Iteration theorems for subversions of forcing classes
aa r X i v : . [ m a t h . L O ] J un ITERATION THEOREMS FOR SUBVERSIONS OF FORCINGCLASSES
GUNTER FUCHS AND COREY BACAL SWITZER
Abstract.
We prove various iteration theorems for forcing classes related tosubproper and subcomplete forcing, introduced by Jensen. In the first part,we use revised countable support iterations, and show that 1) the class of sub-proper, ω ω -bounding forcing notions, 2) the class of subproper, T -preservingforcing notions (where T is a fixed Souslin tree) and 3) the class of subproper,[ T ]-preserving forcing notions (where T is an ω -tree) are iterable with revisedcountable support. In the second part, we adopt Miyamoto’s theory of niceiterations, rather than revised countable support. We show that this approachallows us to drop a technical condition in the definitions of subcompletenessand subproperness, still resulting in forcing classes that are iterable in this way,preserve ω , and, in the case of subcompleteness, don’t add reals. Further, weshow that the analogs of the iteration theorems proved in the first part forRCS iterations hold for nice iterations as well. Introduction
This article brings together two important threads in forcing iteration theory:variations on revised countable support and Jensen’s subversions of forcing classes.In the first half we pursue a Boolean algebraic approach to iterating subproperand subcomplete forcing notions while preserving certain properties of the forcingnotions being iterated. This is based on what has been done in [12]. In the secondhalf we contrast this approach with one involving nice iterations in the sense ofMiyamoto [14]. Here, we use partial preorders and obtain the same iteration andpreservation theorems. In this setting, there is a further upshot that we can removeone of the more technical conditions in the definition of subproperness, thus gettingan iteration theorem for a (seemingly) more general class of forcing notions. Wecan similarly omit that condition from the definition of subcomplete forcing notionswhile maintaining the crucial properties of the forcing class, namely not addingreals, preserving Souslin trees and diamond sequences, and being iterable via niceiterations.In the final section we provide applications of the theorems from the first twosections, constructing multiple new models of the axiom
SCFA .The definitions of the classes of subproper and subcomplete forcing (definedin the next section) result from modifying the definitions of proper and σ -closedforcing in a way we call “subversion”, resulting in properly larger forcing classeswhich include forcing notions that badly fail to be proper, such as Namba forcing Date : June 25, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Iterated forcing, revised countable support, subcomplete forcing.The first author’s research for this paper was supported in part by PSC CUNY research grant61567-00 49. (under CH ), Pˇr´ıkr´y forcing and the forcing to shoot clubs through stationary subsetsof κ ∩ cof( ω ) for regular κ > ℵ , see [11].Both classes come with associated forcing axioms and the subcomplete forcingaxiom, SCFA , which is Martin’s axiom on ℵ for subcomplete forcing is particularlystriking as it has much of the strength of MM , for instance implying failure ofvarious square principles, see Jensen [9] and Fuchs [3], [4], but is consistent with ♦ . Models of SCFA + ¬ CH came up during the investigation of the first author andMinden in [5], and the present paper grew out of discussions relating to this.The basic issue is as follows. In trying to uncover consequences of SCFA one oftenfalls into two situations. Either some consequence of MM can be shown to followalready from SCFA , usually by simply showing that the forcing used for the MM application is actually subcomplete, or else this consequence is incompatible with ♦ , and hence it cannot follow from SCFA . An example for the first type of situation,Jensen’s aforementioned observation that the forcing to shoot a club in ordertype ω through a stationary subset of κ ∩ cof( ω ), for regular κ > ω , is subcomplete,allows one to conclude that SCH follows from
SCFA , [11, Corollary 7.4], using theknown arguments based on Martin’s Maximum. As an example for the second typeof situation,
SCFA does not imply Souslin’s Hypothesis, since Souslin trees exist inany model with a diamond sequence. Another example is that
SCFA does not implythat the nonstationary ideal on ω is saturated (as Martin’s Maximum does), againsince SCFA is compatible with ♦ , which, in turn, implies the failure of saturation.The basic question is whether such non-implications are only obstructed by dia-mond (or CH ), i.e. is it enough to add the failure of CH , say, to SCFA to resurrectthe consequences of MM that are not already consequences of SCFA ? In short, does
SCFA + ¬ CH imply MM ?Of course, we did not expect the answer to this question to be affirmative, butthe fact that this was a question illustrates how little we knew about models of SCFA + ¬ CH .In the final section we will apply our iteration theorems to produce variousmodels of SCFA + ¬ CH with different constellations of cardinal characteristics of thecontinuum inconsistent with MM , thus answering the provocative question abovein the negative, as expected. This is also interesting on its own right, since itshows that there are strong forcing axioms compatible with various constellationsof cardinal characteristics of the continuum often studied in set theory of the reals.A sampling of results along this line is given below. Theorem 1.1.
Assuming the consistency of a supercompact cardinal the followingare consistent with
SCFA + ¬ CH .(1) Souslin’s Hypothesis fails.(2) d < c .(3) MA ℵ ( σ − linked) holds while MA ℵ fails. The key to proving these results is the proof of iteration theorems for theseclasses. Specifically we show that certain iterations of subproper forcing notionspreserving a fixed Souslin tree S preserve S , that certain iterations of ω ω -boundingsubproper forcing notions are ω ω -bounding and that certain iterations of subproperforcing notions not adding branches through a fixed ω -tree T do not add branchesthrough T . We can then, starting in a model with a supercompact cardinal, runthe usual argument, based on Baumgartner’s construction of a model of the properforcing axiom, to produce a model of the forcing axiom for the relevant forcing TERATION THEOREMS FOR SUBVERSIONS OF FORCING CLASSES 3 class. For example, if we do this for the class of subproper forcing notions thatpreserve a particular Souslin tree S , then in the resulting model, there obviously isa Souslin tree, and CH fails, because Cohen forcing is in the class, and SCFA holds,since every subcomplete forcing preserves Souslin trees. So this will be a model of
SCFA + ¬ CH in which Souslin’s Hypothesis fails.We prove these iteration theorems in two different ways. First using RCS itera-tions, generalizing Jensen’s techniques, and second with nice iterations in the senseof Miyamoto, [14], who carried out these arguments in the context of semiproperforcing. In the latter case we drop a technical condition on the definition ofsubproper and subcomplete forcing. We dub these classes ∞ -subproper and ∞ -subcomplete forcing notions. We feel both proofs of the iteration theorems giveinformation and perspective the other does not. It also sheds light on the differencebetween different styles of RCS iterations in this novel context. Since the pre-cise relationship between RCS iterations and nice iterations is still not completelyunderstood this may be of independent interest.As such this article is broken into two parts. In Section 2, we treat RCS iterationsand develop more fully the theory of nicely subproper iterations. This includes theabovementioned iteration theorems. In Section 3 we reconsider these theorems,this time using nice iterations. We introduce ∞ -subproper and ∞ -subcompleteforcing notions, study their general properties, and we review the machinery ofnice iterations needed to prove the iteration theorems in this context. Note thatwhile the two approaches use the same generic word “nice”, it means somethingvery different in Sections 2 and 3. It is simply an unfortunate coincidence that theestablished terminology in the literature conflicts in this way.In Section 4, we give the aforementioned applications to the study of forcingaxioms, and we conclude with some remarks and open questions in Section 5.2. RCS iterations
In this section, we will prove preservation theorems for iterations of subproperforcing notions with revised countable support, and variations thereof. We use adefinition of subproperness that uses a slightly different Hull Property, followingJensen [12, § §
4] and is some-what close to what Jensen would call “very subproper”. Namely, in place of thecardinality of a poset, we use its density, defined as follows.
Definition 2.1.
Given a poset P , δ ( P ) is the smallest cardinal κ such that thereis a dense subset of P that has cardinality κ .There are other, maybe more natural, measures of the size of a poset, introducedin [2]. We could use those as well, and work with the resulting variations of sub-properness, but since we don’t have any applications of these variations thus far, wechose not to do so. The density of a partial order is related to its chain condition: Observation 2.2.
For any poset P , P is δ ( P ) + -c.c.Proof. Let A ⊆ P be an antichain, and let D ⊆ P be a dense set of cardinality δ ( P ).Define f : A −→ D by choosing, for each a ∈ A , an f ( a ) ≤ a with f ( a ) ∈ D . Then f is injective, and hence, card( A ) ≤ card( D ) < δ ( P ) + . (cid:3) The following definition is due to Jensen.
FUCHS AND SWITZER
Definition 2.3.
A transitive model N of ZFC − is full if there is an ordinal γ > L γ ( N ) satisfies ZFC − and N is regular in L γ ( N ), meaning that if a ∈ N and f ∈ L γ ( N ) is a function f : a −→ N , then ran( f ) ∈ N .We are now ready to state Jensen’s definition of subproperness. Here, when B is a complete Boolean algebra, we use the notation B + for B \ { } . Definition 2.4.
A complete Boolean algebra B is subproper if every sufficientlylarge cardinal θ verifies the subproperness of B , meaning that the following holds: B ∈ H θ , and if τ > θ is such that H θ ⊆ N = L Aτ | = ZFC − , and σ : ¯ N ≺ N , where ¯ N is countable and full, and ¯ S = h ¯ θ, ¯ B , ¯ a, ¯ s, ¯ λ , . . . , ¯ λ n i ∈ ¯ N , S = h θ, B , a, s, λ , . . . , λ n i = σ ( ¯ S ), where ¯ a ∈ B + and λ i > δ ( B ) is regular, for 1 ≤ i ≤ n ,then there is a c ∈ B + such that c ≤ a and such that whenever G ⊆ B is genericwith c ∈ G , then there is a σ ′ ∈ V[ G ] such that(a) σ ′ : ¯ N ≺ N .(b) σ ′ ( ¯ S ) = σ ( ¯ S ).(c) ( σ ′ ) − “ G is ¯ B -generic over ¯ N .(d) Letting ¯ λ = On ∩ ¯ N , for all i ≤ n , we have that sup( σ ′ )“¯ λ i = sup σ “¯ λ i .Our definition differs slightly from that used in [12], in that we don’t require τ (in the notation of the definition) to be regular. This ensures that the resultingdefinition is locally based, in the sense of [9, §
2, p. 6], and is in line with thedefinition of subcompleteness employed by Jensen in [9, p. 3]. There are severalways of defining subproperness directly for posets rather than complete Booleanalgebras, but since we are going to work with complete Boolean algebras here, thedefinition given will do. We will return to the poset definition in Section 3. Jensenhas sometimes employed a slight strengthening of condition (d) above. We will alsoreturn to this later.2.1.
The subproperness extension lemma.
The iteration theorems in this sec-tion are based on one main lemma, which we will prove here. It is based on whatJensen calls the One Step Lemma, but slightly more abstract. We call it the Sub-properness Extension Lemma, in analogy to the context of proper forcing. In orderto formulate it, we need some terminology regarding iterated forcing using theBoolean algebraic approach.If B is a complete Boolean algebra, then we write A ⊆ B to express that A isa complete subalgebra of B , meaning that A is a complete Boolean algebra, andfurthermore, that for X ⊆ A , W A X = W B X and V A X = V B X . In this situation,the retraction h B , A : B −→ A is defined by h B , A ( b ) = ^ { a ∈ A | b ≤ B a } . Further, if G ⊆ A is a generic filter, then G generates the filter G = { b ∈ B | ∃ a ∈ G a ≤ b } on B , and writing b ⇒ c for ¬ b ∨ c , G induces an equivalence relation on B defined by identifying b and b ′ iff ( b ⇒ b ′ ) ∧ ( b ′ ⇒ b ) ∈ G . We write b/G for theequivalence class of b under that equivalence relation, and we write B /G for thefactor algebra. The ordering on B /G is given by b/G ≤ B /G c/G iff ( b ⇒ c ) ∈ G .We write ˙ G A for the canonical name for the A -generic ultrafilter. Lemma 2.5 (Subproperness Extension Lemma) . Let B be a complete Booleanalgebra, and let A ⊆ B be a complete subalgebra of B . Let h = h A : B −→ A bethe retraction. Let δ = δ ( B ) . Suppose that (cid:13) A “ ˇ B / ˙ G A is subproper, as verified by TERATION THEOREMS FOR SUBVERSIONS OF FORCING CLASSES 5 ˇ θ ,” where B ∈ H θ . Let N = L Aτ be a ZFC − model with H θ ⊆ N and θ < τ , andlet ¯ N be countable, transitive and full. Let ¯ S = h ¯ θ, ¯ A , ¯ B , ¯ s, ¯ λ , . . . , ¯ λ n i ∈ ¯ N , and S = h θ, A , B , s, λ , . . . , λ n i ∈ N , where λ i is regular, for ≤ i ≤ n . Let ˙ σ , ˙ t , ˙¯ b and ˙ b be A -names, and let a ∈ A + be a condition that forces with respect to A :(1) ˙ σ : ˇ¯ N ≺ ˇ N ,(2) ˙ σ ( ˇ¯ S ) = ˇ S (so ¯ A and ¯ B are complete Boolean algebras in ¯ N ),(3) ˙ t ∈ ˇ¯ N , ˙¯ b ∈ ˇ¯ B , ˙ b ∈ ˇ B , and ˇ h (˙ b ) ∈ ˙ G A ,(4) ˙ σ − “ ˙ G A is ˇ¯ N -generic for ¯ A ,(5) ˙ σ (˙¯ b ) = ˙ b .Then there are a condition c ∈ B + such that h ( c ) = a and a B -name ˙ σ such thatwhenever I is B -generic with c ∈ I , letting σ = ˙ σ I , G = I ∩ A and σ = ˙ σ G , thefollowing conditions hold:(1) σ : ¯ N ≺ N ,(2) σ ( ¯ S ) = S ,(3) σ ( ˙ t G ) = σ ( ˙ t G ) ,(4) σ (˙¯ b G ) = σ (˙¯ b G ) = ˙ b G ,(5) ˙ b G ∈ I ,(6) σ − “ I is ¯ B -generic over ¯ N ,(7) Letting ¯ λ = On ∩ ¯ N , for all i ≤ n , sup σ “¯ λ i = sup σ “¯ λ i .Proof. We follow the proof of [10, §
2, Lemma 2]. Let G be any A -generic filter with a ∈ G . Then in V[ G ], B /G is subproper, as verified by θ . Let t = ˙ t G , ¯ b = ˙¯ b G , b = ˙ b G and σ = ˙ σ G . Then σ : ¯ N ≺ N , σ ( ¯ S ) = S , h ( b ) ∈ G , t ∈ ¯ N and ¯ G = σ − “ G is¯ N -generic for A . Let σ ∗ : ¯ N [ ¯ G ] ≺ N [ G ]be the unique embedding extending σ such that σ ∗ ( ¯ G ) = G .We have that H V [ G ] θ = H θ [ G ] ⊆ N [ G ], and ¯ N [ ¯ G ] is full in V[ G ]. We also havethat b/G = 0. To see this, note that since b ∈ V, it makes sense to write b = ˇ b G .By [10, p. 91, Fact 3], we know that h ( b ) = J ˇ b/ ˙ G A = 0 K . So, since h ( b ) ∈ G , thismeans that J ˇ b/ ˙ G A = 0 K ∈ G , which means precisely that b/G = 0.So, since B /G is subproper in V[ G ], there is a condition d ∈ B /G with d ≤ b/G such that whenever H is generic for B /G over V[ G ], then in V[ G ][ H ], there is anelementary embedding σ ′ : ¯ N [ ¯ G ] ≺ N [ G ] with σ ′ ( ¯ S ) = S , ¯ H = ( σ ′ ) − “ H is ¯ B / ¯ G -generic over ¯ N [ ¯ G ] and for all i ≤ n , sup σ ∗ “¯ λ i = sup σ ′ “¯ λ i . We may moreover insistthat σ ′ maps any finite list of members of ¯ N [ ¯ G ] the same way σ ∗ does. Thus, werequire that σ ′ ( t ) = σ ∗ ( t ), σ ′ (¯ b ) = σ ∗ (¯ b ) = b and σ ′ ( ¯ G ) = σ ∗ ( ¯ G ) = G .Let us temporarily fix such an H , and let I = G ∗ H . Let σ = σ ′ ↾ ¯ N . It followsthat σ ′ ( τ ¯ G ) = ( σ ( τ )) G , for τ ∈ ¯ N ¯ A , since σ ′ ( ¯ G ) = G .Then σ ′ , G and I clearly satisfy conditions (1), (2), (3), (4), (5) and (7) above.It follows also that (6) is satisfied, that is, ¯ I = σ − “ I is ¯ B -generic over ¯ N : since¯ G is ¯ A -generic over ¯ N and ¯ H is ¯ A / ¯ G -generic over ¯ N [ ¯ G ], it follows that ¯ G ∗ ¯ H is ¯ B -generic over ¯ N . But, for ¯ b ∈ ¯ B , we have that ¯ b ∈ ¯ G ∗ ¯ H iff ¯ b/ ¯ G ∈ ¯ H iff σ ′ (¯ b/ ¯ G ) = σ (¯ b ) /G ∈ H iff σ (¯ b ) ∈ G ∗ H = I . Thus, ¯ G ∗ ¯ H = σ − “ I = ¯ I is¯ B -generic over ¯ N , as claimed.So there are a name π in V[ G ] B /G with σ = π H and a condition d ∈ ( B /G ) + that forces over V[ G ] with respect to B /G that π has the properties listed. FUCHS AND SWITZER
Now, all of this is true in V[ G ] whenever G is A -generic over V, with a ∈ G ,and so, there are names ˙ d, ˙ π ∈ V A such that d = ˙ d G and π = ˙ π G , and a forces thesituation described. Let ˙ σ be a B -name such that ˙ σ G ∗ H = ( ˙ π G ) H = σ .The only thing that’s missing is the condition c ∈ B with h ( c ) = a such thatwhenever c ∈ I , I is B -generic over V, G = I ∩ A and σ = ˙ σ I , and σ = ˙ σ G ,it follows that (1)-(7) hold. To find the desired condition, first note that we maychoose the name ˙ d in such a way that (cid:13) A ˙ d ∈ ˇ B / ˙ G A and a = J ˙ d = 0 K A . Namely,given the original ˙ d such that a forces that ˙ d ∈ (ˇ B / ˙ G A ) + and all the other statementslisted above, there are two cases: if a = 1l A , then since a ≤ J ˙ d = 0 K , it alreadyfollows that a = J ˙ d = 0 K and (cid:13) A ˙ d ∈ ˇ B / ˙ G A . If a < A , then let ˙ e ∈ V A be a namesuch that (cid:13) A ˙ e = 0 ˇ B / ˙ G A , and mix the names ˙ d and ˙ e to get a name ˙ d ′ such that a (cid:13) A ˙ d ′ = ˙ d and ¬ a (cid:13) A ˙ d ′ = ˙ e . Then ˙ d ′ is as desired. Clearly, (cid:13) A ˙ d ′ ∈ ˇ B / ˙ G A . Since a (cid:13) A ˙ d ′ = ˙ d , it follows that a ≤ J ˙ d ′ = 0 K , and since ¬ a (cid:13) A ˙ d ′ = ˙ e , it follows that ¬ a ≤ J ˙ d ′ = 0 K = ¬ J ˙ d ′ = 0 K , so J ˙ d ′ = 0 K ≤ a . So we could replace ˙ d with ˙ d ′ .Then, by [11, §
0, Fact 4], there is a unique c ∈ B such that (cid:13) A ˇ c/ ˙ G A = ˙ d , andit follows by [11, §
0, Fact 3] that h ( c ) = J ˇ c/ ˙ G A = 0 K A = J ˙ d = 0 K A = a as wished. (cid:3) RCS and nicely subproper iterations.
We adopt Jensen’s approach toRCS iterations. Thus, an iteration of length α is a sequence h B i | i < α i of completeBoolean algebras such that for i ≤ j < α , B i ⊆ B j , and such that if λ < α is alimit ordinal, then B λ is generated by S i<λ B i , meaning that B λ is the completionof the collection of all infima and suprema of subsets of S i<λ B i . In this setting, ~b = h b i | i < λ i is a thread in ~ B ↾ λ if for every i ≤ j < λ , b i = h B j , B i ( b j ) and b j = 0. B λ is an inverse limit of ~ B ↾ λ if for every thread ~b in ~ B ↾ λ , b ∗ := V B λ i<λ b i = 0, and ifthe set of such b ∗ is dense in B λ . This characterizes B λ up to isomorphism. If h ξ i | i < ¯ λ i is monotone and cofinal in λ and ~b = h b i | i < ¯ λ i is such that for every i < ¯ λ , b i ∈ B ξ i and for every i ≤ j < ¯ λ , h B ξj , B ξi ( b j ) = b i , then we will consider ~b to be athread in ~B ↾ λ as well, since it gives rise to a thread ~c = h c i | i < λ i in the originalsense via the definition c i = h B ξj , B i ( b j ) where j is such that ξ j ≥ i , and vice versa,the restriction of a thread in the original sense to a cofinal index set determinesthe entire thread, so that these two notions are equivalent. If ~ B = h B i | i < α i isan iteration as above, then α is the length of ~ B .The direct limit takes B λ as the minimal completion of S i<λ B i and is charac-terized by the property that S i<λ B i \ { } is dense in B λ . Another way of lookingat it is that it is generated by the eventually constant threads.The RCS limit is defined as the inverse limit, except that only RCS threads ~b are used: ~b = h b i | i < λ i is an RCS thread in ~ B ↾ λ if it is a thread in ~ B ↾ λ and thereis an i < λ such that either, for all j < λ with i ≤ j , b i = b j , or b i (cid:13) B i cf(ˇ λ ) = ˇ ω . Definition 2.6.
Let ~ B be an iteration of length α .Then ~ B is direct if for every i + 1 < α , B i = B i +1 . There is a slightly confusing misprint in the statement of that fact. It should read: “Let A ⊆ B ,and let (cid:13) A ˙ b ∈ ˇ B / ˙ G A , where ˙ b ∈ V A . There is a unique b ∈ B such that (cid:13) A ˙ b = ˇ b/ ˙ G A .” That’swhat the proof given there shows. TERATION THEOREMS FOR SUBVERSIONS OF FORCING CLASSES 7
It is standard if it is direct and for every i + 1 < α , letting δ i = δ ( B i ), (cid:13) B i +1 ˇ δ i ≤ ω .It is an RCS iteration if for every limit λ , B λ is the RCS limit of ~ B ↾ λ .Let Γ = { B | ϕ Γ ( B , p ) } be a class of complete Boolean algebras (defined by someformula ϕ Γ in some parameter p ).Then an iteration ~ B is an iteration of forcings in Γ if for every i + 1 < α , (cid:13) B i “ˇ B i +1 / ˙ G i ∈ Γ,” (i.e., (cid:13) B i ϕ Γ ( B i +1 / ˙ G i , ˇ p )).Γ is standard RCS iterable if whenever ~ B is a standard iteration of forcings in Γ,then for every h ≤ i < α , if G h is generic for B h , then in V[ G h ], B i /G h ∈ Γ (i.e., ϕ Γ ( B i /G h , p ) holds in V[ G h ]).In the context of a given iteration ~ B as above, if i < α and b ∈ B j , for some j < α , we’ll just write h i ( b ) for h B j , B i ( b ). We’ll write lh( ~ B ) = α , the length of theiteration. The following fact summarizes the basic properties of RCS iterations. Fact 2.7 ([11, p. 142]) . Let ~ B = h B i | i < α i be an RCS iteration.(1) If λ < α and cf( λ ) = ω , then B λ is the inverse limit of ~ B ↾ λ .(2) If λ < α and for every i < λ , (cid:13) B i cf(ˇ λ ) > ω , then S i<λ B i is dense in B λ (that is, B λ is formed using only eventually constant threads, making it thedirect limit).(3) If i < λ and G is B i -generic, then the above are true in V[ G ] about theiteration h B i + j /G | j < α − i i . The following fact gives us some information about the chain conditions satisfiedby direct limits in an iteration.
