute property in transitive models. For the converse, suppose ( M, E ) is a model of ZFC ( L ( Q MM )) . Since Q is definable from Q MM we can assume ( M, E ) is an ω -model and ω ( M,E ) = ω . Suppose some ordinal a in ( M, E ) isnon-well-founded. To reach a contradiction it suffices to show that the set of such a is L ( Q MM ) -definable in ( M, E ) . Let ϕ ( x, y, z ) be the first order formula of thelanguage of set theory which says: 57 x = h x , x i , y = h y , y i• x , x < ω , y , y < a • x = x • x < x → y < y .Let us first check that Q MM xyϕ ( x, y, a ) holds in ( M, E ) . Let ( a n ) be a decreasingsequence (in V ) of elements of a . Let X be the set of pairs h n, a n i , where n < ω .By construction, any pair h x, y i in [ X ] satisfies ϕ ( x, y, a ) . Thus Q MM xyϕ ( x, y, a ) holds in ( M, E ) . For the converse, suppose Q MM xyϕ ( x, y, b ) holds in ( M, E ) .Let Y be an infinite set such that every h x, y i in [ Y ] satisfies ϕ ( x, y, b ) . Everytwo pairs in Y have a different natural number as the first component. So we canchoose pairs from Y where the first components increase. But then the secondcomponents decrease and b has to be non-well-founded. Theorem 8.4.
A structure is a model of ZFC ( L ω ω ) if and only if it is isomorphicto a transitive a model M of ZFC such that M ω ⊆ M .Proof. Suppose first M is a transitive a model of ZFC such that M ω ⊆ M . Thenwe can eliminate L ω ω because the semantics of L ω ω is absolute in transitivemodels and the assumption M ω ⊆ M guarantees that all the L ω ω -formulas of thelanguage of set theory are elements of M . For the converse, suppose ( M, E ) is amodel of ZFC ( L ω ω ) . Since Q is definable in L ω ω , we may assume that ( M, E ) is an ω -model and ω ( M,E ) = ω . Suppose ( a n ) is a sequence (in V ) of elements of M . Let ϕ ( x, y, u , u , . . . , z , z , . . . ) be the L ω ω -formula ^ n ( x = u n → y = z n ) . Note that ( M, E ) satisfies ∀ x ∈ ω ∃ yϕ ( x, y, , , . . . , a , a , . . . ) . If we apply the Schema of Replacement of ZFC ( L ω ω ) , we get an element b of M which has all the a n as its elements. By a similar application of the Schema ofSeparation we get { a n : n ∈ ω } ∈ M . Thus M is closed under ω -sequences andin particular it is well-founded. 58y a similar argument one can see that the only model of the class size theoryZFC ( L ∞ ω ) is the class size model V itself. This somewhat extreme exampleshows that by going far enough along this line eventually gives everything. Onecan also remark that the class of models of ZFC ( L ω ω ) is exactly the same as theclass of models of ZFC ( L ω ω ) . This is because in transitive models M such that M ω ⊆ M also the truth of L ω ω -sentences is absolute. So despite their otherwisehuge difference, the logics L ω ω and L ω ω do not differ in the current context.Second order logic is again an interesting case. Note that ZFC ( L ) is by nomeans the same as the so-called second order ZFC, or ZFC as it is denoted.We have not changed the Separation and Replacement Schemas into a second or-der form, we have just allowed second order formulas to be used in the schemasinstead of first order formulas. So, although the models of ZFC are, up to iso-morphism, of the form V κ , and are therefore, a fortiori, also models of ZFC ( L ) ,we shall see below that models of ZFC ( L ) need not be of that form. Theorem 8.5.
Assume V = L . A structure is a model of ZFC ( L ) if and only if itis isomorphic to a model M of ZFC of the form L κ where κ is inaccessible.Proof. First of all, if V = L and L κ | = ZFC, where κ is inaccessible, then trivially L κ | = ZFC ( L ) . For the converse, suppose ( M, E ) | = ZFC ( L ) . Because Q MM isdefinable in L , we may assume ( M, E ) is a transitive model ( M, ∈ ) .We first observe that the model M satisfies V = L . To this end, suppose α ∈ M and x ∈ M is a subset of α . Let β be minimal β such that x ∈ L β .There is a binary relation on α , second order definable over M , with order type β . By the second order Schema of Separation this relation is in the model M . So M | = “ x ∈ L β ” . Hence M | = V = L . Let M = L α . It is easy to see that α has tobe an inaccessible cardinal.Note that if exists, then is in every transitive model of ZFC ( L ) .If there is an inaccessible cardinal κ and we add a Cohen real, then ZFC ( L ) has a transitive model M which is not of the form V α , namely the V κ of the groundmodel. By the homogeneity of Cohen-forcing this model is a model of ZFC ( L ) but, of course, it is not V κ of the forcing extension. This is a consequence ofthe homogeneity of Cohen forcing. Note that M is not a model of ZFC , the second order ZFC, in which the Separation and Replacement Schemas of ZFCare replaced by their second order versions, making ZFC a finite second ordertheory. Here we have an example where ZFC = ZFC ( L ) .59 Open Questions
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