Fact 2.8 (Baumgartner, see [16, Theorem 3.13]) . Let h B i | i < λ i be an iterationsuch that for every α < λ , B α is <λ -c.c., and such that the set of α < λ such that B α is the direct limit of ~ B ↾ α is stationary. Then the direct limit of ~ B is <λ -c.c. A variation of the RCS iteration theorem for subproper forcing [12, §
4, pp. 2,Thm. 5] says:
Theorem 2.9 (Jensen) . The class of complete subproper Boolean algebras is stan-dard RCS iterable.
In detail, Jensen proved the version of this theorem for subcomplete forcingin [12, §
3, Theorem 2], and states that the version for subproper forcing can bereproven easily (see [12, §
4, p. 19]).Jensen [8] uses a more flexible notion of iteration of subcomplete forcing notions,and an elegant proof of a generalization of the main iteration theorem of thatwork is given in [12]. We follow the latter presentation here, albeit in the contextof subproper forcing. The following is a version of [12, §
3, p. 9] translated fromthe subcomplete to the subproper context. We also work with δ ( B i ) rather thancard( B i ). Definition 2.10 (after Jensen) . Let Γ be a class of complete Boolean algebras. Astandard iteration ~ B = h B i | i < α i is nicely Γ if, letting δ i = δ ( B i ), for i < α , thefollowing hold:(1) Suppose i + 1 < α . Then (cid:13) B i ˇ B i +1 / ˙ G i ∈ Γ.(2) Suppose λ < α is a limit ordinal of countable cofinality.(a) If ~b is a thread in ~ B ↾ λ , then V i<λ b i = 0 in B λ . FUCHS AND SWITZER (b) If for every i < λ , B i ∈ Γ, then B λ ∈ Γ.(3) Suppose that λ < α is a limit ordinal such that for every i < λ , (cid:13) B i cf(ˇ λ ) >ω . Then S i<λ B i is dense in B λ , that is, B λ is the direct limit of ~ B ↾ λ .(4) Let i < α . Then, if G i is B i -generic, (1)-(3) hold in V[ G i ] for h B i + j /G i | j < α − i i .Finally, we say that Γ is nicely iterable if whenever ~ B = h B i | i < α i is a nicely Γiteration, then for every h ≤ δ < α , if G h is B h -generic, then in V[ G h ], B δ /G h ∈ Γ.That is, in a nicely Γ iteration, we already know that belonging to Γ propagatesto limit stages of countable cofinality, but we have almost no restrictions as to howthose limit stages are formed. The following observation shows that iterations ofsubproper forcing as in Theorem 2.9 are nicely subproper.
Observation 2.11.
Every standard RCS iteration of subproper forcings is nicelysubproper.Proof.
Let i < α , and let G i be B i -generic. We have to verify that conditions(1)-(3) of Definition 2.10 hold of h B i + j /G i | j < α − i i in V[ G i ].Condition (1) is trivial: let j + 1 < α − i . Since (cid:13) B i + j ˇ B i + j +1 / ˙ G i + j is subproper,it clearly has to be that (cid:13) B i + j /G i (ˇ B i + j +1 / ˇ G i ) / ˙ G B i + j / ˇ G i is subproper in V[ G i ]. Itcan be shown similarly that h B i + j /G i | j < α − i i is a standard iteration in V[ G i ].For condition (2), let λ < α − i be a limit ordinal that has countable cofinalityin V[ G i ]. By Fact 2.7, part (3), the first two parts of that fact apply to h B i + j /G i | j < α − i i in V[ G i ]. Thus, B λ /G i is the inverse limit of h B i + j /G i | j < λ i . Thisimplies condition (2)(a). And by Theorem 2.9, B λ /G i is subproper in V[ G i ], socondition (2)(b) holds.Finally, condition (3) hold because part (2) of Fact 2.7 applies to h B i + j /G i | j < α − i i in V[ G i ], by part (3) of that fact. (cid:3) The following is a version of a theorem that Jensen proved for subcompleteforcing in [12, §
3, pp. 9-11]. By Observation 2.11, it generalizes Theorem 2.9.
Theorem 2.12 (Jensen) . The class of subproper Boolean algebras is nicely iterable.Proof.
The proof is a virtual repetition of the argument of Jensen’s proof of [12, §
4, Theorem 5, pp. 3-12], incorporating the changes necessitated by working with δ ( B i ) (as in [12, §
3, p. 2, Theorem 2]), except that it is somewhat simpler, becausethe limit of countable cofinality case is vacuous now. We have checked that theproof goes through, and so has Jensen (see [12, §
4, last three lines on p. 19]). (cid:3)
Iterating subproper Souslin tree preserving forcing.
The main idea forthis section stems from Miyamoto [14, Lemma 5.0], even though we do not employhis “nice iterations” here. In the original setting, a Souslin tree T is fixed, andit is shown that nice limits of nice iterations of semi-proper forcing notions thatpreserve T also preserve T . The corresponding theorem holds for subproper forcingas well, and one can work with RCS iterations rather than nice iterations too, aswe shall show. However, we will first prove a different version of this preservationfact, because we want to establish a proof template that we will reuse in differentsituations later. The proof of that fact will be slightly more complicated and hencemore suitable for these later variations. TERATION THEOREMS FOR SUBVERSIONS OF FORCING CLASSES 9
Definition 2.13.
A forcing notion P preserves Souslin trees if for every Souslintree T , (cid:13) P “ ˇ T is Souslin.”The main difference between this concept and the preservation of a fixed Souslintree, when forming iterations of such forcing notions, is that iterands in an iterationof Souslin tree preserving forcing notions are required to preserve the Souslin treesthat may have been added by earlier stages of the iteration, not only one fixed T , orsome collection of Souslin trees in the ground model. We will give the proof of thefollowing theorem in considerable detail, in order to establish a point of referencefor later variations of the argument. Theorem 2.14.
The class of subproper Boolean algebras that are Souslin tree pre-serving is standard RCS iterable.Proof.
Let ~ B = h B i | i < α i be a standard RCS iteration of forcings that aresubproper and Souslin-tree preserving. We prove by induction on δ : whenever h ≤ δ < α and G h is B h -generic, then in V[ G h ], B δ /G h is subproper and Souslintree preserving.It suffices to focus on the Souslin tree preservation, since by Theorem 2.12, B δ /G h is subproper in V[ G h ] whenever h ≤ δ < α and G h is B h -generic.The successor case is trivial, as is the case h = δ . So let δ be a limit ordinal, andlet h < δ .Note that by Observation 2.11, ~ B is nicely subproper. We will use this, ratherthan that ~ B is an RCS iteration, whenever possible. Doing so will make it easier toslightly generalize the theorem later. Case 1: there is an i < δ such that cf( δ ) ≤ δ ( B i ).Fix such an i . Then whenever i < j < δ , (cid:13) j cf(ˇ δ ) ≤ ω . It suffices to prove(A) if i < j < δ and G j is B j -generic, then in V[ G j ], B δ /G j is Souslin treepreserving.For if we have done so, then the full claim follows: let h ≤ i , and let G h ⊆ B h begeneric. By Theorem 2.12, we know that B δ /G h is subproper in V[ G h ]. Thus, itsuffices to prove that if T is some Souslin tree in V[ G h ] and H is B δ /G h -generic overV[ G h ], then T is a Souslin tree in V[ G h ][ H ]. But letting G i +1 = ( G h ∗ H ) ∩ B i +1 ,we know inductively that T is Souslin in V[ G i +1 ] = V[ G h ][ G i +1 /G h ], and so, by( A ), T is Souslin in V[ G i +1 ∗ (( G h ∗ H ) /G i +1 )] = V[ G h ][ H ].To prove ( A ), we would now have to fix a j < δ , a G j ⊆ B j that’s generic, anda T such that in V[ G j ], T is Souslin. We’d have to prove in V[ G j ] that B δ /G j preserves T as a Souslin tree. But the iteration h B j + i /G j | i < δ − j i is RCS (seeFact 2.7) in V[ G j ], and it satisfies everything in V[ G j ] that we assumed about ~ B inV, with the addition of the fact that in V[ G j ], cf( δ ) ≤ ω .Thus, it suffices to show:(B) if cf( δ ) ≤ ω then B δ is Souslin tree preserving.For the argument, carried out in V[ G j ] would prove ( A ).Note that by arguing in V[ G j ], but pretending V[ G j ] is V, we effectively absorbed T into V. It is this step that’s not necessary if one only wants to preserve one fixedground model Souslin tree.To prove (B), let us fix a Souslin tree T . Note that if cf( δ ) = ω , then for every i < δ , (cid:13) B i cf(ˇ δ ) = ˇ ω , as B i preserves ω .Thus, B δ is the direct limit of ~ B ↾ δ , by part (3) of Definition 2.10, that is, S i<δ B i is dense in B δ in this case. Let us denote this dense set by X .If, on the other hand, cf( δ ) = ω , then since ~ B is an RCS iteration, then we knowby Fact 2.7 that the set { V i<δ t i | h t i | i < δ i is a thread in ~ B ↾ δ } is dense in B δ . Incase cf( δ ) = ω , let X be that dense subset of B δ .Let π : ω −→ δ be cofinal, with π (0) = 0.Let ˙ A be a B δ -name for a maximal antichain in T , and let a ∈ X be a condition.We will find a countable ζ and a condition extending a that forces that ˙ A ⊆ ˇ T | ˇ ζ .Here, T | ζ is the union of the levels of T below ζ .Let N = L Aτ , with H θ ⊆ N , such that θ verifies the subproperness of each B i ,for i ≤ δ . Let S = h θ, δ, ~ B , X, ˙ A, T, π, a i . Let M ≺ N with S ∈ M , M countableand such that, letting σ : ¯ N −→ M be the inverse of the Mostowski collapse (sothat ¯ N is transitive), ¯ N is full – it is easy to see that this situation can be arranged.Let ˜ δ = sup( M ∩ δ ), and let Ω = M ∩ ω = crit( σ ) = ω ¯ N . Fix an enumeration h s n | n < ω i of T (Ω), the Ω-th level of T .Let σ − ( S ) = ¯ S = h ¯ θ, ¯ δ, ~ ¯ B , ¯ X, ˙¯ A, ¯ T , ¯ π, ¯ a i . Let h ν i | i < ω i be a sequence ofordinals ν i < ω ¯ N such that if we let ¯ γ i = ¯ π ( ν i ), it follows that h ¯ γ i | i < ω i iscofinal in ¯ δ , and such that ν = 0, so that ¯ γ = 0. Hence, letting γ i = σ (¯ γ i ),we have that sup i<ω γ i = sup( M ∩ δ ) = ˜ δ . Moreover, whenever σ ′ : ¯ N ≺ N issuch that σ ′ (¯ π ) = π , it follows that for every i < ω , σ ′ (¯ γ i ) = γ i = π ( ν i ), since σ ′ (¯ γ i ) = σ ′ (¯ π ( ν i )) = σ ′ (¯ π )( ν i ) = π ( ν i ).By induction on n < ω , construct sequences h ˙ σ n | n < ω i , h c n | n < ω i , h ˙ b n | n < ω i and h ˙¯ b n | n < ω i with c n ∈ B γ n , ˙ σ n ∈ V B n , ˙ b n , ˙¯ b n ∈ V B γn , such that forevery n < ω , c n forces the following statements with respect to B γ n :(1) ˙ σ n : ˇ¯ N ≺ ˇ N ,(2) ˙ σ n ( ˇ¯ S ) = ˇ S , and for all k < n , ˙ σ n (˙¯ b k ) = ˙ σ k (˙¯ b k ),(3) ˙ σ n (˙¯ b n ) = ˙ b n and ˙¯ b n ∈ ˇ¯ X (and so, ˙ b n ∈ ˇ X ),(4) ˇ h γ n (˙ b n ) ∈ ˙ G B γn ,(5) ˙ σ − n “ ˙ G B γn is ˇ¯ N -generic for ˇ¯ B γ n ,(6) ˙ b n / ˙ G γ n forces wrt. ˇ B δ / ˙ G γ n that there is a node t < s n − with t ∈ ˙ A (for n > c n − = h γ n − ( c n ) (for n > b n ≤ ˇ B δ ˙ b n − (for n > n = 0, we set c = 1l, ˙ σ = ˇ σ and ˙ b = ˇ a and ˙¯ b = σ − (˙ b ).Clearly then, (1)-(6) are satisfied for n = 0 (and (7)-(8), as well as the second partof (2), are vacuous for n = 0).Now suppose m = n −
1, and ˙ σ l , c l , ˙ b l and ˙¯ b l have been defined so that (1)-(8)are satisfied for l ≤ m .An application of Lemma 2.5 (to B γ m ⊆ B γ n ) yields a condition c n ∈ B γ n anda B γ n -name ˙ σ n such that h γ m ( c n ) = c m and whenever I is B γ n -generic over Vwith c n ∈ I , G = I ∩ B γ m and σ n = ˙ σ In it follows that σ n : ¯ N ≺ N , σ − n “ I is¯ B ¯ γ n -generic over ¯ N , σ n ( ¯ S ) = σ m ( ¯ S ), σ n (¯ b k ) = σ m (¯ b k ) for k ≤ m , where σ m = ˙ σ Gm and for k ≤ m , ¯ b k = ˙¯ b I ∩ B γk k as well as b k = ˙ b I ∩ B γk k . Moreover, we can arrangethat h γ n (˙ b Gm ) ∈ I . For this last property, let ˙ b from the statement of Lemma 2.5 TERATION THEOREMS FOR SUBVERSIONS OF FORCING CLASSES 11 be a B γ m -name for h γ n (˙ b m ) and ˙¯ b a name for the preimage of b m under σ n . Sinceinductively, c m forces that ˇ h γ m (˙ b m ) ∈ ˙ G γ m , assumption (3) of the lemma is satisfied,and we get that h γ n (˙ b Gm ) = ˙ b G ∈ I , as wished.With these definitions, all the conditions (1)-(8) are satisfied, as long as theydon’t concern ˙ b n and ˙¯ b n . To define ˙ b n and ˙¯ b n , let I and G as described in theprevious paragraph. Let σ n = ˙ σ In , b m = ˙ b Im , M n = ran( σ n ), ¯ I = σ − n “ I , σ ∗ n :¯ N [ ¯ I ] ∼ ←→ M n [ I ] ≺ N [ I ] with σ n ⊆ σ ∗ n and σ ∗ n ( ¯ I ) = I .Working in V[ I ], we know that b m /I forces wrt. B δ /I that ˙ A/I is a maximalantichain in T (since b m ≤ ˙ b I = a ). Thus, the set D = { t ∈ T | ∃ y ∈ X ∃ ¯ t ≤ T t y ≤ δ b m ∧ h γ n ( y ) ∈ I and y/I (cid:13) B δ /I ˇ¯ t ∈ ˙ A/I } is dense in T . Note that D ∈ M n [ I ]. Working in M n [ I ], let A ⊆ D be a maximalantichain, so A ∈ M n [ I ]. Since B γ n preserves T as a Souslin tree, it follows that A is countable in M n [ I ], and so, A ⊆ M n [ I ]. Since D is dense in T , A is (in V[ I ])a maximal antichain in T , and A ⊆ T | Ω (note that crit( σ n ) = ω ¯ N = Ω). Hence,there is a t ∈ A with t < T s m . Since A ⊆ D , it follows that t ∈ D . Working in M n [ I ] again, let b n ∈ X , ¯ t ≤ T t witness that t ∈ D , i.e., b n ≤ δ b m , h γ n ( b n ) ∈ I and b n /I forces wrt. B δ /I that ˇ¯ t ∈ ˙ A/I . Let ¯ b n = ( σ ∗ m ) − ( b n ).Since all of this holds in V[ I ] whenever I is B γ n -generic over V and c n ∈ I , thereare B γ m -names ˙¯ t for ¯ t and ˙ b n for b n such that c n forces all of this wrt. B γ n . Inparticular, c n forces that ˙¯ t ∈ ˙ A and ˙¯ t < T s m . So (1)-(8) are satisfied. This finishesthe recursive construction.By (7), the sequence h c n | c < ω i is a thread, and so, c = V n<ω c n ∈ B + δ : ifcf( δ ) = ω , then this follows from part (2)(a) of Definition 2.10. And if cf( δ ) = ω ,then γ := sup n<ω γ n < δ and cf( γ ) = ω , so again, c ∈ B + γ ⊆ B + δ , for the samereason.We claim that c forces that ˙ A is bounded in ˇ T . To see this, let G be B δ -genericover V, with c ∈ G . For n < ω , let b n = ˙ b Gn .(C) For all n < ω , b n ∈ G .Proof of (C). Let n < ω . We have that h γ n ( b n ) ∈ G ∩ B γ n , by (4). For every l ≥ n , b l ≤ b n , by (8), so h γ l ( b l ) ≤ h γ l ( b n ). Since h γ l ( b l ) ∈ G ∩ B γ l , this implies that for l ≥ n , h γ l ( b n ) ∈ G ∩ B γ l . This holds for l < n as well, because in that case, h γ l ( b n ) ≥ h γ n ( b n ) ∈ G .Recall that ˜ δ = sup l<ω γ l . Let M n = ran( σ n ). It follows that sup( δ ∩ M n ) = ˜ δ (since σ n (¯ π ) = π ).Now, if cf( δ ) = ω , then ˜ δ < δ , and we know that b n ∈ X = S i<δ B i , so there isan i < δ such that b n ∈ B i . But since b n ∈ M n , the same is true in M n , and thismeans that there is an i < ˜ δ such that b n ∈ B i . But then, letting l > n be suchthat γ l > i , it follows that b n = h γ l ( b n ) ∈ G ∩ B γ l , so b n ∈ G , as claimed.If, on the other hand, cf( δ ) = ω , then since b n ∈ X , there is a thread h t i | i < δ i in ~ B ↾ δ such that b n = V i<δ t i . But then, for every l < ω , t γ l = h γ l ( b n ) ∈ G ∩ B γ l ⊆ G .By genericity (and thus V-completeness) of G , this implies that b n = V i<δ t i = V l<ω t γ l ∈ G . Note that we used that ~ B is an RCS iteration here. (cid:3) Note in particular that b = a ∈ G . Since this is true whenever c ∈ G , c (cid:13) B δ ˇ a ∈ ˙ G , which implies that c ≤ a . Moreover, c forces that ˙ A is bounded in ˇ T :working in V[ G ] again, where G ∋ c is B δ -generic, we have that for every n < ω ,there is a t n ∈ A = ˙ A G with t n < T s n , by (6). So A cannot contain a node a at alevel greater than Ω, because the predecessor of such an a at level Ω would have tobe of the form s m , for some m , and s m > a m . So a, a m ∈ A would be comparable.Thus, c forces wrt. B δ that ˙ A ⊆ ˇ T | ˇΩ. Case 2: for all i < δ , cf( δ ) > δ ( B i ).We may also assume that cf( δ ) > ω , for otherwise, cf( δ ) ≤ ω and the argumentof case 1 goes through (recall that we proved (B)).It follows as in [11, p. 143, claim (2)] that for i < δ , card( i ) ≤ δ ( B i ). But then,it follows that δ is regular, for otherwise, if i = cf( δ ) < δ , it would follow thatcf( δ ) = i ≤ δ ( B i ) < cf( δ ).Thus, δ is a regular cardinal, and δ ≥ ω . Hence, S δω , the set of ordinals lessthan δ with cofinality ω , is stationary in δ . For γ ∈ S δω , since B γ , being subproper,preserves ω , it follows that for every i < γ , (cid:13) B i cf(ˇ γ ) > ω . Thus, since ~ B is nicelysubproper, it follows by part (3) of Definition 2.10 that B γ is the direct limit of ~ B ↾ γ . Moreover, since for i < δ , δ ( B i ) < δ = cf( δ ), it follows by Observation 2.2that B i is <δ ( B i ) + -c.c., and hence <δ -c.c.It follows by Fact 2.8 that the direct limit of ~ B ↾ δ is <δ -c.c.Again, since for all i < δ , B i is <δ -c.c., it follows that B i forces that the cofinalityof δ is uncountable. So since ~ B is nicely subproper, it follows that B δ is the directlimit of ~ B ↾ δ , and hence that B δ is <δ -c.c.Now let h < δ , let G h ⊆ B h be generic, and let T ∈ V[ G h ] be a Souslin tree.We have to show that T is still Souslin in V[ G h ][ H ], whenever H ⊆ B δ /G h isgeneric over V[ G h ]. But if there were an uncountable antichain of T in V[ G h ][ H ] =V[ G h ∗ H ], then since B δ is δ -c.c. and δ > ω , it would follow that this antichainexists already in V[( G h ∗ H ) ∩ B i ], for some i ∈ [ h, δ ). Thus, B i /G h would fail tobe Souslin tree preserving in V[ G h ], contradicting our inductive assumption. (cid:3) Definition 2.15.
Let T be a Souslin tree and P a notion of forcing. P preservesthe Souslinness of T if (cid:13) P “ ˇ T is Souslin.” Theorem 2.16.
Let T be a Souslin tree. Then the class of subproper Booleanalgebras that preserve the Souslinness of T is standard RCS iterable.Proof. Letting ~ B = h B i | i < α i be a standard RCS iteration of forcings that aresubproper and preserve T as a Souslin tree, we prove by induction on δ : whenever h ≤ δ < α and G h is B h -generic, then in V[ G h ], B δ /G h is subproper and preserves T as a Souslin tree.For this, the proof of Theorem 2.14 goes through almost without change. Somesteps of the argument are somewhat simpler in the present context, because theSouslin tree T is in the ground model. (cid:3) Looking over the proof of Theorem 2.14, one sees that the assumption that theiteration in question uses revised countable support was only used at stages of theiteration that acquire countable cofinality. Thus, we obtain nice iterability resultsfor the corresponding forcing classes.As before, the previous RCS iteration theorems imply:
TERATION THEOREMS FOR SUBVERSIONS OF FORCING CLASSES 13
Observation 2.17.
Every standard RCS iteration of subproper and Souslin treepreserving forcings is nicely subproper and Souslin tree preserving. Similarly, if T isa fixed Souslin tree then every standard RCS iteration of subproper and Souslinnessof T preserving forcings is nicely subproper and Souslinness of T preserving. Thus, we can generalize Theorems 2.14 and 2.16 as follows.
Theorem 2.18.
The class of subproper and Souslin tree preserving Boolean alge-bras is nicely iterable, and so is the class of subproper Boolean algebras that preservesome fixed Souslin tree T .Proof. Recall that in the proof of Theorem 2.14, we used the fact that the iterationwas just nicely subproper, rather than an RCS iteration, wherever possible. Theonly places in the argument that used that we were dealing with an RCS iterationoccurred at stages of the iteration that acquired countable cofinality. But thesestages are trivial in a nicely subproper T -preserving iteration. (cid:3) Nicely subproper iterations of [ T ] -preserving forcing.Definition 2.19. Let T be an ω -tree. Then [ T ] denotes the set of cofinal branchesof T , that is, the set of branches of T that have order type ω . We say that a forcingnotion P is [ T ] -preserving to express that P cannot add new cofinal branches through T , that is, that (cid:13) P [ ˇ T ] ⊆ ˇV. As with the preservation of a fixed Souslin tree, thereis a more general version of this preservation property: P is branch preserving iffor every ω -tree S , P is [ S ]-preserving. Theorem 2.20.
The class of subproper and branch preserving forcing notions isstandard RCS iterable.Proof.
We follow along the lines of the proof of Theorem 2.14. So let ~ B = h B i | i < α i be a standard RCS iteration of forcings that are subproper and branchpreserving. We prove by induction on δ : whenever h ≤ δ < α and G h is B h -generic,then in V[ G h ], B δ /G h is subproper and branch preserving. Case 1: δ is a limit ordinal and there is an i < δ such that cf( δ ) ≤ δ ( B i ).As before, it suffices to prove(B) if cf( δ ) ≤ ω then B δ adds no cofinal branch to an ω -tree.So let us fix an ω -tree T . Depending on whether the cofinality of δ is ω or ω ,let X = { V i<δ t i | ~t is a thread in ~ B ↾ δ } or X = S i<δ B i .Towards a contradiction, let ˙ B be a B δ -name such that some condition a ∈ X forces that ˙ B is a new cofinal branch, that is, that ˙ B / ∈ V B γ for all γ < δ .Let π : ω −→ δ be cofinal, with π (0) = 0. Let N = L Aτ , with H θ ⊆ N , such that θ verifies the subproperness of each B i , for i ≤ δ . Let S = h θ, δ, ~ B , X, ˙ B, T, π, a i .Let M ≺ N with S ∈ M , M countable and such that, letting σ : ¯ N −→ M bethe inverse of the Mostowski collapse, where ¯ N is full. Let ˜ δ = sup( M ∩ δ ), andlet Ω = M ∩ ω = crit( σ ) = ω ¯ N . Fix an enumeration h s n | n < ω i of T (Ω).Let σ − ( S ) = ¯ S = h ¯ θ, ¯ δ, ~ ¯ B , ¯ X, ˙¯ B, ¯ T , ¯ π, ¯ a i . Let h ν i | i < ω i be a sequence ofordinals ν i < ω ¯ N such that if we let ¯ γ i = ¯ π ( ν i ), it follows that h ¯ γ i | i < ω i is cofinalin ¯ N , and such that ν = 0, so that ¯ γ = 0. Let γ i = σ (¯ γ i ).By induction on n < ω , construct sequences h ˙ σ n | n < ω i , h c n | n < ω i , h ˙ b n | n < ω i and h ˙¯ b n | n < ω i with c n ∈ B γ n , ˙ σ n ∈ V B n , ˙ b n , ˙¯ b n ∈ V B γn , such that forevery n < ω , c n forces the following statements: (1) ˙ σ n : ˇ¯ N ≺ ˇ N ,(2) ˙ σ n ( ˇ¯ S ) = ˇ S , and for all k < n , ˙ σ n (˙¯ b k ) = ˙ σ k (˙¯ b k ),(3) ˙ σ n (˙¯ b n ) = ˙ b n and ˙¯ b n ∈ ˇ¯ X (and so, ˙ b n ∈ ˇ X ),(4) ˇ h γ n (˙ b n ) ∈ ˙ G B γn ,(5) ˙ σ − n “ ˙ G B γn is ˇ¯ N -generic for ˇ¯ B γ n ,(6) for some t ∈ T (Ω) with t < s n − , ˙ b n forces wrt. B δ that ˇ t ∈ ˙ B , and inparticular, ˙ b n forces that ˇ s n − / ∈ ˙ B (for n > c n − = h γ n − ( c n ) (for n > b n ≤ ˇ B δ ˙ b n − (for n > n = 0, we set c = 1l, ˙ σ = ˇ σ and ˙ b = ˇ a and ˙¯ b = σ − (˙ b ).Now suppose m = n −
1, and ˙ σ l , c l , ˙ b l and ˙¯ b l have been defined, so that (1)-(8) are satisfied for l ≤ m . An application of Lemma 2.5 (to B γ m ⊆ B γ n ) yields c n ∈ B γ n and a B γ n -name ˙ σ n such that (1)-(8) are satisfied at level n , as long asthey don’t refer to ˙ b n and ˙¯ b n , as before. As before, we can also arrange that c n forces that ˇ h γ n (˙ b m ) ∈ ˙ G γ n .To define ˙ b n and ˙¯ b n , let I be B γ n -generic with c n ∈ I , and let G = I ∩ B γ m . Let σ n = ˙ σ In , b m = ˙ b Gm , M n = ran( σ n ), ¯ I = σ − n “ I , σ ∗ n : ¯ N [ ¯ I ] ≺ M n [ I ] with σ n ⊆ σ ∗ n and σ ∗ n ( ¯ I ) = I . We have that h γ n ( b m ) ∈ I .Working in V[ I ], we know that b m /I forces wrt. B δ /I that ˙ B/I is a new cofinalbranch in ˇ T (that is, a branch that did not exist in V[ I ]). Consider the set of z ∈ B γ n such that there exist t , t ∈ T and y , y ≤ δ b m such that • t , t are incomparable in T , • z = h γ n ( y ) = h γ n ( y ), • y (cid:13) B δ ˇ t ∈ ˙ B , • y (cid:13) B δ ˇ t ∈ ˙ B .Then D ∈ M n , and D is dense in B γ n below h γ n ( b m ). So there is a z ∈ I ∩ D ,and this is witnessed by some t , t , y , y . Since M n [ I ] ≺ N [ I ], these objects maybe chosen in M n [ I ]. Thus, t , t ∈ T | Ω. Since t and t are incomparable in T , atmost one of them can be below s m . Say i < t i is not below s m . Thenwe can set b n = y i .Since all of this holds in V[ I ] whenever I is B γ n -generic over V and c n ∈ I , thereis a B γ n -name ˙ b n for b n such that c n forces all of this wrt. B γ n . Then (1)-(8) aresatisfied. This finishes the recursive construction.As before, the sequence h c n | n < ω i is a thread, and so, c = V n<ω c n ∈ B + δ . Itfollows that c (cid:13) ˙ b n ∈ ˙ G δ , for all n < ω . In particular, c ≤ a . We claim that c forces that ˙ B is bounded in ˇ T . To see this, let G be B δ -generic over V, with c ∈ G ,and let B = ˙ B G . Let n < ω . Since b n +1 := ˙ b Gn +1 ∈ G , we have by (6) that s n / ∈ B .Since this holds for all n < ω , B contains no node at level Ω of T . So B is boundedin T . This is a contradiction. Case 2: for all i < δ , cf( δ ) > δ ( B i ).Exactly as in the proof of Theorem 2.14, we may reduce to the case that δ > ω is a regular cardinal. It follows that B δ is <δ -c.c. Thus, if G h is B h -generic and B δ /G h added a new cofinal branch to some ω -tree T ∈ V[ G h ], then already someearlier B γ /G h would add this branch, contradicting our inductive hypothesis. (cid:3) TERATION THEOREMS FOR SUBVERSIONS OF FORCING CLASSES 15
Theorem 2.21.
Let T be an ω -tree. Then the class of subproper forcing notionsthat are [ T ] -preserving is standard RCS iterable.Proof. We can simplify the argument of the proof of Theorem 2.20, as before. (cid:3)
Again, we can easily modify the arguments of the proofs of the previous two theo-rems to obtain results about nice iterability. As before, the previous theorems showthat every standard iteration of subproper and branch preserving/[ T ]-preservingforcing notions is nicely subproper and branch preserving/[ T ]-preserving, so thatthe following theorem is a generalization of these theorems. Theorem 2.22.
The class of subproper and branch preserving forcing notionsis nicely iterable, and so is the class of subproper forcing notions that are [ T ] -preserving, for some fixed ω -tree T . RCS and nicely subproper iterations of ω ω -bounding forcing. We willnow follow Abraham’s handbook article [1], where this is carried out for countablesupport iterations of proper forcing notions, in showing that the stages of certainRCS iterations of ω ω -bounding and subproper forcing notions are also subproperand ω ω -bounding.For f, g ∈ ω ω and n ∈ ω , we write f ≤ n g if for all k ≥ n , f ( k ) ≤ g ( k ), andwe write f ≤ ∗ g if there is an n ∈ ω such that f ≤ n g . A forcing notion P is ω ω -bounding if whenever G is P -generic over V and f ∈ ( ω ω ) V[ G ] , then there is a g ∈ V such that f ≤ ∗ g , and in fact, in this case, there is then a g ∈ V such that f ≤ g . If ˙ f is a P -name for a real, then a weakly decreasing sequence ~p = h p i | i < ω i of conditions in P interprets ˙ f if there is an f ∗ ∈ ω ω such that for every n < ω , p n (cid:13) P ˙ f ↾ ˇ n = ˇ f ∗ ↾ ˇ n . In this case, we say that ~p interprets ˙ f as f ∗ , andwe write f ∗ = intp ( ˙ f , ~p ) to express this. If g ∈ ω ω and ˙ f is a P -name for a real,then we say that a weakly decreasing sequence ~p in P interprets ˙ f and respects g if intp ( ˙ f , ~p ) ≤ g . Theorem 2.23 ([1, Theorem 3.2]) . Let P be an ω ω -bounding forcing notion. Let ˙ f be a P -name for a real, let κ be a sufficiently large cardinal, and let H κ ⊆ N | = ZFC − , where N is transitive. Let M ≺ N be countable, with P , ˙ f ∈ M . Supposethat g ∈ ω ω dominates all reals in M and ~p ∈ M is such that intp ( ˙ f , ~p ) ≤ g .Then there is a condition p ∈ P ∩ M and a real h ∈ ω ω ∩ M such that h ≤ g and p (cid:13) P ˙ f ≤ ˇ h . In particular, p (cid:13) P ˙ f ≤ ˇ g . Moreover, for any n < ω , there is such acondition p with p ≤ p n . We will use the concept of a derived sequence . Suppose that A ⊆ B are completeBoolean algebras, and h = h B , A . In this situation, let ˙ f be a B -name for a real,and let ~b be a sequence in B + that interprets ˙ f , i.e., such that intp ( ˙ f ,~b ) exists.Let us fix a well-order of B , and let G be A -generic over V. Recall that b/G = 0iff h ( b ) ∈ G . Define the derived sequence ~a recursively, as follows: if h ( b i ) ∈ G ,then a i = b i . Note that in this case, h ( b j ) ∈ G for all j < i , so a j = b j for all j ≤ i . If not, then let a i be the least element of B + (with respect to the fixedwell-order) such that h ( a i ) ∈ G , a i ≤ a i − and a i decides ˙ f ↾ ˇ i . We write δ G ( ~b, ˙ f )for the derived sequence, and δ G ( ~b, ˙ f ) /G for the sequence h a i /G | i < ω i . Notethat by construction, δ G ( ~b, ˙ f ) /G is a weakly decreasing sequence in ( B /G ) + . Thefollowing lemma on derived sequences is completely general and has nothing to dowith subproperness vs. properness. Lemma 2.24 ([1, Lemma 3.3]) . Let A ⊆ B be complete Boolean algebras, where A is ω ω -bounding, and let h = h B , A . Let ˙ f ∈ V B be a name for a real, b ∈ B + , andsuppose that:(1) ~b is a weakly decreasing sequence below b in B + that interprets ˙ f ,(2) M ≺ N is countable, where for some sufficiently large cardinal κ , H κ ⊆ N | = ZFC − , N transitive, with A , B , ˙ f ,~b, b ∈ M ,(3) g ∈ ω ω bounds the reals of M and intp ( ˙ f ,~b ) ≤ g .Then there is a condition = a ≤ h ( b ) with a ∈ M such that a forces with respectto A that if ~a = δ ˙ G A (ˇ ~b ) , then ~a is below ˇ b , interprets ˇ˙ f and respects ˇ g , meaning that intp ( ˇ˙ f, ~a ) ≤ ˇ g . The following lemma is a version of [1, Lemma 1.1] for complete Boolean algebras.
Lemma 2.25.
Let A ⊆ B be complete Boolean algebras, and let G be A -genericover V . Suppose that in V[ G ] , there is a sequence h r i | i < ω i such that r i ∈ B and r i +1 /G ≤ B /G r i /G , for all i < ω . Then, in V[ G ] , there is a sequence h g i | i < ω i such that for all i < ω , g i ∈ G = { b ∈ B | ∃ a ∈ G a ≤ b } and, letting s i = r i ∧ g i , for i < ω , it follows that ~s is weakly decreasing in B .Proof. Recall that in this situation, G is a V-complete filter in B , and that thequotient B /G = B /G consists of equivalence classes with respect to the equivalencerelation that identifies b , b ∈ B iff ( b ⇒ b ) ∧ ( b ⇒ b ) ∈ G (where b ⇒ b = ¬ b ∨ b ). b/G denotes the equivalence class of b with respect to this equivalencerelation. The partial order on B /G is defined by b /G ≤ B /G b /G iff ( b ⇒ b ) ∈ G .Now let g i = V j
Lemma 2.26.
Let B ⊆ B ⊆ B be complete Boolean algebras. Note that h B , B = h B , B ↾ B . So let us write h for h B , B and h for h B , B .Suppose that B is subproper and ω ω -bounding, and that (cid:13) B “ ˇ B / ˙ G B is sub-proper and ω ω -bounding.” Let ˙ f ∈ V B be a name for a real, and let δ = δ ( B ) .Let N = L Aτ be such that H θ ⊆ N , where θ is sufficiently large, and fix s ∈ N .Let a ∈ B +0 , ¯ θ, ¯ B , ¯ B , ¯ B , ¯ s ∈ ¯ N , where ¯ N is countable, transitive and full, and ¯ B ⊆ ¯ B ⊆ ¯ B are complete Boolean algebras in ¯ N .Let g ∈ ω ω bound all the reals in ¯ N .Let ¯ S = h ¯ θ, ¯ B , ¯ B , ¯ B , ˙¯ f, ¯ s, ¯ λ , . . . , ¯ λ n i and S = h θ, B , B , B , ˙ f , s, λ , . . . , λ n i ,where λ i > δ is regular. Let ˙ σ , ˙ t , ˙¯ b , ˙ b be B -names and suppose that a forces:(A1) ˙ σ : ˇ¯ N ≺ ˇ N ,(A2) ˙ σ ( ˇ¯ S ) = ˇ S ,(A3) ˙ σ (˙¯ b ) = ˙ b and ˙¯ b ∈ ˇ¯ B (and so, ˙ b ∈ ˇ B ),(A4) ˇ h (˙ b ) ∈ ˙ G B ,(A5) ˙ t ∈ ˇ¯ N , TERATION THEOREMS FOR SUBVERSIONS OF FORCING CLASSES 17 (A6) ˙ σ − “ ˙ G B is ˇ¯ N -generic for ˇ¯ B ,(A7) Letting M = ran( ˙ σ ) , there is in M [ ˙ G B ] a decreasing sequence ~b of condi-tions in B +2 , below ˙ b , such that ~b/ ˙ G B is decreasing in ( B / ˙ G B ) + , and suchthat intp ( ˇ˙ f,~b ) ≤ ˇ g .Then there are a condition c ∈ B +1 with h ( c ) = a and a B -name ˙ σ such thatwhenever I is B -generic with c ∈ I , letting σ = ˙ σ I , G = I ∩ B and σ = ˙ σ G , thefollowing conditions hold:(1) σ : ¯ N ≺ N ,(2) σ ( ¯ S ) = S ,(3) σ (˙¯ b G ) = σ (˙¯ b G ) and σ ( ˙ t G ) = σ ( ˙ t G ) ,(4) h (˙ b G ) ∈ I ,(5) σ − “ I is ¯ B -generic over ¯ N ,(6) Letting ¯ λ = On ∩ ¯ N , we have that for i ≤ n , sup σ “¯ λ i = sup σ “¯ λ i .(7) Letting M = ran( σ ) , there is in M [ G ] a decreasing sequence ~c of conditionsin B +2 below ˙ b G such that ~c/I is decreasing in ( B /I ) + and intp ( ˙ f , ~c ) ≤ g .Proof. Let G be B -generic over V, with a ∈ G . Let σ = ˙ σ G , M = ran( σ ), b = ˙ b G , etc. Then σ : ¯ N ≺ M , and σ lifts to σ ∗ : ¯ N [ ¯ G ] ≺ M [ G ], where¯ G = σ − “ G . Since M [ G ] ≺ N [ G ], we will be able to apply Lemma 2.24 inV[ G ].Note that ω ω ∩ M [ G ] = ω ω ∩ ¯ N [ ¯ G ]. Since B is ω ω -bounding, ¯ N thinks that ¯ B is ω ω -bounding, and since ¯ G is ¯ B -generic over ¯ N , it follows that every real of ¯ N [ ¯ G ]is bounded by some real of ¯ N , which is bounded by g . Thus, the reals of ¯ N [ ¯ G ] arebounded by g , and hence, the reals of M [ G ] are bounded by g .The B -name ˙ f can be viewed as a B /G -name in the obvious way. Let’s write˙ f /G for this name. Then ˙ f /G is a B /G -name for a real.By assumption (A7), let ~b be a decreasing sequence of conditions in B , below b ,such that ~b/G is decreasing in B /G , and such that intp ( ˙ f ,~b ) ≤ g . This meansthat intp ( ˙ f /G ,~b/G ) ≤ g .Thus, by Lemma 2.24, applied to A = B /G , B = B /G , the name ˙ f /G ,the model M [ G ], the condition b/G and the sequence ~b/G , there is a condition d ∈ B +1 ∩ M [ G ] such that d/G ≤ h B /G , B /G ( b/G ) and such that d/G forceswith respect to B /G that if ~a = δ ˙ G B /G ( ~b/G ), then ~a is below ˇ b/G in ( B /G ) + ,interprets ( ˙ f /G )ˇand respects g .Let ˙ d be a B -name for d such that a forces (with respect to B ) that all of theseproperties hold. Let ˙¯ d be a B -name such that a forces that ˙ σ ∗ ( ˙¯ d ) = ˙ d .We can now apply Lemma 2.5 to B ⊆ B , ˙ σ , ˙¯ d , ˙ d getting a condition c ∈ B with h ( c ) = a and a B -name ˙ σ such that whenever I is B -generic over V with c ∈ I and G = I ∩ B , it follows that(1) σ : ¯ N ≺ N ,(2) σ ( ¯ S ) = S ,(3) σ (˙¯ b G ) = σ (˙¯ b G ) and σ ( ˙ t G ) = σ ( ˙ t G ),(4) σ ( ˙¯ d G ) = σ ( ˙¯ d G ) = ˙ d G ,(5) ˙ d G ∈ I ,(6) σ − “ I is ¯ B -generic over ¯ N , (7) for all i ≤ n , sup σ “¯ λ i = sup σ “¯ λ i .where δ = δ ( B ). To conclude the proof, we verify that we also have: h (˙ b G ) ∈ I ,and letting M = ran( σ ), there is in M [ G ] a decreasing sequence ~c of conditions in B below ˙ b G such that ~c/I is decreasing in B /I and intp ( ˙ f , ~c ) ≤ g .In the present situation, we have I ⊆ B is generic over V, G = I ∩ B is B -generic over V, and I/G = { x/G | x ∈ I } is B /G -generic over V[ G ]. Moreover, I = G ∗ ( I/G ) = { x ∈ B | x/G ∈ I/G } .Let b = ˙ b G ∈ B , b ′ = h ( b ) and d = ˙ d G ∈ B . We have that d/G ≤ h B /G , B /G ( b ), since this was forced by a . But h B /G , B /G ( b ) = h ( b ) /G (see[16, Prop. 4.9]). Thus, d/G ≤ b ′ /G . Since d ∈ I , we have that d/G ∈ I/G , and so, b ′ /G ∈ I/G . But since I = { x ∈ B | x/G ∈ I/G } , this means that b ′ = h ( b ) ∈ I ,as wished.Further, what d/G forces over V[ G ] is true in V[ G ][ I/G ] = V[ I ]. Thus, let ~a = δ I/G ( ~b/G ). Then ~a is below b/G in ( B /G ) + , interprets ˙ f /G and respects g .Note that ~a ∈ M [ G ], and ~a/ ( I/G ) is weakly decreasing in (( B /G ) / ( I/G )) + , whichwe can identify with B /I .For every i < ω , let a ′ i ∈ B be such that a i = a ′ i /G . We then have that for every i < ω , a i / ( I/G ) = ( a ′ i /G ) / ( I/G ), which we may identify with a ′ i /I . Thus, h a ′ i | i < ω i is a sequence in B +2 such that for all i < ω , a ′ i +1 /I ≤ B /I a ′ i /I , and a ′ i /I = 0.By replacing a ′ i with a ′ i ∧ b if necessary, we may assume that a ′ i ≤ b , for all i – notethat a i = a ′ i /G = ( a ′ i ∧ b ) /G since a i /G ≤ b/G .By Lemma 2.25, there is a sequence h q i | i < ω i in V[ I ] such that each q i ∈ I ∗ = { x ∈ B | ∃ y ∈ I y ≤ x } , and such that if we let c i = a ′ i ∧ q i , then ~c is weaklydecreasing in B . Since a ′ i /I = 0, we know that h B , B ( a ′ i ) ∈ I . If q i ≥ q ′ i ∈ I , thenwe have that h B , B ( a ′ i ∧ q ′ i ) = h B , B ( a ′ i ) ∧ q ′ i ∈ I (we used here [11, P. 87, secondbullet point] and the fact that q ′ i ∈ B ), and so, ( a ′ i ∧ q ′ i ) /I = 0. Since q i ≥ q ′ i , itfollows that c i /I = 0. And since c i ≤ a ′ i for all i , it follows that ~c interprets ˙ f andrespects g , and ~c is below b , as wished. (cid:3) We can now follow the proof templates of Theorems 2.14 and 2.20 to obtain ouriteration theorem for ω ω -bounding subproper forcing. Theorem 2.27.
The class of subproper ω ω -bounding Boolean algebras is standardRCS iterable.Proof. Let ~ B = h B i | i < α i be a standard RCS iteration of subproper and ω ω -bounding complete Boolean algebras. We prove by induction on δ : whenever h ≤ δ < α and G h is B h -generic, then in V[ G h ], B δ /G h is subproper and Souslin treepreserving.As before, it suffices to focus on the ω ω -bounding property, and it suffices tofocus on the case that δ is a limit ordinal. Case 1: there is an i < δ such that cf( δ ) ≤ δ ( B i ).Let i < δ have this property. As before, it suffices to prove:(A) if i < j < δ and G j is B j -generic, then in V[ G j ], B δ /G j is ω ω -bounding.This further reduces to showing(B) if cf( δ ) ≤ ω then B δ is ω ω -bounding.As before, we can define a dense set X ⊆ B δ , depending on the cofinality of δ : if cf( δ ) = ω , then X = S i<δ B i , and if cf( δ ) = ω , then { V i<δ t i | h t i | i < δ i is a thread in ~ B ↾ δ } . TERATION THEOREMS FOR SUBVERSIONS OF FORCING CLASSES 19
Let π : ω −→ δ be cofinal, with π (0) = 0.Let ˙ f be a B δ -name for a member of ω ω , and let a ∈ B δ be a condition. Wehave to find g ∈ ω ω and a condition extending a that forces that ˙ f ≤ ˇ g . Since X is dense in B δ , we may assume that a ∈ X .Let N = L Aτ , with H θ ⊆ N , such that θ verifies the subproperness of each B i , for i ≤ δ . Let S = h θ, δ, ~ B , X, ˙ f , π, a i . Let M ≺ N with S ∈ M , M countable andsuch that, letting σ : ¯ N −→ M be the inverse of the Mostowski collapse (so that¯ N is transitive), ¯ N is full. Let σ − ( S ) = ¯ S = h ¯ θ, ¯ δ, ~ ¯ B , ¯ X, ˙¯ f, ¯ π, ¯ a i . Let h ν i | i < ω i be a sequence of ordinals ν i < ω ¯ N such that if we let ¯ γ i = ¯ π ( ν i ), it follows that h ¯ γ i | i < ω i is monotone and cofinal in ¯ δ , and such that ν = 0, so that ¯ γ = 0. Hence,letting γ i = σ (¯ γ i ), we have that sup i<ω γ i = sup( M ∩ δ ). Moreover, whenever σ ′ : ¯ N ≺ N is such that σ ′ (¯ π ) = π , it follows that σ ′ (¯ γ i ) = γ i = π ( ν i ), as before.Working inside M , construct a decreasing sequence h a i | i < ω i that interprets˙ f as some f ∗ ∈ M (so ~a starts with the given condition fixed above). Let g boundthe reals of M , with f ∗ ≤ g .By induction on n < ω , construct sequences h ˙ σ n | n < ω i , h c n | n < ω i , h ˙ b n | n < ω i and h ˙¯ b n | n < ω i with c n ∈ B γ n , ˙ σ n ∈ V B n , ˙ b n , ˙¯ b n ∈ V B γn , such that forevery n < ω , c n forces the following statements:(1) ˙ σ n : ˇ¯ N ≺ ˇ N .(2) ˙ σ n ( ˇ¯ S ) = ˇ S , and for all k < n , ˙ σ n (˙¯ b k ) = ˙ σ k (˙¯ b k ).(3) ˙ σ n (˙¯ b n ) = ˙ b n and ˙¯ b n ∈ ˇ¯ B ¯ δ (and so, ˙ b n ∈ ˇ B δ ). Moreover, ˙¯ b n ∈ ˇ¯ X , so ˙ b n ∈ ˇ X .(4) ˇ h γ n (˙ b n ) ∈ ˙ G B γn .(5) ˙ σ − n “ ˙ G B γn is ˇ¯ N -generic for ˇ¯ B γ n .(6) Letting M n = ran( ˙ σ n ), there is in M n [ ˙ G B γn ] a decreasing sequence of con-ditions in ˇ B δ , below ˙ b n , such that ~b/ ˙ G B δ is decreasing in (ˇ B δ / ˙ G B γn ) + , andsuch that intp ( ˇ˙ f,~b ) ≤ ˇ g .(7) ˙ b n decides ˙ f ↾ ˇ n (with respect to B δ ), and ˙ b n (cid:13) ˇ B δ ˙ f ↾ ˇ n ≤ ˇ g ↾ ˇ n .(8) c n − = h γ n − ( c n ) (for n > b n ≤ ˇ B δ ˙ b n − (for n > n = 0, we set c = 1l, ˙ σ = ˇ σ and ˙ b = ˇ a and ˙¯ b = σ − (˙ b ).Clearly then, (1)-(7) are satisfied for n = 0 – for (3) and (6), note that we picked ~a in X . Conditions (8)-(10), as well as the second part of (2), are vacuous for n = 0.Now suppose m = n −
1, and ˙ σ l , c l , ˙ b l and ˙¯ b l have been defined, so that (1)-(10)are satisfied for l ≤ m .We want to apply Lemma 2.26 in the present situation. Here is the conversion:Lemma 2.26 Current situation B , B , B B γ m , B γ n , B δ ˙ σ ˙ σ m ˙ t h ˙¯ b , . . . , ˙¯ b m i ˙¯ b, ˙ b ˙¯ b m , ˙ b m a c m The assumptions (1)-(7) stated in the lemma are then satisfied, by our inductiveassumption, and the lemma then guarantees the existence of certain objects, whichwe convert to the current situation as follows:
Lemma 2.26 Current situation c c n ˙ σ ˙ σ n We are thus left to define ˙ b n and ˙¯ b n . To do this, let I be B γ n -generic over V,and let G = I ∩ B γ m , with c n ∈ I . Working in V[ I ], let σ n = ˙ σ In , σ m = ˙ σ Gm , t = ˙ t G , b k = ˙ b Gk and ¯ b k = ˙¯ b Gk , for k ≤ m . Since ˙ σ n and c n were chosen accordingto Lemma 2.26, we then have that σ n : ¯ N ≺ N , σ n ( ¯ S ) = S , σ n (¯ b k ) = b k for k ≤ m , h γ m ( b m ) ∈ I , ¯ I = σ − n “ I is ¯ B ¯ γ n -generic over ¯ N , and C Nδ ( B γn ) (ran( σ m )) = C Nδ ( B γn ) (ran( σ n )). So let M n = ran( σ n ), and let σ ∗ n : ¯ N [ ¯ I ] ≺ M [ I ] be such that σ n ⊆ σ ∗ n and σ ∗ n ( ¯ I ) = I . By conclusion (7) of Lemma 2.26, there is in M n [ I ] asequence ~r = h r i | i < ω i decreasing in B δ , below b m , such that ~r/I is decreasingin ( B δ /I ) + and intp ( ˙ f , ~r ) ≤ g . Let ˙ b n be a B γ n -name such that ˙ b In = r n , andsuch that c n forces that ˙ b n decides ˙ f ↾ ˇ n and forces ˙ f ↾ ˇ n ≤ ˇ g ↾ ˇ n . Finally, let ˙¯ b n be a B γ n -name for ( σ ∗ n ) − (˙ b n ).This concludes the construction of the sequences ~ ˙ σ , ~c , ~b and ~ ¯ b .Now, the sequence h c n | c < ω i is a thread, and it follows as before that c = V n<ω c n ∈ B + δ .We claim that c forces that ˙ f ≤ ˇ g . To see this, let G be B δ -generic over V, with c ∈ G . Let n < ω . Let b n := ˙ b Gn ∈ G . We show as before:(C) For all n < ω , b n ∈ G . Now by point (7) in our recursive construction, b n decides ˙ f ↾ ˇ n to be totallybounded by g ↾ n . This shows that ˙ f G ≤ g . Case 2: for all i < δ , cf( δ ) > δ ( B i ).It follows as before that B δ the direct limit of ~ B ↾ δ , and is <δ -c.c.But then it follows that if G is B δ -generic over V, then any real in V[ G ] is alreadyin V[ G ∩ B γ ], for some γ < δ , and hence is bounded by a real in V, since we knowinductively that B γ is ω ω -bounding. (cid:3) As before, the previous RCS iteration theorems imply:
Observation 2.28.
Every standard RCS iteration of subproper and ω ω -boundingforcings is nicely subproper and ω ω -bounding. And as before, we can generalize Theorem 2.27 as follows.
Theorem 2.29.
The class of subproper and ω ω -bounding Boolean algebras is nicelyiterable. Nice Iterations
In this section, we will adopt Miyamoto’s nice iterations from [14], and provepreservation theorems for generalizations of subproper and subcomplete forcing.Initially, we proved that subcomplete forcing can be iterated in this way, but thenrealized that we could drop a condition in the definition of subcompleteness. Wethen observed that the corresponding simplification works for the class of subproperforcing notions as well. Later, we learned that Miyamoto [15] (unpublished) alsomade this latter observation for subproper forcing.
TERATION THEOREMS FOR SUBVERSIONS OF FORCING CLASSES 21
Subcompleteness and ∞ -subcompleteness. Let us start by giving thedefinition of subcompleteness, as well as its simplification. Recall the definitionsof the density of a partial order (Definition 2.1) and fullness (Definition 2.3). Inorder to discuss these variations, we will present the definitions in the more gen-eral framework of Fuchs [2]. We will work with the “hull condition” version ofsubcompleteness. In order to formulate it, we use the following terminology.
Definition 3.1.
Let N = L Aτ = h L τ [ A ] , ∈ , A ∩ L τ [ A ] i be a ZFC − model, ε anordinal and X ∪{ ε } ⊆ N . Then C Nε ( X ) is the smallest Y ≺ N such that X ∪ ε ⊆ Y . Definition 3.2.
Let ε be an ordinal. A forcing P is ε - subcomplete if there is acardinal θ > ε which verifies the ε -subcompleteness of P , which means that P ∈ H θ ,and for any ZFC − model N = L Aτ with θ < τ and H θ ⊆ N , any σ : ¯ N −→ Σ ω N such that ¯ N is countable, transitive and full and such that P , θ, ε ∈ ran( σ ), any¯ G ⊆ ¯ P which is ¯ P -generic over ¯ N , and any ¯ s ∈ ¯ N , the following holds. Letting σ ( h ¯ ε, ¯ θ, ¯ P i ) = h ε, θ, P i , and setting ¯ S = h ¯ s, ¯ ε, ¯ θ, ¯ P i , there is a condition p ∈ P suchthat whenever G ⊆ P is P -generic over V with p ∈ G , there is in V[ G ] a σ ′ suchthat(1) σ ′ : ¯ N ≺ N ,(2) σ ′ ( ¯ S ) = σ ( ¯ S ),(3) ( σ ′ )“ ¯ G ⊆ G ,(4) C Nε (ran( σ ′ )) = C Nε (ran( σ )).In this parlance, P is subcomplete iff it is δ ( P )-subcomplete. It is easy to seethat increasing ε weakens the condition of being ε -subcomplete. Thus, we referto the version of subcompleteness obtained by dropping the hull condition 4 as ∞ -subcompleteness . This makes sense if one interprets ∞ as On ∩ N in Definition3.1. Since in our context, N is a ZFC − model of the form L Aτ , it follows then that C N ∞ ( ∅ ) = N , and hence that the hull condition is vacuous.It was pointed out in [2] that the hull condition is somewhat unnatural, be-cause the density of a partial order is not preserved under forcing equivalence. Itwas shown there that the ε -subcompleteness of a partial order, however, is pre-served under forcing equivalence, and it is easy to see that the same is true of ∞ -subcompleteness. Another flaw of the concept of subcompleteness that was ad-dressed in [2] is that it is unclear whether factors of subcomplete forcing notions aresubcomplete. The result of [2] that factors of ε -subcomplete forcing notions are ε -subcomplete carries over to ∞ -subcompleteness; in fact, it simplifies slightly, sinceone does not need to worry about proving that the factor satisfies the hull condition.As a result the proof of this fact below is much simpler than the corresponding onein [2]. Theorem 3.3.
Let P be a poset, and let ˙ Q be a P -name for a poset, such that P ∗ ˙ Q is ∞ -subcomplete. Then P is ∞ -subcomplete.Proof. Let θ be large enough to verify that P ∗ ˙ Q is ∞ -subcomplete. We claim thatit is also large enough to verify that P is subcomplete. To see this, let N = L τ [ A ],be a ZFC − model with τ > θ regular, and H θ ⊆ N . Fix a parameter s ∈ N and let σ : ¯ N ≺ N with ¯ N countable, transitive and full so that σ ( P , ˙ Q , θ, s ) = P , ˙ Q , θ, s . Let G ⊆ P be generic over ¯ N . Let ( p, ˙ q ) be a condition witnessing the ∞ -subcompleteness of P ∗ ˙ Q and let ( p, ˙ q ) ∈ G ∗ H be P ∗ ˙ Q -generic over V. Workin V[ G ∗ H ] and let σ ′ : ¯ N ≺ N be an embedding so that σ ′ ( P , ˙ Q , θ, s ) = P , ˙ Q , θ, s and σ ′ “ ¯ G ⊆ G . Fixing an enumeration of ¯ N in order type ω , we can consider thetree T G of finite initial segments of an elementary embedding σ : ¯ N ≺ N with σ ( P , ˙ Q , θ, s ) = P , ˙ Q , θ, s and so that σ ′ “ ¯ G ⊆ G . Note that this tree is in fact inV[ G ]. Moreover, in V[ G ∗ H ] it’s ill-founded since σ ′ generates an infinite branch.But then by the absoluteness of ill-foundedness, T G is ill-founded in V[ G ]. So thereis an infinite branch in V[ G ] and this branch witnesses that P is ∞ -subcomplete. (cid:3) The following observation underlines the simplicity and elegance of the conceptof ∞ -subcomplete forcing. Proposition 3.4 (Essentially Lemma 2.3 of [2]) . ∞ -subcomplete forcings are closedunder forcing equivalence.Proof. The argument of [2, Lemma 2.3] goes through. Let P be ∞ -subcompleteand Q be forcing equivalent to P . By this, we mean that P and Q have the sameforcing extensions - this can be expressed in a first order way. To show that Q is ∞ -subcomplete, let θ be large enough to verify the ∞ -subcompleteness of P andassume P ( Q ∪ P ) ∈ H θ . Let σ : ¯ N ≺ N, s, ¯ Q , ¯ H , etc be as in the definition of ∞ -subcompleteness, where ¯ H is ¯ Q -generic over ¯ N and σ ( ¯ Q ) = Q . We may assumethat P ∈ ran( σ ), and write ¯ P = σ − ( P ). By elementarity, ¯ N believes that ¯ P and¯ Q are forcing equivalent, and hence, there is a ¯ G which is ¯ P -generic over ¯ N , suchthat ¯ N [ ¯ G ] = ¯ N [ ¯ H ]. Since P is ∞ -subcomplete, there is a condition p ∈ P so thatif G ∋ p is P -generic over V then in V [ G ] there is an embedding σ ′ : ¯ N ≺ N asin the definition of ∞ -subcompleteness, so σ ′ “ ¯ G ⊆ G . We may also assume that σ ′ ( ¯ Q ) = Q . σ ′ then lifts to an embedding ˜ σ : ¯ N [ ¯ G ] ≺ N [ G ] with ˜ σ ( ¯ G ) = G . Byelementarity of ˜ σ , letting H = ˜ σ ( ¯ H ), it follows that H is Q -generic over N , andsince N contains all subsets of Q , it follows that H is Q -generic over V. Moreover,by elementarity of ˜ σ , we see that N [ G ] = N [ H ], so G ∈ V[ H ], so σ ′ ∈ V[ H ]. Andclearly, since H = ˜ σ ( ¯ H ), it follows that ˜ σ “ ¯ H ⊆ H , that is, σ ′ “ ¯ H ⊆ H . Since σ ′ ∈ V[ H ], there is a condition q ∈ H that forces the existence of an embeddinglike σ ′ , showing that Q is ∞ -subcomplete. (cid:3) It is unclear if the corresponding fact is true for subcomplete forcing notionssince there are forcing equivalent notions with radically different densities.We do not know whether the iteration theorems for nicely subcomplete iterations,which give great leeway in how limit stages of the iteration are formed, can becarried out without some version of the hull or suprema condition. But if weuse Miyamoto’s method of forming limits in nice iterations, it turns out that wedo not need any hull or suprema conditions. All other preservation propertiesof subcomplete forcing notions that we know of actually do not need the hull orsuprema condition either, and thus are preservation properties of ∞ -subcompleteforcing. We list some in the following observation. Observation 3.5.
Let P be ∞ -subcomplete.(1) P preserves stationary subsets of ω .(2) P preserves Souslin trees.(3) P preserves the principle ♦ .(4) P does not add reals. Apart from simplifying the theory, however, we do not have a particular use ofthe concept of ∞ -subcompleteness. In fact, the following question is open: TERATION THEOREMS FOR SUBVERSIONS OF FORCING CLASSES 23
Question 3.6.
Is every ∞ -subcomplete forcing also subcomplete?However, we feel that simplifying a highly technical concept such as subcom-pleteness is worthwhile in its own right.The same “ ∞ ”-modification made to subcomplete forcing notions can be madeto subproper forcing notions as well. The proofs that ∞ -subproperness is invariantunder forcing equivalence and that factors of ∞ -subproper forcing notions are ∞ -subproper follow the proofs of the corresponding results given above, so we leavethe details to the interested reader and just list this definition and fact. Note thatthe definition of ∞ -subproperness results from dropping the suprema condition (d)from Definition 2.4. We repeat it in full below for completeness, and since weuse partial orders rather than Boolean algebras, as the former will be used in theiteration theorem. Definition 3.7.
A forcing notion P is ∞ - subproper if every sufficiently large cardi-nal θ verifies the ∞ -subproperness of P , meaning that the following holds: P ∈ H θ ,and if τ > θ is such that H θ ⊆ N = L Aτ | = ZFC − , and σ : ¯ N ≺ N , where ¯ N iscountable and full, and ¯ S = h ¯ θ, ¯ P , ¯ p, ¯ s i ∈ ¯ N , S = h θ, P , p, s i = σ ( ¯ S ), where ¯ p ∈ P ,for 1 ≤ i ≤ n , then there is a q ∈ P such that q ≤ p and such that whenever G ⊆ P is generic with q ∈ G , then there is a σ ′ ∈ V[ G ] such that(a) σ ′ : ¯ N ≺ N .(b) σ ′ ( ¯ S ) = σ ( ¯ S ).(c) ( σ ′ ) − “ G is P -generic over ¯ N . Lemma 3.8. ∞ -subproper forcing notions are closed under factors and forcingequivalence. Let us make a remark on the suprema condition that is part of Definition 2.4and is omitted in Definition 3.7. When motivating his definition of subproperness,Jensen writes in [10, §
0, p. 3] on this condition:
We needed (d) to handle certain regular limit points in the iter-ation. The experts on the subject may well be able to modify oreliminate (d).
This prediction came true, as Miyamoto showed, in unpublished work, for thecase of subproperness, which we learned after proving our iteration theorems fornice iterations. Our contribution is that this can be done for subcompleteness aswell, and that some subclasses of subproper forcing, exhibiting more preservationproperties, without requiring the hull or suprema condition, can be iterated in thisway as well. The additional preservation properties of the subclasses we consider arethe same properties that Miyamoto imposed on the class of semiproper forcing inhis iteration theorems from [14] and [13]. Our proofs in this section are adaptationsof Miyamoto’s arguments in the context of semiproper forcing. The honor of beingconsidered an expert on the subject by Jensen is entirely Miyamoto’s.3.2.
The theory of nice iterations.
We collect here first the relevant facts anddefinitions from [14]. For a more in depth discussion, including proofs, see thatarticle. For basic notions of projection etc, see the introduction there. For asequence x we denote its length by l ( x ). Definition 3.9 (Iterations) . Let ν be a limit ordinal. A sequence of separativepartial preorders of length ν , h ( P α , ≤ α , α ) | α < ν i is called a general iteration ifffor any α ≤ β < ν the following holds(1) For any p ∈ P β , p ↾ α ∈ P α and 1 β ↾ α = 1 α .(2) For any p ∈ P α and any q ∈ P β , if p ≤ α q ↾ α then p ⌢ q ↾ [ α, β ) ∈ P β and p ⌢ q ↾ [ α, β ) ≤ β q (3) For any p, q ∈ P β , if p ≤ β q then p ↾ α ≤ α q ↾ α and p ≤ β p ↾ α ⌢ q ↾ [ α, β ).(4) If β is a limit ordinal and p, q ∈ P β , p ≤ β q if and only if for all α < βp ↾ α ≤ α q ↾ α .A general iteration h ( P α , ≤ α , α ) | α < ν i is an iteration iff for every limit ordinal β < ν and all p, q ∈ P β , p ≤ β q iff for all α < β , p ↾ α ≤ α q ↾ α .We will use the following fact (and the notation introduced there). Fact 3.10 (see [14, Prop. 1.3]) . Let h ( P α , ≤ α , α ) | α < ν i be a general iteration,and let α ≤ β < ν . Then(1) Let G β be P β -generic over V . Set G β ↾ α = { p ↾ α | p ∈ G β } , G β ↾ [ α, β ) = { p ↾ [ α, β ) | p ∈ G β } , and let P α,β = P β / ( G β ↾ α ) = { p ↾ [ α, β ) | p ∈ P β and p ↾ α ∈ G β ↾ α } be equipped with the ordering p ≤ α,β q iff there is an r ∈ G β ↾ α such that r ⌢ p ≤ β r ⌢ q . Then G β ↾ α is P α -generic over V and G β ↾ [ α, β ) is P α,β -generic over V[ G β ↾ α ] .(2) If G α is P α -generic over V and H is P α,β = P β /G α -generic over V[ G α ] ,then G α ∗ H = { p ∈ P α | p ↾ α ∈ G α and p ↾ [ α, β ) ∈ H } is P β -generic over V , ( G α ∗ H ) ↾ α = G α and ( G α ∗ H ) ↾ [ α, β ) = H .(3) Let G β be P β -generic over V . Then a condition p ∈ P β is in G β iff p ↾ α ∈ G β ↾ α and p ↾ [ α, β ) ∈ G β ↾ [ α, β ) . In what follows we suppress the notation ≤ α , α and associate a partial preorderwith its underlying set. We have the following useful proposition. Proposition 3.11 (Proposition 1.7 of [14]) . Let h P α | α < ν i be an iteration and β < ν limit. Then for any p ∈ P β and any P β generic G β we have p ∈ G β if andonly if for all α < β p ↾ α ∈ G β ↾ α . From now on we always assume sequence ~P = h P α | α < ν i is an iteration. Letus quickly define the relevant definitions: nested antichain, S ∠ T , mixtures and( T, β )-nice. We refer the reader to [14] for an in depth discussion of these ideas andtheir significance.
Definition 3.12 (The machinery of Nice Iterations) . A nested antichain in ~ P is atriple h T, h T n | n < ω i , h suc nT | n < ω ii so that(1) T = S n<ω T n (2) T consists of a unique element of some P α for α < ν For each n < ω we have that(3) T n ⊆ S { P α | α < ν } and suc nT : T n → P ( T n +1 )(4) For a ∈ T n and b ∈ suc nT ( a ), l ( a ) ≤ l ( b ) and b ↾ l ( a ) ≤ a Here, ( P , ≤ ,
1) is a partial preorder if ( P , ≤ ) is reflexive and transitive, and if 1 is a greatestelement. There may be several such greatest elements since P is not required to be antisymmetric.If p ≤ q and q ≤ p , then we write p ≡ q . TERATION THEOREMS FOR SUBVERSIONS OF FORCING CLASSES 25 (5) For a ∈ T n the set of all b ↾ l ( a ) so that b ∈ suc nT ( a ) forms a maximalantichain below a in P l ( a ) . In particular any two elements in this set areincompatible and it is non-empty.(6) T n +1 = S { suc nT ( a ) | a ∈ T n } Given such a nested antichain T in ~ P with { a } = T , for a condition p ∈ P β with β < ν we say that p is a mixture of T up to β if for all i < β the condition p ↾ i forces that(1) p ↾ [ i, i + 1) ≡ a ↾ [ i, i + 1) if i < l ( p ) and a ↾ i ∈ ˙ G i , the canonical namefor the P i -generic.(2) p ↾ [ i, i + 1) ≡ s ↾ [ i, i + 1) if there is ( r, s ) so that r, s ∈ T with s ∈ suc nT ( r )for some n and l ( r ) ≤ i < l ( s ) and s ↾ i ∈ ˙ G i .(3) p ↾ [ i, i + 1) ≡ i +1 ↾ [ i, i + 1) if there is a sequence h a n | n < ω i so that a ∈ T and for all n < ω a n +1 ∈ suc nT ( a n ) and l ( a n ) ≤ i and a n ∈ ˙ G i ↾ l ( a n ).If β is a limit ordinal we say that a sequence p of length β (not necessarily in P β ) is ( T, β )- nice if for all α < β , p ↾ α is a mixture of T up to α .Finally, given two nested antichains S and T in ~ P we define S ∠ T (“ S hooks T ”) if for every n < ω and all b ∈ S n there is a a ∈ T n +1 so that l ( a ) ≤ l ( b ) and b ↾ l ( a ) ≤ a .We will need the following characterization of mixtures. Fact 3.13 (see [14, Prop. 2.5]) . Let T be a nested antichain in an iteration h P α | α < ν i , β < ν and p ∈ P β . Then p is a mixture of T up to β iff the following hold:(1) Let T = { a } and µ = min( l ( a ) , β ) . Then a ↾ µ ≡ p ↾ µ .(2) For any a ∈ T , letting µ = min( l ( a ) , β ) , we have that a ↾ µ ≤ p ↾ µ .(3) If n < ω , a ∈ T n , b ∈ suc nT ( a ) and l ( a ) ≤ β , then, letting µ = min( β, l ( b )) ,we have that b ↾ µ ≡ b ↾ l ( a ) ⌢ p ↾ [ l ( a ) , µ ) .(4) For any i ≤ β and any q ∈ P i with q ≤ i p ↾ i , if q forces with respect to P i that there is a sequence h a n | n < ω i such that a ∈ T , and for all n < ω , a n +1 ∈ suc nT ( a n ) , l ( a n ) ≤ i and a n ∈ ˙ G i ↾ l ( a n ) , then q ⌢ β ↾ [ i, β ) ≡ q ⌢ p ↾ [ i, β ) . This previous definition combines Definitions 2.0, 2.4 and 2.10 of [14]. Thefollowing is Definition 3.6 in Miyamoto’s article.
Definition 3.14 (Nice Iterations) . An iteration h P α | α < ν i is called nice if(1) For any i such that i + 1 < ν if p ∈ P i and τ is a P i name such that p (cid:13) i “ τ ∈ P i +1 and τ ↾ i ∈ ˙ G i ” then there is a q ∈ P i +1 so that q ↾ i = p and p (cid:13) i τ ↾ [ i, i + 1) ≡ q ↾ [ i, i + 1).(2) For any limit ordinal β < ν and any sequence x of length β , x ∈ P β ifand only if there is a nested antichain T in h P α | α < β i such that x is( T, β )-nice.We will use the following facts.
Lemma 3.15 (Lemma 2.7 of [14]) . Let ν be a limit ordinal and A ⊆ ν be cofinal.Suppose that T is a nested antichain in an iteration h P α | α < ν i and p is asequence of length ν such that p is ( T, ν ) -nice. Then for any β < ν and any s ∈ P β strengthening p ↾ β we get a nested antichain S so that (1) If T = { a } and S = { b } then l ( b ) ∈ A and l ( a ) , β ≤ l ( b ) .(2) For any b ∈ S , l ( b ) ∈ A .(3) r = s ⌢ p ↾ [ β, ν ) is ( S, ν ) -nice. Lemma 3.16 (Lemma 2.11 of [14]) . Let h P α | α < ν i be an iteration with limitordinal ν and A ⊆ ν a cofinal subset of ν . If ( T, U, p, q, r ) satisfy the following: T and U are nested antichains, p and q are sequences of length ν with p ( T, ν ) -nice and q ( U, ν ) -nice and r ∈ T so that q ↾ l ( r ) ≤ r and for all α ∈ [ l ( r ) , ν ) , q ↾ α ≤ r ⌢ p ↾ [ l ( r ) , α ) ; then there is a nested antichain S in h P α | α ≤ ν i so that q is ( S, ν ) -nice, if { b } = S then l ( r ) ≤ l ( b ) and b ↾ l ( r ) ≤ r , for all s ∈ S , l ( s ) ∈ A and S ∠ T . We also recall the definition of a fusion structure.
Definition 3.17 (Fusion Structure) . Let ~ P = h P α | α < ν i be an iterationwith limit ordinal ν . Given a nested antichain T in ~ P ↾ ν we call a structure h q ( a,n ) , T ( a,n ) | a ∈ T n , n < ω i a fusion structure if for all n < ω and a ∈ T n thefollowing hold:(1) T ( a,n ) is a nested antichain in h P α | α < ν i .(2) q ( a,n ) ∈ P ν is a mixture of T ( a,n ) up to ν .(3) a ≤ q ( a,n ) ↾ l ( a ) and if { p } = T ( a,n )0 then l ( a ) = l ( p ).(4) For any b ∈ suc nT ( a ) , T ( b,n +1) ∠ T ( a,n ) so q ( b,n +1) ≤ q ( a,n ) .If p ∈ P ν is a mixture of T up to ν then we call p the fusion of the fusion structure. Proposition 3.18 (Proposition 3.5 of [14]) . Let h P α | α ≤ ν i be an iteration withlimit ordinal ν . If p ∈ P ν is a fusion of a fusion structure h p ( a,n ) , T ( a,n ) | a ∈ T n , n < ω i then there is a sequence h a n | n < ω i such that p forces the followinghold:(1) a ∈ T , and for all n < ω a n +1 ∈ suc nT ( a n ) , q n ∈ ˙ G ν ↾ l ( a n ) and p ( a n ,n ) ∈ ˙ G ν .(2) If β = sup { l ( a n ) | n < ω } then p ( a n ,n ) ↾ β ∈ ˙ G ν ↾ β and p ( a n ,n ) ↾ [ β, ν ) ≡ ν ↾ [ β, ν ) . Nice iterations of ∞ -subcomplete forcing. First we prove that ∞ -sub-complete forcing is preserved under nice iterations. We use the following notationalconvention: if h P α | α ≤ ν i is an iteration then for i ≤ j ≤ ν the poset P i,j , whichis defined in Fact 3.10, depends on the P i -generic chosen so we will identify it withits P i name P j / ˙ G i .The special case i = 0 and j = ν of following theorem implies that if everysuccessor stage of a nice iteration is forced to be ∞ -subcomplete, then so is theiteration. Theorem 3.19.
Let ~ P = h P α | α ≤ ν i be a nice iteration so that P = { } and forall i with i + 1 < ν , (cid:13) i P i,i +1 is ∞ -subcomplete. Then for all j ≤ ν the followingstatement ϕ ( j ) holds: if i ≤ j , p ∈ P i , ˙ σ ∈ V P i , θ is a sufficiently large cardinal, τ is an ordinal, H θ ⊆ N = L τ [ A ] | = ZFC − , ¯ N is a transitive model, ¯ s, ¯ ~ P , ¯ i, ¯ j ∈ ¯ N , ¯ G ¯ i , ¯ G ¯ i, ¯ j ⊆ ¯ N ,and p forces with respect to P i that the following assumptions hold:(A1) ˙ σ (ˇ¯ ~ P , ˇ¯ i, ˇ¯ j, ˇ¯ θ, ˇ¯ G ¯ i ) = ˇ ~ P , ˇ i, ˇ j, ˇ θ, ˙ G i , TERATION THEOREMS FOR SUBVERSIONS OF FORCING CLASSES 27 (A2) the following holds in V : ¯ G ¯ i is ¯ P ¯ i -generic over ¯ N and ¯ G ¯ i, ¯ j is ¯ P ¯ i, ¯ j -genericover ¯ N [ ¯ G ¯ i ] , where ¯ P ¯ i, ¯ j = ¯ P j / ¯ G ¯ i ,(A3) ˙ σ : ˇ¯ N [ ˇ¯ G ¯ i ] ≺ ˇ N [ ˙ G i ] is countable, transitive and full. then there is a condition p ∗ ∈ P j such that p ∗ ↾ i = p and whenever G j ∋ p ∗ is P j -generic, then in V[ G j ] , there is a σ ′ such that, letting σ = ˙ σ G i , the followinghold:(a) σ ′ (¯ s, ¯ ~ P , ¯ i, ¯ j, ¯ θ, ¯ G ¯ i ) = σ (¯ s, ¯ ~ P , ¯ i, ¯ j, ¯ θ, ¯ G ¯ i ) ,(b) ( σ ′ )“ ¯ G ¯ i, ¯ j ⊆ G i,j ,(c) σ ′ : ¯ N [ ¯ G ¯ i ] ≺ N [ G i ] . Let us stress that our proof is similar to that of [14, Lemma 4.3], adapting forthe case of ∞ -subcomplete forcings in place of semiproper forcings. Proof.
The proof is by induction on j . So let us assume that ϕ ( j ′ ) holds for every j ′ < j . Fix some i ≤ j . Since nothing is to be shown when i = j , let i < j . Inparticular, the case j = 0 is trivial.Let us fix y ∈ P i , ˙ σ ∈ V P i , θ , τ , A , N , ¯ N , ¯ s, ¯ ~ P , ¯ i, ¯ j ∈ ¯ N , ¯ G ¯ i , ¯ G ¯ i, ¯ j ⊆ ¯ N so thatassumptions (A1)-(A3) hold.When it causes no confusion, if a is in the range of some elementary embedding σ : ¯ N ≺ N then we will always let σ ( a ) = a . Often we will not note this explicitly.Case 1: j is a limit ordinal.Let { t n | n < ω } enumerate the elements of ¯ N . Throughout this proof we willidentify the t n ’s with their check names when it causes no confusion. Without lossof generality we may assume that t = ∅ . Also let { p n | n < ω } enumerate theelements of ¯ G ¯ i ∗ ¯ G ¯ i, ¯ j , where we assume that ¯ p = 1 ¯ P ¯ j . It follows then that p forcesthat ˙ σ ( p ) = ˇ1 j , so we write p = 1 j .Note that since in some forcing extension, there is an elementary embeddingfrom ¯ N to N = L τ [ A ], we may assume that there is a definable well order of ¯ N ,call it ≤ ¯ N . As noted in [14], by Lemma 3.15, in ¯ N , ¯ p is a mixture up to ¯ j ofsome nested antichain in ¯ ~ P ↾ ¯ j whose root has length ¯ i . Letting ¯ W be the ≤ ¯ N -leastone, we know that p forces that ˙ σ ( ˇ¯ W ) is the L τ [ A ]-least nested antichain W in ~ P ↾ j such that p is a mixture of W up to j , since we know that p forces that ¯ p , ¯ i, ¯ j aremapped to p , i, j by ˙ σ , respectively.We will define a nested antichain h T, h T n | n < ω i , h suc nT | n < ω ii , a fusion struc-ture hh q ( t,n ) , T ( a,n ) i | n < ω, a ∈ T n i in h P α | α ≤ j i and a sequence h ˙ σ ( a,n ) | n <ω, a ∈ T n i so that the following conditions hold.(1) T = { p } , q ( p, = p , T ( p, = W and ˙ σ ( p, = ˙ σ .Further, for any n < ω and a ∈ T n :(2) q ( a,n ) ∈ P j and ˙ σ ( a,n ) is a P l ( a ) -name,(3) a forces the following statements with respect to P l ( a ) :(a) ˙ σ ( a,n ) : ¯ N [ ¯ G ¯ i ] ≺ N [ ˙ G i ],(b) ˙ σ ( a,n ) ( θ, ¯ ~ P , s, ν, ¯ i, ¯ j, ¯ G ¯ i ) = θ, ~ P , τ, ˙ σ (¯ s ) , ν, i, j, ˙ G i ,(c) q ( a,n ) ≤ j ˙ σ ( a,n ) ( p n ), and q ( a,n ) ∈ ran( ˙ σ ( a,n ) ).(4) for some ¯ q ( a,n ) ∈ ¯ P ¯ j and ¯ T ( a,n ) ∈ ¯ N , we have that q ( a,n ) ∈ ¯ G ¯ i ∗ ¯ G ¯ i, ¯ j , q ( a,n ) ≤ p n and a (cid:13) ˙ σ ( a,n ) ( h q ( a,n ) , T ( a,n ) , l ( a ) i ) = h q ( a,n ) , T ( a,n ) , l ( a ) i . If b ∈ suc nT ( a ) and m ≤ n , then(5) b (cid:13) l ( b ) ˙ σ ( b,n +1) ( t m ) = ˙ σ ( a,n ) ( t m ) and ˙ σ ( b,n +1) ( p m ) = ˙ σ ( a,n ) ( p m ).(6) b ↾ l ( a ) (cid:13) l ( a ) q ( b,n +1) , T ( b,n +1) ∈ ran( ˙ σ ( a,n ) ).First let’s see that constructing such objects is sufficient to prove the existenceof a condition p ∗ as in the statement of the theorem. So suppose that we haveconstructed a nested antichain h T, h T n | n < ω i , h suc nT | n < ω ii , a fusion structure hh q ( t,n ) , T ( a,n ) i | n < ω, a ∈ T n i in h P α | α ≤ j i and a sequence h ˙ σ ( a,n ) | n < ω, a ∈ T n i , so that (1) through (6) above are satisfied. Let q ∗ ∈ P j be a fusion of thefusion structure, and let p ∗ = p ⌢ q ∗ ↾ [ i, j ). By (1), we have that q ∗ ↾ i ≡ p , so p ∗ ≡ q ∗ and p ∗ ↾ i = p , as required. To see that p ∗ is as wished, let G j be P j -generic overV with p ∗ ∈ G j . We have to show that in V[ G j ], there is a σ ′ so that conclusions(a)-(c) are satisfied. Since p ∗ ≡ q ∗ , we have that q ∗ ∈ G j . Work in V[ G j ]. ByProposition 3.18 there is a sequence h a n | n < ω i ∈ V[ G j ] so that for all n < ω , a n +1 ∈ suc nT ( a n ), a n ∈ G j ↾ l ( a n ) and q ( a n ,n ) ∈ G j . Let σ n be the evaluation of˙ σ ( a n ,n ) by G j . Then we define σ ′ : ¯ N → N to be the map such that σ ′ ( t n ) = σ n ( t n ).We claim that σ ′ satisfies the conclusions (a)-(c).Condition (a) says that σ ′ moves the parameters ¯ s, ¯ ~ P , ¯ i, ¯ j, ¯ θ and ¯ G ¯ i the same way σ = ˙ σ G i does. But this is true of every σ n , hence also of σ ′ .Condition (b) says that ( σ ′ )“ ¯ G ¯ i, ¯ j ⊆ G i,j . So let ¯ p ∈ ¯ G ¯ i, ¯ j . We have to show that σ ′ (¯ p ) ∈ G i,j . Recall ¯ G ¯ i ∗ ¯ G ¯ i, ¯ j = { p n | n < ω } . Let n be such that ¯ p = ¯ p n ↾ [¯ i, ¯ j ).By (3)(c), we have that q ( a n ,n ) ≤ j σ n ( p n ), so since q ( a n ,n ) ∈ G j , it follows that σ n ( p n ) ∈ G j as well. By (5) and the definition of σ ′ , we have that σ n ( p n ) = σ ′ ( p n ).It follows that σ ′ (¯ p ) = σ ′ (¯ p n ↾ [¯ i, ¯ j )) = σ ′ (¯ p n ) ↾ [ i, j ) ∈ G i,j , as claimed.Condition (c) says that σ ′ : ¯ N ≺ N is elementary. Since any one formula canonly use finitely many parameters, and σ ′ ↾ { t , . . . , t n } = σ n ↾ { t , . . . , t n } , this istrue by (5).Therefore it remains to show that the construction described above can actuallybe carried out. This is done by recursion on n . The recursion proceeds as follows.At stage n + 1 of the construction, we assume that T m , T ( a,m ) , q ( a,m ) and ˙ σ ( a,m ) have been defined, for all m ≤ n and all a ∈ T m . Also, for m < n and a ∈ T m ,we assume that suc mT ( a ) has been defined. Our inductive hypothesis is that for all m ≤ n and all a ∈ T m , conditions (2)-(4) hold, and that for all m < n , all a ∈ T m and all b ∈ suc mT ( a ), conditions (5)-(6) are satisfied. In order to define T n +1 , we willspecify suc nT ( a ), for every a ∈ T n , which implicitly defines T n +1 = S a ∈ T n suc nT ( a ).Simultaneously, we will define, for every such a and every b ∈ suc nT ( a ), the objects T ( b,n +1) , q ( b,n +1) and ˙ σ ( b,n +1) in such a way that whenever a ∈ T n and b ∈ suc nT ( a ),(5)-(6) are satisfied by a and b , and (2)-(4) are satisfied by b and n + 1 (instead of a and n ).For stage 0 of the construction, notice that (1) gives the base case where n = 0and in this case (2)-(4) are satisfied, p forces that q ( p, = p = ˙ σ ( p ) and p forcesthat W = T ( p, has a preimage under ˙ σ , namely ¯ W .At stage n + 1 of the construction, work under the assumptions described above.Fixing a ∈ T n , we have to define suc nT ( a ). To this end let D be the set of allconditions b for which there are a nested antichain S in ~ P ↾ j , and objects ˙ σ b , u , ¯ u and ¯ S satisfying the following:(D1) b ∈ P l ( b ) and l ( b ) < j .(D2) l ( a ) ≤ l ( b ) and b ↾ l ( a ) ≤ a . TERATION THEOREMS FOR SUBVERSIONS OF FORCING CLASSES 29 (D3) S ∠ T ( a,n ) , ¯ S ∈ ¯ N , S ∈ N .(D4) u ∈ P j , u ≤ q ( a,n ) and u is a mixture of S up to j .(D5) u ∈ ¯ G ¯ i ∗ ¯ G ¯ i, ¯ j , and u ≤ p n +1 .(D6) b ↾ l ( a ) (cid:13) l ( a ) S, u, l ( b ) ∈ ran( ˙ σ ( a,n ) ).(D7) b forces the following statements with respect to P l ( b ) :(a) ˙ σ b ( θ, ¯ i, ¯ j, ¯ ~ P , ¯ G ¯ i , s, u, ¯ S ) = θ, i, j, ~ P , ˙ G i , ˙ σ (¯ s ) , u, S .(b) ∀ m ≤ n ˙ σ b ( t m ) = ˙ σ ( a,n ) ( t m ) and ˙ σ b ( p m ) = ˙ σ ( a,n ) ( p m ),(c) ˙ σ b : ¯ N [ ¯ G ¯ i ] ≺ N [ ˙ G i ],Note that if b ∈ D and b ′ ≤ l ( b ) b , then b ′ ∈ D as well. It follows that D ↾ l ( a ) := { b ↾ l ( a ) | b ∈ D } is open in P l ( a ) . Thus, it suffices to show that D ↾ l ( a ) is predensebelow a in P l ( a ) . For if we know this, D ↾ l ( a ) is dense below a , and we may choosea maximal antichain A ⊆ D ↾ l ( a ) (with respect to P l ( a ) ), which then is a maximalantichain in P l ( a ) below a . Thus, for every c ∈ A , we may pick a condition b ( c ) ∈ D such that b ( c ) ↾ l ( a ) = c , and define suc nT ( a ) = { b ( c ) | c ∈ A } (in order to satisfyDefinition 3.12, part (5)). Now, for every b ∈ suc nT ( a ), let S , ˙ σ b , u and ¯ u witnessthat b ∈ D , i.e., let them be chosen in such a way that (D1)-(D7) hold. Set T ( b,n +1) = S , ˙ σ ( b,n +1) = ˙ σ b , q ( b,n +1) = u , ¯ q ( b,n +1) and ¯ T ( b,n +1) = ¯ S . Then a , b satisfy (5)-(6) at stage n , and b satisfies (2)-(4) at stage n + 1.To see that D ↾ l ( a ) is predense below a , let G l ( a ) be P l ( a ) -generic over V with a ∈ G l ( a ) . We have to find a b ∈ D so that b ↾ l ( a ) ∈ G l ( a ) . Work in V [ G l ( a ) ]. Let σ n = ( ˙ σ ( a,n ) ) G l ( a ) . Since (3) holds at stage n , we have that σ n : ¯ N [ ¯ G ¯ i ] ≺ N [ G i ] and σ n ( θ, ¯ ~ P , ¯ s, ν, ¯ i, ¯ j, ¯ G ¯ i ) = θ, ~ P , σ (¯ s ) , ν, i, j, G i . We also have objects ¯ q ( a,n ) , ¯ T ( a,n ) , l ( a )satisfying condition (4), so that ¯ q ( a,n ) ∈ ¯ G ¯ i ∗ ¯ G ¯ i, ¯ j , ¯ T ( a,n ) ∈ ¯ N , ¯ q ( a,n ) ≤ ¯ p n , σ n (¯ q ( a,n ) , ¯ T ( a,n ) ) = q ( a,n ) , T ( a,n ) and σ n ( l ( a )) = l ( a ).First we find the requisite u and ¯ u . By elementarity, q ( a,n ) is a mixture of T ( a,n ) up to j . Recall that σ n : h L ¯ τ [ ¯ A ][ ¯ G ¯ i ] , ∈ , ¯ A i ≺ h L τ [ A ][ G i ] , ∈ , A i , so in particular,¯ σ := σ n ↾ L ¯ τ [ ¯ A ] : h L ¯ τ [ ¯ A ] , ∈ , ¯ A i ≺ h L τ [ A ] , ∈ , A i , and ¯ σ (¯ q ( a,n ) , ¯ T ( a,n ) ) = q ( a,n ) , T ( a,n ) .Clearly, in L τ [ A ], it is true that q ( a,n ) is a mixture of T ( a,n ) up to j , so it is true in L ¯ τ [ ¯ A ] that ¯ q ( a,n ) is a mixture of ¯ T ( a,n ) up to ¯ j , and by absoluteness, this it true inV as well.Let ¯ T ( a,n )0 = { ¯ a } . Let’s write ¯ G ¯ j = ¯ G ¯ i ∗ ¯ G ¯ i, ¯ j , and for k ≤ ¯ j , let’s set ¯ G k =¯ G ¯ j ↾ k . By Fact 3.13.(1), ¯ a ≡ ¯ q ( a,n ) ↾ l (¯ a ) ∈ ¯ G l (¯ a ) , since l (¯ q ( a,n ) ) = ¯ j > l (¯ a ).So ¯ a ∈ ¯ G l (¯ a ) . Let ¯ r ∈ ¯ T (¯ a,n )1 , that is, ¯ r ∈ suc T (¯ a,n ) (¯ a ), be such that ¯ r ↾ l ( a ) ∈ ¯ G l (¯ a ) . There is such a ¯ r by Definition 3.12(5). By Fact 3.13.(3), again since l (¯ r ) < ¯ j = l (¯ q ( a,n ) ), it follows that ¯ r ≡ ¯ r ↾ l (¯ a ) ⌢ ¯ q ( a,n ) ↾ [ l (¯ a ) , l (¯ j )). Since ¯ r ↾ l (¯ a ) ∈ ¯ G l (¯ a ) , this implies that ¯ r ↾ [ l (¯ a ) , l (¯ r )) ≡ ¯ q ( a,n ) ↾ [ l (¯ a ) , l (¯ r )) (in the partial order¯ P l (¯ a ) ,l (¯ r ) = ¯ P l (¯ r ) / ¯ G l (¯ a ) ), and ¯ q ( a,n ) ↾ [ l (¯ a ) , l (¯ r )) ∈ ¯ G ↾ [ l (¯ a ) , l (¯ r )). So we have that¯ r ↾ l (¯ a ) ∈ ¯ G l (¯ a ) and ¯ r ↾ [ l (¯ a ) , l (¯ r )) ∈ ¯ G ↾ [ l (¯ a ) , l (¯ r )). By Fact 3.10.(3), this impliesthat ¯ r ∈ ¯ G l (¯ r ) .It follows that r ⌢ q ( a,n ) ↾ [ l ( r ) , ¯ j ) ∈ ¯ G ¯ j , again using Fact 3.10.(3). Let u ∈ ¯ G ¯ j strengthen both r ⌢ q ( a,n ) ↾ [ l ( r ) , ¯ j ) and p n +1 . By Lemma 3.16, applied in ¯ N ,there is a nested antichain ¯ S ∠ ¯ T ( a,n ) such that ¯ u is a mixture of ¯ S up to ¯ j andsuch that letting ¯ S = { ¯ d } , we have that l (¯ r ) ≤ l ( ¯ d ) and ¯ d ↾ l (¯ r ) ≤ ¯ r . Let S, d , u = σ n ( ¯ S, ¯ d , ¯ u ), and let w ∈ G l ( a ) force this. Since a ∈ G l ( a ) , we may choose w so that w ≤ a . Note that
S, d , u are in N (and hence in V), since ¯ S, ¯ d , ¯ u ∈ ¯ N .We are going to apply our inductive hypothesis ϕ ( l ( d )), noting that l ( d ) < j ,to i = l ( a ) ≤ l ( d ), the filters ¯ G l ( a ) , ¯ G l ( a ) ,l ( ¯ d ) , the models ¯ N , N , the condition w (in place of p ), the name ˙ σ ( a,n ) (in place of ˙ σ and the parameter ¯ s ′ ∈ ¯ N which wewill specify below). No matter which ¯ s ′ we choose, by the inductive hypothesis,there is a condition w ∗ ∈ P l ( d ) with w ∗ ↾ l ( a ) = w and a name ˙ σ ′ such that w ∗ forces with respect to P l ( a ) :(a) ˙ σ ′ (ˇ¯ s ′ , ˇ¯ ~ P , ˇ l ( a ) , ˇ l ( d ) , ˇ¯ θ, ˙¯ G l ( a ) ) = ˙ σ ( a,n ) (ˇ¯ s ′ , ˇ¯ ~ P , ˇ l ( a ) , ˇ l ( d ) , ˇ¯ θ, ˙¯ G l ( a ) ),(b) ( ˙ σ ′ )“ ˙¯ G l ( a ) ,l ( ¯ d ) ⊆ ˙ G l ( a ) ,l ( d ) ,(c) ˙ σ ′ : ˇ¯ N [ ˙¯ G l ( a ) ] ≺ ˇ N [ ˙ G l ( a ) ].By choosing ¯ s ′ appropriately, and temporarily fixing H as above, we may insurethat it is forced that ˙ σ ′ moves any finite number of members of ¯ N the same way˙ σ ( a,n ) does. Thus, we may insist that w ∗ forces that ˙ σ ′ (¯ u, ¯ d , ¯ S ) = ˙ σ ( a,n ) (¯ u, ¯ d , ¯ S ).Recall that w forced that ˙ σ ( a,n ) (¯ u, ¯ d , ¯ S ) = u, d , S . Hence, since w ∗ ↾ l ( a ) = w , weget that w ∗ forces that ˙ σ ′ (¯ u, ¯ d , ¯ S ) = u, d , S as well.In addition, we may insist that σ ′ moves the parameters ¯ i , ¯ j , ¯ ~ P , ¯ θ , ¯ s , ¯ p , . . . ,¯ p n , t , . . . , t n the same way ˙ σ ( a,n ) does. Note that already a forced with respect to P l ( a ) that ¯ i, ¯ j, ¯ ~ P , ¯ θ are mapped to i, j, ~ P , θ by ˙ σ ( a,n ) .Now, setting b = w ∗ , ˙ σ b = ˙ σ ′ , conditions (D1)-(D7) are satisfied, that is, b ∈ D .Most of these are obvious; let me just remark that b forces that ˙ σ b ( ¯ G ¯ i ) = ˙ G i because it forces that ˙ σ b (¯ i ) = i and ˙ σ b ( ¯ G l ( a ) ) = ˙ G l ( a ) . Condition (D6) holdsbecause b ↾ l ( a ) = w . For the same reason, we have that b ↾ l ( a ) ∈ G l ( a ) , completingthe proof that D ↾ l ( a ) is predense below a . This concludes the treatment of case 1. Case 2: j is a successor ordinal.Let j = k + 1. Since we assumed i < j , it follows that i ≤ k . Inductively, weknow that ϕ ( k ) holds. Note that ¯ j is of the form ¯ k + 1, where p forces with respectto P i that ˙ σ (¯ k ) = k , and if we let ¯ G ¯ k = ¯ G ¯ j ↾ ¯ k , then the assumptions (A1)-(A3) aresatisfied by p ∈ P i , ˙ σ ∈ V P i , θ , τ , A , N , ¯ N , ¯ s, ¯ ~ P , ¯ i, ¯ k ∈ ¯ N , ¯ G ¯ i , ¯ G ¯ i, ¯ k ⊆ ¯ N and k . By ϕ ( k ), we obtain a condition p ∗∗ ∈ P k with p ∗∗ ↾ i = y and a P k -name ˙¯ σ such that p ∗∗ forces(a1) ˙¯ σ (ˇ¯ s, ˇ¯ ~ P , ˇ¯ i, ˇ¯ k, ˇ¯ j, ˇ¯ θ, ˇ¯ G ¯ i ) = σ (ˇ¯ s, ˇ¯ ~ P , ˇ¯ i, ˇ¯ k, ˇ¯ j, ˇ¯ θ, ˇ¯ G ¯ i ),(b1) ˙¯ σ “ ˇ¯ G ¯ i, ¯ k ⊆ ˙ G i,k ,(c1) ˙¯ σ : ˇ¯ N [ ˇ¯ G ¯ i ] ≺ ˇ N [ ˙ G i ].It follows then that p ∗∗ forces that ˙¯ σ “ ¯ G ¯ k ⊆ ˙ G k , and hence that ˙¯ σ lifts to anelementary embedding from ¯ N [ ¯ G ¯ k ] ≺ N [ ˙ G k ] that maps ¯ G ¯ k to ˙ G k . Let ˙˜ σ be a P k -name such that p ∗∗ forces that ˙˜ σ is that lifted embedding.Temporarily fix a P k -generic filter H that contains p ∗∗ . In V[ H ], the forcing P k,k +1 = P k,j = P j /H is ∞ -subcomplete. Letting ˜ σ = ˙˜ σ H , we have that ˜ σ :¯ N [ ¯ G ¯ k ] ≺ N [ H ], and thus, since ¯ N [ ¯ G ¯ k ] is full, there is a condition q in P k,j suchthat q forces the existence of an elementary embedding σ ′ with(a2) σ ′ (ˇ¯ s, ˇ¯ ~ P , ˇ¯ i, ˇ¯ k, ˇ¯ j, ˇ¯ θ, ˇ¯ G ¯ i , ˇ¯ G ¯ k ) = ˜ σ (ˇ¯ s, ˇ¯ ~ P , ˇ¯ i, ˇ¯ k, ˇ¯ j, ˇ¯ θ, ˇ¯ G ¯ i , ˇ¯ G ¯ k ),(b2) ( σ ′ )“ ˇ¯ G ¯ k, ¯ j ⊆ ˙ G k,j ,(c2) σ ′ : ˇ¯ N [ ˇ¯ G ¯ k ] ≺ ˇ N [ ˙ H ]. TERATION THEOREMS FOR SUBVERSIONS OF FORCING CLASSES 31
Since this holds in V[ H ] whenever p ∗∗ ∈ H , there is a P k -name τ which is essentiallya name for q above - more precisely, τ is such that p ∗∗ forces that τ ∈ P j , τ ↾ k ∈ ˙ G k and τ ↾ [ k, j ) has the properties of q , as listed above. Since the iteration is nice, thereis a condition p ∗ ∈ P j such that p ∗ ↾ k = p ∗∗ and p ∗ forces that τ ↾ [ k, j ) ≡ p ∗ ↾ [ k, j );see Definition 3.14, part (1). We claim that p ∗ is as wished.First, note that p ∗ ↾ i = ( p ∗ ↾ k ) ↾ i = p ∗∗ ↾ i = p . Now, let G j be a P j -generic filterwith p ∗ ∈ G j . We have to show that in V[ G j ], there is a σ ′ such that, letting σ = ˙ σ G i , the following hold:(a) σ ′ (¯ s, ¯ ~ P , ¯ i, ¯ j, ¯ θ, ¯ G ¯ i ) = σ (¯ s, ¯ ~ P , ¯ i, ¯ j, ¯ θ, ¯ G ¯ i ),(b) ( σ ′ )“ ¯ G ¯ i, ¯ j ⊆ G i,j ,(c) σ ′ : ¯ N [ ¯ G ¯ i ] ≺ N [ G i ].But this follows, because V[ G j ] = V[ G k ][ G k,j ], where p ∗ ↾ [ k, j ) ∈ G k,j , where p ∗∗ ∈ G k , writing H for G k , puts us in the situation described above. More-over, p ∗ ↾ [ k, j ) ∈ G k,j and p ∗ ↾ [ k, j ) ≡ ˙ q G j , where ˙ q is a name for the condition q mentioned above. Thus, there is a σ ′ in V[ G j ] such that the conditions (a2)-(c2)listed above hold in V[ G j ]. Remembering that ˜ σ lifts ¯ σ and ¯ σ moves the requiredparameters as prescribed (by (a1)-(c1)), it follows that (a)-(c) are satisfied. (cid:3) Nice iterations of ∞ -subproper forcing. Next we prove a similar theoremfor ∞ -subproper forcing. After we proved Theorem 3.20 we learned that Miyamoto(unpublished) had also proved this result earlier. What we call ∞ -subproper here,Miyamoto calls “preproper.” Theorem 3.20 (Miyamoto) . Let ~ P = h P α | α ≤ ν i be a nice iteration so that P = { } and for all i with i + 1 < ν , (cid:13) i P i,i +1 is ∞ -subproper. Then for all j ≤ ν the following statement ϕ ( j ) holds:if i ≤ j , p ∈ P i , ˙ σ, ˙¯ G ¯ i ∈ V P i , q ∈ P j , θ is a sufficiently large cardinal, τ is anordinal, H θ ⊆ N = L τ [ A ] | = ZFC − ¯ N is a countable, full, transitive model whichelementarily embeds into N so that ¯ s, ¯ ~ P , ¯ i, ¯ j ∈ ¯ N and p forces with respect to P i thatthe following assumptions hold:(A1) ˙ σ (ˇ¯ s, ˇ¯ ~ P , ˇ¯ i, ˇ¯ j, ˇ¯ θ, ˇ¯ q ) = ˇ s, ˇ ~ P , ˇ i, ˇ j, ˇ q (A2) ˙¯ G ¯ i is the pointwise image of the generic under ˙ σ and is ¯ P i -generic over ˇ¯ N (A3) ˙ σ : ˇ¯ N [ ˙¯ G ¯ i ] ≺ ˇ N [ ˙ G i ] is countable, transitive and full. then there is a condition p ∗ ∈ P j such that p ∗ ↾ [ i, j ) ≤ q ↾ [ i, j ) , p ∗ ↾ i = p andwhenever G j ∋ p ∗ is P j -generic, then in V[ G j ] , there is a σ ′ such that, letting σ = ˙ σ G i , the following hold:(a) σ ′ (¯ s, ¯ ~ P , ¯ i, ¯ j, ¯ θ, ¯ G ¯ i , ¯ q ) = σ (¯ s, ¯ ~ P , ¯ i, ¯ j, ¯ θ, ¯ G ¯ i , q ) , (b) ( σ ′ ) − G i,j := ¯ G i,j is P i,j -generic over ¯ N [ ¯ G ¯ i ] ,(c) σ ′ : ¯ N [ ¯ G ¯ i ] ≺ N [ G i ] . Many parts of this proof are verbatim the same as in the previous iterationtheorem but we repeat them for the convenience of the reader.
Proof.
Like in the previous proof, we induct on j . So let us assume that ϕ ( j ′ ) holdsfor every j ′ < j . Fix some i ≤ j . Since nothing is to be shown when i = j , let i < j . In particular, the case j = 0 is trivial. Let us fix p ∈ P i , ˙ σ, ˙¯ G ¯ i ∈ V P i , q ∈ P j , and without loss suppose p ≤ q ↾ i . Alsome fix θ , τ , A , N , ¯ N , ¯ s, ¯ ~ P , ¯ i, ¯ j ∈ ¯ N , ¯ G ¯ i , ¯ G ¯ i, ¯ j ⊆ ¯ N so that assumptions (A1)-(A3)hold.When it causes no confusion, we will again employ our “bar” convention.Case 1: j is a limit ordinal.Let { t n | n < ω } enumerate the ¯ P i -names in ¯ N . Without loss of generality wemay assume that t is the check name for ∅ . Also let { D n | n < ω } enumerate thenames in ¯ N for the dense open subsets of ¯ P i,j . Without loss assume that ¯ q is forcedto be in D .As before we may assume that there is a definable well ordering of the universe, ≤ ¯ N , of ¯ N . As noted in [14], by Lemma 3.15, in ¯ N , ¯ p is a mixture up to ¯ j of somenested antichain in ¯ ~ P ↾ ¯ j whose root has length ¯ i . Letting ¯ W be the ≤ ¯ N -least one,we know that p forces that ˙ σ ( ˇ¯ W ) is the L τ [ A ]-least nested antichain W in ~ P ↾ j suchthat p is a mixture of W up to j , since we know that p forces that ¯ p , ¯ i, ¯ j aremapped to p , i, j by ˙ σ , respectively.We will define a nested antichain h T, h T n | n < ω i , h suc nT | n < ω ii , a fusion struc-ture hh q ( a,n ) , T ( a,n ) i | n < ω, a ∈ T n i in h P α | α ≤ j i and a sequence h ˙ σ ( a,n ) | n <ω, a ∈ T n i so that the following conditions hold.(1) T = { p } , q ( p, = x , T ( p, = W and ˙ σ ( p, = ˙ σ .Further, for any n < ω and a ∈ T n :(2) q ( a,n ) ≤ q and q ( a,n ) ∈ P j and ˙ σ ( a,n ) is a P l ( a ) -name,(3) a forces the following statements with respect to P l ( a ) :(a) ˙ σ ( a,n ) : ˇ¯ N [ ˙¯ G ¯ i ] ≺ ˇ N [ ˙ G i ],(b) ˙ σ ( a,n ) (ˇ θ, ˇ¯ ~ P , s, ˇ¯ i, ˇ¯ j, ˙¯ G ¯ i , ˇ¯ q ) = ˇ θ, ˇ ~ P , ˇ s, ˇ i, ˇ j, ˙ G i , ˇ q (c) q ( a,n ) ∈ ran( ˙ σ ( a,n ) ) and its preimage is in D n .(4) for some ¯ q ( a,n ) ∈ ¯ P ¯ j and ¯ T ( a,n ) ∈ ¯ N , we have that q ( a,n ) ∈ ¯ G ¯ i ∗ ¯ G ¯ i, ¯ j , q ( a,n ) ∈ D n and a (cid:13) ˙ σ ( a,n ) ( h q ( a,n ) , T ( a,n ) , l ( a ) i ) = h q ( a,n ) , T ( a,n ) , l ( a ) i .If b ∈ suc nT ( a ) and m ≤ n , then(5) b (cid:13) l ( b ) ˙ σ ( b,n +1) ( t m ) = ˙ σ ( a,n ) ( t m ) and ˙ σ ( b,n +1) ( D m ) = ˙ σ ( a,n ) ( D m ).(6) b ↾ l ( a ) (cid:13) l ( a ) q ( b,n +1) , T ( b,n +1) ∈ ran( ˙ σ ( a,n ) ).First let’s see that constructing such objects is sufficient to prove the existence of acondition p ∗ as in the statement of the theorem. Suppose that we have constructedsequences satisfying (1) through (6) above. Let q ∗ ∈ P j be a fusion of the fusionstructure, and let p ∗ = p ⌢ q ∗ ↾ [ i, j ). By (1), we have that q ∗ ↾ i ≡ p , so p ∗ ≡ q ∗ and p ∗ ↾ i = p , as required. To see that p ∗ is as wished, let G j be P j -generic over V with p ∗ ∈ G j . We have to show that in V[ G j ], there is a σ ′ so that conclusions (a)-(c) aresatisfied. Since p ∗ ≡ q ∗ , we have that q ∗ ∈ G j . Work in V[ G j ]. By Proposition 3.18there is a sequence h a n | n < ω i ∈ V [ G j ] so that for all n < ω , a n +1 ∈ suc nT ( a n ), a n ∈ G j ↾ l ( a n ) and q ( a n ,n ) ∈ G j . Let σ n be the evaluation of ˙ σ ( a n ,n ) by G j . Then,as in the iteration theorem for subcompleteness, define σ ′ : ¯ N → N to be the mapsuch that σ ′ ( t n ) = σ n ( t n ). We claim that σ ′ satisfies the conclusions (a)-(c). Indeedthe verification of this fact exactly mirrors the case of subcompleteness with onedifference: we need to ensure that the pointwise preimage of G i,j is ¯ P i,j -generic TERATION THEOREMS FOR SUBVERSIONS OF FORCING CLASSES 33 over ¯ N [ ¯ G ¯ i ]. However this requirement is taken care of in the construction since thepre-image of q ( a,n ) is in the evaluation of D n by (3) (c).Thus it remains to see that such a construction can be carried out. This is doneby induction on n , in a manner similar to the previous proof. Like last time, atstage n + 1 of the construction, we assume that T m , T ( a,m ) , q ( a,m ) and ˙ σ ( a,m ) havebeen defined, for all m ≤ n and all a ∈ T m . Also, for m < n and a ∈ T m , weassume that suc mT ( a ) has been defined. Our inductive hypothesis is that for all m ≤ n and all a ∈ T m , conditions (2)-(4) hold, and that for all m < n , all a ∈ T m and all b ∈ suc mT ( a ), conditions (5)-(6) are satisfied. In order to define T n +1 , we willspecify suc nT ( a ), for every a ∈ T n , which implicitly defines T n +1 = S a ∈ T n suc nT ( a ).Simultaneously, we will define, for every such a and every b ∈ suc nT ( a ), the objects T ( b,n +1) , q ( b,n +1) and ˙ σ ( b,n +1) in such a way that whenever a ∈ T n and b ∈ suc nT ( a ),(5)-(6) are satisfied by a and b , and (2)-(4) are satisfied by b and n + 1 (instead of a and n ).For stage 0 of the construction, notice that (1) gives the base case where n = 0and in this case (2)-(4) are satisfied, p forces that q ( p, = q = ˙ σ ( q ) ∈ D and p forces that W = T ( p, has a preimage under ˙ σ , namely ¯ W .At stage n + 1 of the construction, work under the assumptions described above.Fixing a ∈ T n , we have to define suc nT ( a ). To this end let D be the set of allconditions b for which there are a nested antichain S in ~ P ↾ j , and objects ˙ σ b , u , ¯ u and ¯ S satisfying the following:(D1) b ∈ P l ( b ) and l ( b ) < j .(D2) l ( a ) ≤ l ( b ) and b ↾ l ( a ) ≤ a .(D3) S ∠ T ( a,n ) , ¯ S ∈ ¯ N , S ∈ N .(D4) u ∈ P j , u ≤ x ( a,n ) and u is a mixture of S up to j .(D5) u ↾ ¯ i ∈ ¯ G ¯ i , and u ∈ D n +1 (in ¯ N [ ¯ G ¯ i ]).(D6) b ↾ l ( a ) (cid:13) l ( a ) S, u, l ( b ) ∈ ran( ˙ σ ( a,n ) ).(D7) b forces the following statements with respect to P l ( b ) :(a) ˙ σ b (ˇ θ, ˇ¯ i, ˇ¯ j, ˇ¯ ~ P , ˙¯ G ¯ i , ˇ s, ˇ u, ˇ¯ S, ˇ¯ q ) = ˇ θ, ˇ i, ˇ j, ˇ ~ P , ˙ G i , ˙ σ (¯ s ) , ˇ u, ˇ S, ˇ q .(b) ∀ m ≤ n ˙ σ b ( t m ) = ˙ σ ( a,n ) ( t m ) and ˙ σ b ( D m ) = ˙ σ ( a,n ) ( D m ),(c) ˙ σ b : ˇ¯ N [ ˙¯ G ¯ i ] ≺ ˇ N [ ˙ G i ],Note that if b ∈ D and b ′ ≤ l ( b ) b , then b ′ ∈ D as well. It follows that D ↾ l ( a ) := { b ↾ l ( a ) | b ∈ D } is open in P l ( a ) . Thus, it suffices to show that D ↾ l ( a ) is predensebelow a in P l ( a ) . For if we know this, D ↾ l ( a ) is dense below a , and we may choosea maximal antichain A ⊆ D ↾ l ( a ) (with respect to P l ( a ) ), which then is a maximalantichain in P l ( a ) below a . Thus, for every c ∈ A , we may pick a condition b ( c ) ∈ D such that b ( c ) ↾ l ( a ) = c , and define suc nT ( a ) = { b ( c ) | c ∈ A } . Now, for every b ∈ suc nT ( a ), let S , ˙ σ b , u and ¯ u witness that b ∈ D , i.e., let them be chosen in sucha way that (D1)-(D7) hold. Set T ( b,n +1) = S , ˙ σ ( b,n +1) = ˙ σ b , q ( b,n +1) = u , ¯ q ( b,n +1) and ¯ T ( b,n +1) = ¯ S . Then a , b satisfy (5)-(6) at stage n , and b satisfies (2)-(4) atstage n + 1.To see that D ↾ l ( a ) is predense below a , let G l ( a ) be P l ( a ) -generic over V with a ∈ G l ( a ) . We have to find a b ∈ D so that b ↾ l ( a ) ∈ G l ( a ) . Work in V [ G l ( a ) ]. Let σ n = ( ˙ σ ( a,n ) ) G l ( a ) . Since (3) holds at stage n , we have that σ n : ¯ N [ ¯ G ¯ i ] ≺ N [ G i ]and σ n ( θ, ¯ ~ P , ¯ s, ν, ¯ i, ¯ j, ¯ G ¯ i ) = θ, ~ P , σ (¯ s ) , ν, i, j, G i . We also have objects ¯ q ( a,n ) , ¯ T ( a,n ) , l ( a ) satisfying condition (4), so that ¯ q ( a,n ) ∈ ¯ G ¯ i ∗ ¯ G ¯ i, ¯ j , ¯ T ( a,n ) ∈ ¯ N , ¯ q ( a,n ) ≤ ¯ p n , σ n (¯ q ( a,n ) , ¯ T ( a,n ) ) = q ( a,n ) , T ( a,n ) and σ n ( l ( a )) = l ( a ).Let’s first find u and ¯ u again. By elementarity, q ( a,n ) is a mixture of T ( a,n ) upto j . We have that σ n : h L ¯ τ [ ¯ A ][ ¯ G ¯ i ] , ∈ , ¯ A i ≺ h L τ [ A ][ G i ] , ∈ , A i , so in particular,¯ σ := σ n ↾ L ¯ τ [ ¯ A ] : h L ¯ τ [ ¯ A ] , ∈ , ¯ A i ≺ h L τ [ A ] , ∈ , A i , and ¯ σ (¯ q ( a,n ) , ¯ T ( a,n ) ) = q ( a,n ) , T ( a,n ) .Clearly, in L τ [ A ], it is true that q ( a,n ) is a mixture of T ( a,n ) up to j , so it is true in L ¯ τ [ ¯ A ] that ¯ q ( a,n ) is a mixture of ¯ T ( a,n ) up to ¯ j , and by absoluteness, this it true inV as well.Let ¯ T ( a,n )0 = { ¯ a } . Since we’re working in V[ G l ( a ) ], we have access to all ¯ G k for k ≤ l ( a ) by considering the pointwise pre image of σ ( a,n ) . By induction these areall generic over ¯ N [ G ¯ i ]. Moreover, by Fact 3.13.(1), ¯ a ≡ ¯ q ( a,n ) ↾ l (¯ a ) ∈ ¯ G l (¯ a ) , since l (¯ q ( a,n ) ) = ¯ j > l (¯ a ). So ¯ a ∈ ¯ G l (¯ a ) . Let ¯ r ∈ ¯ T (¯ a,n )1 , that is, ¯ r ∈ suc T (¯ a,n ) (¯ a ), besuch that ¯ r ↾ l ( a ) ∈ ¯ G l (¯ a ) . There is such a ¯ r by Definition 3.12(5). By Fact 3.13.(3),again since l (¯ r ) < ¯ j = l (¯ q ( a,n ) ), it follows that ¯ r ≡ ¯ r ↾ l (¯ a ) ⌢ ¯ q ( a,n ) ↾ [ l (¯ a ) , l (¯ j )). Since¯ r ↾ l (¯ a ) ∈ ¯ G l (¯ a ) , this implies that ¯ r ↾ [ l (¯ a ) , l (¯ z )) ≡ ¯ q ( a,n ) ↾ [ l (¯ a ) , l (¯ r )) (in the partialorder ¯ P l (¯ a ) ,l (¯ r ) = ¯ P l (¯ r ) / ¯ G l (¯ a ) ), and ¯ q ( a,n ) ↾ [ l (¯ a ) , l (¯ r )) ∈ ¯ G ↾ [ l (¯ a ) , l (¯ r )). So we havethat ¯ r ↾ l (¯ a ) ∈ ¯ G l (¯ a ) and ¯ r ↾ [ l (¯ a ) , l (¯ r )) ∈ ¯ G ↾ [ l (¯ a ) , l (¯ r )). By Fact 3.10.(3), thisimplies that ¯ r ∈ ¯ G l (¯ r ) .Now let u strengthen r ⌢ r ( a,n ) ↾ [ l ( r ) , ¯ j ) so that u ∈ D n +1 . By Lemma 3.16,applied in ¯ N , there is a nested antichain ¯ S ∠ ¯ T ( a,n ) such that ¯ u is a mixture of ¯ S up to ¯ j and such that letting ¯ S = { ¯ d } , we have that l (¯ r ) ≤ l ( ¯ d ) and ¯ d ↾ l (¯ r ) ≤ ¯ r .Let S, d , u = σ n ( ¯ S, ¯ d , ¯ u ), and let w ∈ G l ( a ) force this. Since a ∈ G l ( a ) , we maychoose w so that w ≤ a .Note that S, d , u are in N (and hence in V), since ¯ S, ¯ d , ¯ u ∈ ¯ N .We are going to apply our inductive hypothesis ϕ ( l ( d )), noting that l ( d ) < j ,to i = l ( a ) ≤ l ( d ), the filters ¯ G l ( a ) , ¯ G l ( a ) ,l ( ¯ d ) , the models ¯ N , N , the condition w (in place of p ), the name ˙ σ ( a,n ) (in place of ˙ σ and the parameter ¯ s ′ ∈ ¯ N which wewill specify below. No matter which ¯ s ′ choosen, by the inductive hypothesis, thereis a condition w ∗ ∈ P l ( d ) with w ∗ ↾ l ( a ) = w and a name ˙ σ ′ such that w ∗ forces withrespect to P l ( a ) :(a) ˙ σ ′ (ˇ¯ s ′ , ˇ¯ ~ P , ˇ l ( a ) , ˇ l ( d ) , ˇ¯ θ, ˙¯ G l ( a ) ) = ˙ σ ( a,n ) ((ˇ¯ s ′ , ˇ¯ ~ P , ˇ l ( a ) , ˇ l ( d ) , ˇ¯ θ, ˙¯ G l ( a ) ),(b) ( ˙ σ ′− )“ ˙ G l ( a ) ,l ( d ) is generic over ˇ¯ N [ ˙¯ G l ( a ) ], and(c) ˙ σ ′ : ˇ¯ N [ ˙¯ G l ( a ) ] ≺ ˇ N [ ˙ G l ( a ) ].By choosing ¯ s ′ appropriately, and temporarily fixing H as above, we may insurethat it is forced that ˙ σ ′ moves any finite number of members of ¯ N the same way˙ σ ( a,n ) does. Thus, we may insist that w ∗ forces that ˙ σ ′ (¯ u, ¯ d , ¯ S ) = ˙ σ ( a,n ) (¯ u, ¯ d , ¯ S ).Recall that w forced that ˙ σ ( a,n ) (¯ u, ¯ d , ¯ S ) = u, d , S . Hence, since w ∗ ↾ l ( a ) = w , weget that w ∗ forces that ˙ σ ′ (¯ u, ¯ d , ¯ S ) = u, d , S as well.In addition, we may insist that σ ′ moves the parameters ¯ i , ¯ j , ¯ ~ P , ¯ θ , ¯ s , ¯ p , . . . ,¯ p n , t , . . . , t n the same way ˙ σ ( a,n ) does. Note that already a forced with respect to P l ( a ) that ¯ i, ¯ j, ¯ ~ P , ¯ θ are mapped to i, j, ~ P , θ by ˙ σ ( a,n ) .Now, set b = w ∗ , ˙ σ b = ˙ σ ′ . It follows that the conditions (D1)-(D7) are satisfied,that is, b ∈ D . Most of these are straightforward to verifty; let us just remark that b forces that ˙ σ b ( ¯ G ¯ i ) = ˙ G i because it forces that ˙ σ b (¯ i ) = i and ˙ σ b ( ¯ G l ( a ) ) = ˙ G l ( a ) . TERATION THEOREMS FOR SUBVERSIONS OF FORCING CLASSES 35
Condition (D6) holds because b ↾ l ( a ) = w . For the same reason, we have that b ↾ l ( a ) ∈ G l ( a ) , completing the proof that D ↾ l ( a ) is predense below a . This concludesthe treatment of case 1. Case 2: j is a successor ordinal.Let j = k + 1. Since we assumed i < j , it follows that i ≤ k . Inductively, weknow that ϕ ( k ) holds. Note that ¯ j is of the form ¯ k + 1, where p forces with respectto P i that ˙ σ (¯ k ) = k , and if we let ¯ G ¯ k = ¯ G ¯ j ↾ ¯ k , then the assumptions (A1)-(A4) aresatisfied by p ∈ P i , ˙ σ ∈ V P i , q ∈ P j , y ≤ q ↾ i , θ , τ , A , N , ¯ N , ¯ s, ¯ ~ P , ¯ i, ¯ k ∈ ¯ N , and k .By ϕ ( k ), we obtain a condition p ∗∗ ∈ P k with p ∗∗ ↾ i = p and a P k -name ˙¯ σ such that p ∗∗ forces(a1) ˙¯ σ (ˇ¯ s, ˇ¯ ~ P , ˇ¯ i, ˇ¯ k, ˇ¯ j, ˇ¯ θ, ˇ¯ q ) = σ (ˇ¯ s, ˇ¯ ~ P , ˇ¯ i, ˇ¯ k, ˇ¯ j, ˇ¯ θ, ˇ¯ q ),(b1) ˙¯ σ − “ ˙ G i,k is generic over ˇ¯ N [ ˙¯ G ¯ i ](c1) ˙¯ σ : ˇ¯ N [ ˙¯ G ¯ i ] ≺ ˇ N [ ˙ G i ].It follows then that p ∗∗ forces that ˙¯ σ − “ ˙ G k is generic over ¯ N , and hence that ˙¯ σ lifts to an elementary embedding from ¯ N [ ¯ G ¯ k ] ≺ N [ ˙ G k ] that maps ¯ G ¯ k to ˙ G k . Let ˙˜ σ be a P k -name such that p ∗∗ forces that ˙˜ σ is that lifted embedding.Temporarily fix a P k -generic filter H that contains p ∗∗ . In V[ H ], the forcing P k,k +1 = P k,j = P j /H is ∞ -subproper. Letting ˜ σ = ˙˜ σ H , we have that ˜ σ : ¯ N [ ¯ G ¯ k ] ≺ N [ H ], and thus, since ¯ N [ ¯ G ¯ k ] is full, there is a condition r in P k,j such that r forcesthe existence of an elementary embedding σ ′ with(a2) σ ′ (ˇ¯ s, ˇ¯ ~ P , ˇ¯ i, ˇ¯ k, ˇ¯ j, ˇ¯ θ, ˇ¯ q, ˙¯ G ¯ i , ) = ˜ σ (ˇ¯ s, ˇ¯ ~ P , ˇ¯ i, ˇ¯ k, ˇ¯ j, ˇ¯ θ, ˇ¯ q, ˙¯ G ¯ i , ),(b2) ( σ ′ ) − “ ˙ G k,j is generic over ˇ¯ N [ ˙¯ G ¯ k ](c2) σ ′ : ˇ¯ N [ ˙¯ G ¯ k ] ≺ N [ H ].Since this holds in V[ H ] whenever p ∗∗ ∈ H , there is a P k -name τ which is essentiallya name for r above - more precisely, τ is such that p ∗∗ forces that τ ∈ P j , τ ↾ k ∈ ˙ G k and τ ↾ [ k, j ) has the properties of r , as listed above. Since the iteration is nice, thereis a condition p ∗ ∈ P j such that p ∗ ↾ k = p ∗∗ and p ∗ forces that τ ↾ [ k, j ) ≡ p ∗ ↾ [ k, j );see Definition 3.14, part (1). We claim that p ∗ is as wished.First, note that p ∗ ↾ i = ( p ∗ ↾ k ) ↾ i = p ∗∗ ↾ i = p . Now, let G j be a P j -generic filterwith p ∗ ∈ G j . We have to show that in V[ G j ], there is a σ ′ such that, letting σ = ˙ σ G i , the following hold:(a) σ ′ (¯ s, ¯ ~ P , ¯ i, ¯ j, ¯ θ, ¯ q ) = σ (¯ s, ¯ ~ P , ¯ i, ¯ j, ¯ θ, ¯ q ),(b) ( σ ′− )“ G i,j is generic over ¯ N [ ¯ G ¯ i ],(c) σ ′ : ¯ N [ ¯ G ¯ i ] ≺ N [ G i ].But this follows, because V[ G j ] = V[ G k ][ G k,j ], where p ∗ ↾ [ k, j ) ∈ G k,j , where p ∗∗ ∈ G k , so writing H for G k , this is the situation described above. Moreover, p ∗ ↾ [ k, j ) ∈ G k,j and p ∗ ↾ [ k, j ) ≡ ˙ r G j , where ˙ r is a name for the condition r mentioned above.Thus, there is a σ ′ in V[ G j ] such that the conditions (a2)-(c2) listed above holdin V[ G j ]. Remembering that ˜ σ lifts ¯ σ and ¯ σ moves the required parameters asprescribed (by (a1)-(c1)), it follows that (a)-(c) are satisfied. (cid:3) So ∞ -subproper and ∞ -subcomplete forcings are nicely iterable . Let us makeone strengthening of these theorems that will be useful in applications. The key Now we mean this in the sense of nice iterations, it’s unfortunate that the terminology conflictshere. step in both proofs was the construction of the fusion sequence in the limit stageand in particular the u and ¯ u . In both proofs we needed u to be as strong as acertain condition, but in fact looking at the proof, we could have strengthened itfurther if we liked. Thus we get the following theorem: Theorem 3.21.
Let h P α | α ≤ ν i be a nice iteration with ν limit, so that P = { } and either for all i with i +1 < ν , (cid:13) i P i,i +1 is ∞ -subproper or for all such i , (cid:13) i P i,i +1 is ∞ -subcomplete. Let θ , N , etc. be as in either Theorem 3.19 or Theorem 3.20and suppose for all n < ω we have that E n ⊆ P ν satisfies the following: for every p ∈ P ν and every α < ν if in V P α there is a name for an embedding ˙ σ : ¯ N ≺ N with p forced to be in the range of ˙ σ and for any u ≤ p in the range of ˙ σ with u ↾ α ∈ ˙ G α then there is an s ≤ u in the range of ˙ σ so that s ↾ α ∈ ˙ G α and s ∈ E n . Then thereis a q ≤ p forcing that there is a decreasing sequence q ≥ q ≥ ... ≥ q n ≥ ... all in G so that q n +1 ∈ E n . In particular, q forces G ∩ E n = ∅ for all n < ω .Proof. In the case of semiproper forcing this is checked in detail by Miyamoto as[14, Lemma 4.3]. Making the exact same modification he makes in that case to ourproofs of Theorems 3.19 and 3.20 works here. The reader is referred to Miyamoto’spaper for the details. (cid:3)
Preserving Properties of Trees.
In this section we lift some results aboutpreservation of properties of trees from [14] to the context of ∞ -subproper forcing.The proofs are in the same spirit as those given using RCS iterations, but sincethose results do not apply to ∞ -subproper forcing we give them here is the modifiedcontext as well. Lemma 3.22.
Let S = ( S, ≤ S ) be a Souslin tree and P = h P α | α ≤ ν i be anice iteration of ∞ -subproper forcings such that for each i with i + 1 ≤ ν , (cid:13) i P i,i +1 preserves ˇ S then P ν preserves S . We stress that the proof of this theorem is similar to that of Lemma 5.0 andTheorem 5.1 of [14].
Proof.
This is proved by induction on ν . The successor stage is by hypothesis sowe focus on the limit case and the inductive assumption is not just that P i,i +1 preserves S but in fact P i preserves S for all i < ν . From now on assume ν is alimit ordinal. Let ˙ A be a P ν name for an antichain of S and let p ∈ P ν force that ˙ A is maximal. We need to find a q ≤ p forcing that ˙ A is countable. Fix θ sufficientlylarge that ˙ A, P , S ∈ H θ and fix σ : ¯ N ≺ N as in the standard setup. Denote by δ = ω ∩ ¯ N . Note that for all α < δ and all s ∈ S α we may assume that σ ( s ) = s since we may assume S ⊆ H ω . Enumerate the δ th level of S as h s n | n < ω i . Foreach n , define E n = { r ∈ P ν | ∃ s ∈ S such that s < S s n and r (cid:13) s ∈ ˙ A } . We needto check that the E n ’s satisfy the predensity condition stipulated in Theorem 3.21.If we can do this then it follows there is a q forcing that ˙ G ∩ E n = ∅ for all n < ω and hence that the maximal antichain is bounded below δ so countable.To check the predensity condition, fix u ≤ p , in the range of σ , α < ν and a P α -name ˙ σ α which will evaluate to an embedding witnessing the ∞ -subpropernessof P α . Let G α be P α -generic over V , σ α = ( ˙ σ α ) G α and σ α ( u, α, S ) = u, α, S . Wewant to find a r ∈ ¯ N ∩ P ν so that r ≤ u , σ α ( r ↾ α ) = r ↾ α ∈ G α and r ∈ E n .Let D be the set of s ∈ S for which there is a condition ¯ r ∈ P ν which strengthens u and so that ¯ r ↾ ¯ α ∈ ¯ G α and ¯ r (cid:13) s ∈ ˙ A . In symbols D = { s ∈ S | ∃ ¯ r ∈ P ν r ≤ TERATION THEOREMS FOR SUBVERSIONS OF FORCING CLASSES 37 u r ↾ α ∈ ¯ G α and r (cid:13) s ∈ ˙ A } where ¯ G α := σ − α “ G α . Note that since σ α is an ∞ -subcompleteness embedding it lifts to an embedding σ ′ α : ¯ N [ G α ] ≺ N [ G α ] andthis set D is in ¯ N [ G α ]. Moreverover D is a predense subset of S in ¯ N [ G α ] since, bythe maximality of ˙ A for any t ∈ S there is densely many conditions forcing some s ∈ ˙ A compatible with t . Finally since S remains Souslin in V[ G α ] by hypothesisand thus S remains Souslin in ¯ N [ G α ] there is an s ∈ D ∩ ¯ N [ G α ] below s n . Letting r ∈ ¯ N [ G α ] be the witness for this s completes the proof. (cid:3) A similar modification of Lemma 5.2 and Theorem 5.3 of [14] can be used toprove the preservation of “not adding uncountable branches through trees.”
Lemma 3.23.
Let T be an ω -tree and let P = h P α | α ≤ ν i be a nice iterationof ∞ -subproper forcings such that for each i with i + 1 ≤ ν , (cid:13) i “ P i,i +1 does notadd an uncountable branch through ˇ T ” then P ν does not add an uncountable branchthrough T .Proof. The lemma proceeds by induction on ν and is by contradiction. Since thesuccessor case is by assumption, the inductive hypothesis is that, for all i < j P i adds no new cofinal branch through T . Let ˙ B be a P ν name for a branch through T and, towards a contradiction, let p ∈ P ν force that ˙ b is uncountable. We needto find a q ≤ p forcing that actually ˙ b is countable. Fix θ sufficiently large that˙ b, P , T ∈ H θ and fix σ : ¯ N ≺ N as in the standard setup. Denote by δ = ω ∩ ¯ N .Note that for all α < δ and all t ∈ T α we may assume that σ ( t ) = t since we mayassume T ⊆ H ω . Enumerate the δ th level of T as h t n | n < ω i . For each n , define E n = { r ∈ P ν | ∃ t ∈ T <δ such that t (cid:2) T t n and r (cid:13) ˇ t ∈ ˙ B } . We need to check thatthe E n ’s satisfy the predensity condition stipulated in Theorem 3.21. If we can dothis then it follows there is a q ≤ p forcing that ˙ G ∩ E n = ∅ for all n < ω and hencethe branch is bounded below δ so it cannot be uncountable.Fix u ≤ p , in the range of σ , α < ν and a P α -name ˙ σ α which will evaluate to anembedding witnessing the ∞ -subproperness of P α . Let G α be P α -generic over V, σ α = ( ˙ σ α ) G α and σ α ( u, α, T ) = u, α, T . We want to find a r ∈ ¯ N ∩ P ν so that r ≤ u , σ α ( r ↾ α ) = r ↾ α ∈ G α and r ∈ E n . Note that since P α didn’t add an uncountablebranch to T , there are incomparable conditions u , u whose restrictions to α arethe same but force incompatible elements t and t respectively to be in ˙ B . Byelementarity, this situation is true as well in ¯ N using u and therefore there are u , u ≤ u in P ν so that u ↾ α = u ↾ α ∈ ¯ G α but u and u force incomparableelements t and t to be in ˙ B . At least one of these elements is incomparable with t n and both of them are of level less than δ (since they’re in ¯ N ). Therefore at leastone of u , u works. (cid:3) Nice iterations of ∞ -subproper ω ω -bounding forcing. In this sectionwe prove that nice iterations of ω ω -bounding ∞ -subproper forcing notions are ω ω -bounding. Theorem 3.24.
Nice iterations of ω ω -bounding subproper forcing notions are ω ω -bounding. This theorem follows immediately from Theorem 3.25, which we prove below.
Theorem 3.25.
Let h P α | α ≤ ν i be a nice iteration so that P = { } and for all i with i + 1 < ν (cid:13) i P i,i +1 is ∞ -subproper and ω ω -bounding. Then for all j ≤ ν thefollowing statement ϕ ( j ) holds: If i ≤ j , p ∈ P i , ˙ σ, ˙¯ G ¯ i ∈ V P i , q ∈ P j with q ↾ i = p , θ is a sufficientlylarge cardinal, τ is an ordinal, ˙ x is a P j -name, H θ ⊆ N = L τ [ A ] | = ZFC − ¯ N is a countable, full, transitive model which elementarily embeds into N so that ¯ s, ¯ ~ P , ¯ i, ¯ j, ¯˙ x ∈ ¯ N and p forces with respect to P i that the following assumptions hold:(A1) ˙ σ (ˇ¯ s, ˇ¯ ~ P , ˇ¯ i, ˇ¯ j, ˇ¯ θ, ˇ¯ q, ¯˙ x ) = ˇ s, ˇ ~ P , ˇ i, ˇ j, ˇ q, ˙ x (A2) ˙¯ G ¯ i is the pointwise preimage of the generic under ˙ σ and is P i -generic over ¯ N (A3) q forces that ˙ x is a name for an element of Baire space(A4) ˙ σ : ˇ¯ N [ ˙¯ G ¯ i ] ≺ ˇ N [ ˙ G i ] is countable, transitive and full. then there is a condition p ∗ ∈ P j such that p ∗ ↾ [ i, j ) ≤ q ↾ [ i, j ) , p ∗ ↾ i = p andthere is a real y ∈ ω ω so that whenever G j ∋ p ∗ is P j -generic, then in V[ G j ] , thereis a σ ′ such that, letting σ = ˙ σ G i , and ¯ G i = ˙¯ G G i i the following hold:(a) σ ′ (¯ s, ¯ ~ P , ¯ i, ¯ j, ¯ θ, ¯ G ¯ i , ¯ q, ¯˙ x ) = σ (¯ s, ¯ ~ P , ¯ i, ¯ j, ¯ θ, ¯ G ¯ i , ¯ q, ¯˙ x ) , (b) ( σ ′− )“ G i,j := ¯ G i,j is P i,j -generic over ¯ N [ ¯ G ¯ i ] ,(c) p ∗ (cid:13) ˙ x ≤ ∗ ˇ y (d) σ ′ : ¯ N [ ¯ G ¯ i ] ≺ N [ G i ] .Proof. As always with these proofs we induct on j . Thus fix j and assume ϕ ( j ′ )holds for all j ′ ≤ j . We again note that we may assume i < j since i = j is trivial.Since we have already proved that the iteration is subproper we focus on proving ω ω -bounding. Note that the case where j is a successor follows from Theorem3.20 plus the fact that two step iterations of ω ω -bounding forcing notions are again ω ω -bounding, thus assume j is limit.Let ˙ x be a P j -name and fix p, q, ... etc as in the statement of the theorem. Inparticular, assume that p ∈ P i and q ∈ P j is such that q ↾ i = p and q (cid:13) j ˙ x : ω → ω .We need to find a p ∗ ≤ q as in the statement of the theorem and a real y ∈ ω ω ∩ Vso that p ∗ (cid:13) j ˙ x ≤ ˇ y . In fact we’ll show something stronger: that P j is ω ω -bounding. Let p ∈ G i ⊆ P i be generic over V and, working briefly in V[ G i ], let σ : ¯ N ≺ N = L τ [ A ] | = ZFC − with H θ ⊆ N for θ large enough, ¯ N countable,transitive and full etc as in the statement of the theorem (essentially letting p forcethat these objects are in the standard set up). Back in V, let h t n | n < ω i enumeratethe elements of ¯ N , and let W be the L τ [ A ]-least nested antichain so that q is amixture of W up to j with T = { } . As in the previous proofs, we may assumethat W is definable and hence in the range of any embedding we will discuss and¯ W is its preimage in ¯ N . Also let { D n | n < ω } enumerate the names in ¯ N for thedense open subsets of ¯ P i,j . Without loss assume that ¯ q ↾ [¯ i, ¯ j ) is forced to be in D (for instance we can make D be the canonical name for the whole poset). Byassumption we have that σ ( q, P j , ˙ x, W ) = q, P j , ˙ x, W . Note that by elementarity q forces ˙ x to be a real and is a mixture of W up j (defined in ¯ N as the length of theiteration P j ).Note that for any real z ∈ ω ω ∩ ¯ N we get that σ ( z ) = z by the absoluteness of ω .Therefore we do not need the “bar” convention to describe reals. Fix some strictlydecreasing sequence of conditions ~s = h s k | k < ω i so that for all k < ω s k ∈ P j and σ ( s k ) = s k ∈ P j and some real x ∈ ( ω ω ) ¯ N with s k (cid:13) j ˇ x ↾ ˇ k = ˙ x ↾ ˇ k and s ≤ q .Note that by elementarity the “bar” versions of the s ’s force the same to be trueof the name ˙ x . Finally in V[ G i ] fix some y ∈ ( ω ω ) V so that for all z ∈ ω ω ∩ ¯ N [ ¯ G i ] TERATION THEOREMS FOR SUBVERSIONS OF FORCING CLASSES 39 z ≤ ∗ y (such a y exists since ¯ N is countable) and in particular x ≤ y where ¯ G i is the pointwise preimage of G i under σ . This is generic over ¯ N by our inductivehypothesis. An important point is that we can find y ∈ V (as opposed to V[ G i ])exactly by the inductive hypothesis that P i is ω ω -bounding. In fact, fixing this y ,the inductive hypothesis on l < j is that y is ≤ ∗ above ¯ N [ ¯ G i ][ ¯ G i,l ].Much like in the iteration theorem the goal is to build a fusion structure anda sequence of names and show that doing so preserves the required properties.Specifically here we want a fusion structure and a sequence of names hh q ( a,n ) , T ( a,n ) , ˙ x ( a,n ) , ˙ σ ( a,n ) , h ˙ s ( a,n ) k | k < ω ii | n < ω and a ∈ T n i so that the following all hold: q (1 , = q , T (1 , = W , ˇ σ (1 , , ˙ x (1 , = ˇ x and h ˙ s (1 , k | k < ω i = h ˇ s k | k <ω i .For all a ∈ T n we have a (cid:13) l ( a ) “ ˙ σ ( a,n ) : ˇ¯ N ≺ N and ˙ σ ( a,n ) ( θ, P j , ˙ x, h ˙ s k | k < ω i ) = θ, P j , ˙ x, h ˙ s k | k <ω i and l ( a ) , T ( a,n ) , q ( a,n ) , ˙ x ( a,n ) , h ˙ s ( a,n ) k | k < ω i are in the range of ˙ σ ( a,n ) and( ˙ σ ( a,n ) ) − ( q ( a,n ) ) ∈ D n ,” and q ( a,n ) ≤ q . For each k < ω ˙ s ( a,n ) k is a l ( a )-name for an element of P j so that q ( a,n ) ↾ l ( a ) (cid:13) l ( a ) q ( a,n ) = ˙ s ( a,n )0 and ˙ s ( a,n ) k +1 ≤ ˙ s ( a,n ) k and ˙ s ( a,n ) k ↾ l ( a ) ∈ ˙ G ↾ l ( a ) ˙ x ( a,n ) is an l ( a ) name that q ( a,n ) ↾ l ( a ) forces to be an element of Baire spaceand for each k < ω q ( a,n ) ↾ l ( a ) (cid:13) l ( a ) “ ˙ s ( a,n ) k (cid:13) ˙ x ↾ ˇ k = ˙ x ( a,n ) ↾ ˇ k .”For every b ∈ suc Tn ( a ) we have that For all k ≤ n , b (cid:13) l ( b ) ˙ σ ( b,n +1) ( t k ) = ˙ σ ( a,n ) ( t k ) and b (cid:13) l ( b ) ( ˙ σ ( b,n +1) ) − ( q ( a m ,m ) ) = ( ˙ σ ( a,n ) ) − (ˇ q ( a m ,m ) ) for all m ≤ n and all a m ∈ T m extended by a . q ( b,n +1) (cid:13) ˙ x ↾ n ≤ ˇ y ↾ n . Where for z ↾ n ≤ y ↾ n means that for each i < nz ( i ) ≤ y ( i ).Supposing we can construct such a sequence and letting p ∗ be the fusion of thefusion sequence it follows almost immediately that p ∗ (cid:13) ˙ x ≤ ˇ y so we would bedone. Thus it suffices to show that such a sequence can be constructed. This isdone by recursion on n < ω . The case n = 0 is given by 1 and it’s routine to checkthat the parameters given there satisfy 2, 3 and 4. For the inductive step, supposefor some n < ω we have constructed q ( a,n ) , T ( a,n ) , ˙ x ( a,n ) , h ˙ s ( a,n ) k | k < ω i and ˙ σ ( a,n ) for some a ∈ T n satisfying 1 to 4 and suc Tm has been defined for all m ≤ n . Assumealso q ( a,n ) decides ˙ x ↾ n − nT ( a ). To this end, let D be the set of all b of length longer than l ( a ) sothat l ( b ) < j and b ↾ l ( a ) ≤ a There are ˙ σ b , u and h ˙ r bk | k < ω i , ˙ x b and and nested antichain S so that S ∠ T ( a,n ) u ≤ q ( a,n ) is a mixture of S up to j b decides ˙ σ b ( t n +1 ), b (cid:13) l ( b ) S, u, l ( b ) ∈ range ( ˙ σ b ) and b (cid:13) l ( b ) ˙ σ b : ˇ¯ N ≺ N and ∀ m ≤ n ˙ σ b ( t m ) = ˙ σ ( a,n ) ( t m ) and ˙ σ b ( θ, P j , j ) = θ, P j , j and there is a u ∈ D n +1 such that b (cid:13) l ( b ) ˙ σ b ( u ) = u .
3, 4, and 6 all hold with u replacing q ( b,n +1) , h ˙ r bk | k < ω i , replacing h ˙ s ( b,n +1) k | k < ω i , and ˙ x b replacing ˙ x ( b,n +1) . As in the proofs of the two iteration theorems, note that if b ∈ D and b ′ ≤ l ( b ) b ,then b ′ ∈ D as well. It follows that D ↾ l ( a ) := { b ↾ l ( a ) | b ∈ D } is open in P l ( a ) .As a result, again it suffices to show that D ↾ l ( a ) is predense below a in P l ( a ) . Forif we know this, D ↾ l ( a ) is dense below a , and we may choose a maximal antichain A ⊆ D ↾ l ( a ) (with respect to P l ( a ) ), which then is a maximal antichain in P l ( a ) below a . Thus, for every c ∈ A , we may pick a condition b ( c ) ∈ D such that b ( c ) ↾ l ( a ) = c ,and define suc nT ( a ) = { b ( c ) | c ∈ A } with ˙ σ b = ˙ σ ( b ( c ) ,n +1) , u = q ( b ( c ) ,n +1) , h ˙ r bk | k <ω i = h ˙ s ( b ( c ) ,n +1) k | k < ω i , and ˙ x b = ˙ x ( b ( c ) ,n +1) .To prove that D ↾ l ( a ) is dense, let G i,l ( a ) be a P i,l ( a ) generic filter over V[ G i ]and assume a ∈ G i ∗ G i,l ( a ) . Work in V[ G i ][ G i,l ( a ) ]. For readability we write G l ( a ) for G i ∗ G i,l ( a ) . Let σ n be the evaluation of ˙ σ ( a,n ) by G l ( a ) . We need to argue that( D ↾ l ( a )) ∩ G l ( a ) is non-empty. There are two steps to this. First we will findconditions that ensure ∞ -subproperness is preserved, as in the previous iterationtheorems. Then we will use them to construct conditions ensuring ω ω -boundingis preserved. The following 3 paragraphs, which describe how to find the u , areessentially verbatim from the iteration theorem for ∞ -subproper forcing and werepeat them for the reader’s convenience. Once this is found, we will argue furtherto find the name ˙ x b and the ˙ r k ’s.Let’s first find u and ¯ u again. By elementarity, q ( a,n ) is a mixture of T ( a,n ) upto j . We have that σ n : h L ¯ τ [ ¯ A ][ ¯ G ¯ i ] , ∈ , ¯ A i ≺ h L τ [ A ][ G i ] , ∈ , A i , so in particular,¯ σ := σ n ↾ L ¯ τ [ ¯ A ] : h L ¯ τ [ ¯ A ] , ∈ , ¯ A i ≺ h L τ [ A ] , ∈ , A i , and ¯ σ (¯ q ( a,n ) , ¯ T ( a,n ) ) = q ( a,n ) , T ( a,n ) .Clearly, in L τ [ A ], it is true that q ( a,n ) is a mixture of T ( a,n ) up to j , so it is true in L ¯ τ [ ¯ A ] that ¯ q ( a,n ) is a mixture of ¯ T ( a,n ) up to ¯ j , and by absoluteness, this it true inV as well.Let ¯ T ( a,n )0 = { ¯ a } . Since we’re working in V [ G l ( a ) ], we have access to all ¯ G k for k ≤ l ( a ) by considering the pointwise pre image of σ ( a,n ) . By induction these areall generic over ¯ N [ G ¯ i ]. Moreover, by Fact 3.13.(1), ¯ a ≡ ¯ q ( a,n ) ↾ l (¯ a ) ∈ ¯ G l (¯ a ) , since l (¯ q ( a,n ) ) = ¯ j > l (¯ a ). So ¯ a ∈ ¯ G l (¯ a ) . Let ¯ v ∈ ¯ T (¯ a,n )1 , that is, ¯ v ∈ suc T (¯ a,n ) (¯ a ), besuch that ¯ v ↾ l ( a ) ∈ ¯ G l (¯ a ) . There is such a ¯ v by Definition 3.12(5). By Fact 3.13.(3),again since l (¯ v ) < ¯ j = l (¯ q ( a,n ) ), it follows that ¯ v ≡ ¯ v ↾ l (¯ a ) ⌢ ¯ q ( a,n ) ↾ [ l (¯ a ) , l (¯ j )). Since¯ v ↾ l (¯ a ) ∈ ¯ G l (¯ a ) , this implies that ¯ v ↾ [ l (¯ a ) , l (¯ v )) ≡ ¯ q ( a,n ) ↾ [ l (¯ a ) , l (¯ v )) (in the partialorder ¯ P l (¯ a ) ,l (¯ v ) = ¯ P l (¯ v ) / ¯ G l (¯ a ) ), and ¯ q ( a,n ) ↾ [ l (¯ a ) , l (¯ v )) ∈ ¯ G ↾ [ l (¯ a ) , l (¯ v )). So we havethat ¯ v ↾ l (¯ a ) ∈ ¯ G l (¯ a ) and ¯ v ↾ [ l (¯ a ) , l (¯ v )) ∈ ¯ G ↾ [ l (¯ a ) , l (¯ v )). By Fact 3.10.(3), thisimplies that ¯ v ∈ ¯ G l (¯ v ) .Now let u strengthen v ⌢ q ( a,n ) ↾ [ l ( v ) , j ) so that u ∈ D n +1 and decides ˙ x ↾ n .This last clause is the only thing different in this case from the proofs of the iterationtheorems above.By Lemma 3.16, applied in ¯ N , there is a nested antichain ¯ S ∠ ¯ T ( a,n ) such that ¯ u is a mixture of ¯ S up to ¯ j and such that letting ¯ S = { ¯ d } , we have that l (¯ v ) ≤ l ( ¯ d )and ¯ d ↾ l (¯ v ) ≤ ¯ v . Let S, d , u = σ n ( ¯ S, ¯ d , ¯ u ), and let w ∈ G l ( a ) force this. Since a ∈ G l ( a ) , we may choose w so that w ≤ a .Note that S, d , u are in N (and hence in V), since ¯ S, ¯ d , ¯ u ∈ ¯ N .Let l ( d ) = β . Now we force one further step and let G l ( a ) ,β be P l ( a ) ,β -genericover V[ G l ( a ) ] with u ↾ [ l ( a ) , β ) ∈ G l ( a ) ,β . Let G β be the composition of the G l ( a ) and G l ( a ) ,β and let ¯ G β be the pointwise preimage of G β under σ n . Now, workingin ¯ N [ ¯ G β ], define recursively a descending sequence of conditions r k ∈ ¯ P j , so that TERATION THEOREMS FOR SUBVERSIONS OF FORCING CLASSES 41 r k decides ˙ x ↾ k , u = r and if s k ↾ β ∈ ¯ G β then r k = s k . Let x ∈ ¯ N [ ¯ G l ( a ) ] be thereal so that r k (cid:13) ˇ x ↾ ˇ k = ˙ x ↾ ˇ k .Back in V[ G l ( a ) ] let h ˙ r k | k < ω i name the sequence of ¯ r ’s, let ˙ r k = σ n ( ˙ r k ) andlet ˙ x be the name for x . Apply the inductive hypothesis ϕ ( β ), noting that β < j ,to i = l ( a ) ≤ β , the filters ¯ G l ( a ) , ¯ G l ( a ) ,l ( β ) , the models ¯ N , N , the condition w (inplace of p ), the name ˙ σ ( a,n ) (in place of ˙ σ and the parameters h ˙ r k | ; k < ω i , ˙ x ∈ ¯ N .The hypothesis allows us to obtain a condition w ∗ ∈ P β with w ∗ ↾ l ( a ) = w and aname ˙ σ ′ such that w ∗ forces with respect to P l ( a ) :(a) ˙ σ ′ and ˙ σ ( a,n ) move the parameters h ˙ r k | ; k < ω i ˇ, ˇ˙ x , ˇ¯ ~ P , ˇ l ( a ), ˇ β , ˇ¯ θ , ˙¯ G l ( a ) , ˇ¯ u ,ˇ¯ d and ˇ¯ S the same way,(b) ( ˙ σ ′− )“ ˙ G l ( a ) ,β is generic over ˇ¯ N [ ˙¯ G l ( a ) , and(c) ˙ σ ′ : ˇ¯ N [ ˙¯ G l ( a ) ] ≺ ˇ N [ ˙ G l ( a ) ].Note that w forced that ˙ σ ( a,n ) (¯ u, ¯ d , ¯ S ) = u, d , S and hence, since w ∗ ↾ l ( a ) = w ,we get that w ∗ forces that ˙ σ ′ (¯ u, ¯ d , ¯ S ) = u, d , S as well. In addition, we may insistthat σ ′ moves the parameters ¯ i, ¯ j, ¯ ~ P , ¯ θ, ¯ s, ¯ p , . . . , ¯ p n , t , . . . , t n the same way ˙ σ ( a,n ) does. Note that already a forced with respect to P l ( a ) that ¯ i, ¯ j, ¯ ~ P , ¯ θ are mapped to i, j, ~ P , θ by ˙ σ ( a,n ) . Finally, applying the ω ω -bounding part of the inductive hypoth-esis to β and ˙ x we may assume that w ∗ forces that ˙ x ≤ ˇ y .To finish, we claim that w ∗ ∈ D as witnessed by u , h ˙ r k | k < ω i , S ˙ x and ˙ σ ′ .Much of this follows from the previous iteration theorem. Indeed the only thingthat requires checking is 6: that u (cid:13) ˙ x ↾ n ≤ ˇ y ↾ n . But this is now clear sincewe explicitly constructed u to decide ˙ x ↾ n and w ∗ forced s n ≤ u to force that˙ x ↾ n = ˙ x ↾ n and w ∗ forced that ˙ x ≤ ˇ y . This completes the inductive step ofthe construction and hence the proof. (cid:3) Remark . In the second author’s PhD thesis (written under the direction of thefirst author) a general version of Theorem 3.25 is proved that implies that manypreservation results on the reals for proper forcing hold for ∞ -subproper forcing aswell including the Sacks and Laver properties.4. Applications
In this section we provide some applications of the preservation theorems provedin the previous two sections, as announced in the introduction. In what follows,one can choose to iterate using either RCS iterations or nice iterations. If using thelatter, “subcomplete” and “subproper” can be replaced by their “infinity” versions.The main technique used here is as follows. We start in a model with a super-compact cardinal. Suppose we have a subproper forcing P which preserves someproperty that is also preserved by all subcomplete forcing (for instance not killinga fixed Souslin tree S ). Then we can add P into the standard Baumgartner typeiteration to produce a model of SCFA while preserving that property (so S remainsSouslin). Moreover, SCFA will be forced in the final model, but P makes somecontribution as well.Recall that a forcing notion P is σ -linked if it can be written as the countableunion, P = S n<ω P n where for each n < ω P n consists of pairwise compatibleelements. Note that σ -linked forcing notions are ccc. The following is well known. Proposition 4.1. If S is a Souslin tree and P is σ -linked then forcing with P doesnot kill S .Proof. Let P and S be as in the statement and since P is σ -linked it can be written as S n<ω P n . Now suppose ˙ A names a maximal antichain in S and suppose p ∈ P forcesthat ˙ A is uncountable. For each n < ω let A n = { s ∈ S | ∃ q ≤ p q ∈ P n q (cid:13) ˇ s ∈ ˙ A } .Since each P n consists of pairwise compatible elements, it follows that each A n isan antichain (in V). Therefore it’s countable. But that means that S n<ω A n iscountable i.e. the set of all s ∈ S so that there is some condition stronger than p forcing s to be in ˙ A is countable, which contradicts the fact that p forced ˙ A to beuncountable. (cid:3) In the following, we will treat both
SCFA , the subcomplete forcing axiom, and itsbounded version
BSCFA . SCFA states that if P is a subcomplete forcing and h D i | i < ω i is a sequence of dense subsets of P , then there is a filter F ⊆ P such that forevery i < ω , F ∩ D i = ∅ . The bounded version of the axiom, denoted BSCFA , isthe weaker form of the axiom stating that whenever P is a subcomplete forcing and h A i | i < ω i is a sequence of maximal antichains in P , each of which has size at most ℵ , then there is a filter F ⊆ P such that for every i < ω , F ∩ A i = ∅ . This axiomwas originally introduced for proper forcing by Goldstern and Shelah [6], where theconsistency strength of the bounded proper forcing axiom was shown to be exactlya reflecting cardinal. In Fuchs [3], the version for subcomplete forcing was analyzedand shown to have the same consistency strength. In all the applications below SCFA and
BSCFA could be replaced by their “ ∞ ” versions however, since we do notknow if these statements are equivalent or not we leave this out. Theorem 4.2.
Assume that κ is supercompact. Then there is a κ -length iteration P κ of subproper forcing notions so that if G ⊆ P κ is generic over V then in V[ G ] there are Souslin trees, c = ℵ and SCFA holds. If κ is only a reflecting cardinalthen the same conclusion holds true with SCFA replaced by
BSCFA , the boundedsubcomplete forcing axiom.
As mentioned in the beginning of this section, the word “iteration” in the theoremcan be interpreted either way discussed in this paper.
Proof.
By forcing if necessary, assume first that in V that there is a Souslin tree S . Let κ be supercompact. We will define a κ -length iteration, P κ as follows: let f : κ → V κ be a Laver function. At stage α if f ( α ) = ( ˙ P , D ) is a pair of P α namessuch that ˙ P is a subcomplete forcing notion and D is a γ -sequence of dense subsetsof ˙ P for some γ < κ then let ˙ Q α = ˙ P otherwise add a Cohen real. By the iterationtheorems proved in the earlier sections, at limit stages either we can decide to takeRCS limits, in which case we need to collapse each iterand to ℵ as well, or elsenice limits in the sense of the previous section. Either way, since every iterandis either subcomplete or proper, the entire iteration is subproper. Moreover, sincesubcomplete forcing doesn’t kill Souslin trees, and neither does Cohen forcing, sinceit’s σ -linked, the entire iteration doesn’t kill S .A standard ∆-system argument shows that P κ has the κ -c.c. but since P κ col-lapses everything inbetween ω and κ , in the extension κ = ω . Also Cohen reals areadded unboundedly often there are κ many new reals in the extension so κ = 2 ℵ .Finally the usual Baumgartner argument shows that SCFA must be forced as well.For a detailed proof of this in the subcomplete context see Jensen [11, pp.65-66,
TERATION THEOREMS FOR SUBVERSIONS OF FORCING CLASSES 43
Proof of Theorem 5]. There Jensen checks the Baumgartner proof when no realsare added but it’s easily seen to go through in this case as well.For the case of
BSCFA the proof is nearly identical, replacing the argument for
SCFA from the Laver diamond by the one for
BSCFA with a reflecting cardinal, seeFuchs [3, Lemma 3.5]. (cid:3)
Following language used by Jensen in [9], let us refer to any model obtainedby performing the Baumgartner style iteration below a supercompact cardinal, byiterating forcings belonging to some forcing class Γ as the natural model for theforcing axiom for
Γ. The previous theorem then shows that the natural model forthe forcing axiom for the class of all forcings which are either subcomplete or Cohenforcing, satisfies
SCFA + c = ω + “there is a Souslin tree”, because Cohen forcingadds a Souslin tree, and that Souslin tree survives.Observe that all that was used about Cohen forcing in the proof above is that itis σ -linked and adds a real. It follows that we could have ensured that we force withevery σ -linked forcing the Laver diamond guessed as well. As a result essentiallythe same proof gives the following. Theorem 4.3.
Assume that κ is supercompact. Then there is a κ -length iteration P κ of subproper forcing notions so that if G ⊆ P κ is generic over V then in V[ G ] there are Souslin trees, c = ℵ , and both SCFA and MA ℵ ( σ − linked) hold. If κ isonly a reflecting cardinal then the same conclusion holds true with SCFA replaced by
BSCFA . As a result,
SCFA + MA ℵ ( σ − linked) does not imply MA (as MA impliesSouslin’s Hypothesis). Again, the model of the previous theorem can be taken to be the natural modelfor the forcing axiom for the class of all forcing notions that are subcomplete or σ -linked.The ω ω -bounding preservation theorem gives us another result along these lines.Recall that d , the dominating number, is the smallest cardinal κ such that there isa collection D of reals such that every real is dominated by some real in D . Theorem 4.4.
Assume that κ is supercompact. Then there is a κ -length iteration P κ of subproper forcing notions so that if G ⊆ P κ is geneic over V then in V[ G ] we have that d = ℵ < c = ℵ and SCFA holds. Moreover, it can be arranged thatthere are either Souslin trees or not. If κ is only a reflecting cardinal then the sameconclusion holds true with SCFA replaced by
BSCFA .Proof.
The idea is the same as in the previous theorem, replacing Cohen forcingwith some fixed ω ω -bounding forcing which adds a real, for example random forcing.By the preservation theorem, the entire iteration will be ω ω -bounding and so d = ℵ as witnessed by the collection of the ground model reals, which will be dominatingand have size ℵ in the final model. For the “moreover” part, note that sincerandom forcing is σ -linked, the resulting iteration will preserve any given Souslintree, thus giving the consistency of the above with a Souslin tree. However, onecould also choose to force with every Souslin tree as they are guessed by the Laverdiamond. Since forcing with a Souslin tree is ccc and does not add reals, it’s inparticular proper and ω ω -bounding. In the latter case there will be no Souslin treesin the final model. (cid:3) Finally let us note, one more application in this spirit, this one due to Jensen [9, § Theorem 4.5 (Jensen) . Assume that κ is supercompact. Then there is a κ -lengthiteration P κ of subproper forcing notions so that if G ⊆ P κ is generic over V thenin V[ G ] SCFA holds, CH holds but all Aronszajn trees are special. In particular ♦ fails. If κ is only a reflecting cardinal then the same conclusion holds true with SCFA replaced by
BSCFA .Proof.
In [7] Jensen introduces the class of “Dee-subcomplete and < ω -subproperforcing notion” and proves that the associated forcing axiom DSCFA is consistentrelative to a supercompact. This class does not add reals hence, like with
SCFA , thenatural model of this axiom satisfies CH as well. While we omit the definition ofthis class here, we note that Jensen shows that it contains all subcomplete forcing,plus a forcing notion for specializing Aronszajn trees. It follows that under DSCFA both
SCFA and “all Aronszajn trees are special” hold (so ♦ fails). Since this axiomis consistent with CH , we’re done. (cid:3) Conclusion and Open Questions
The possible structure of the continuum under
SCFA remains something of amystery. In particular, the above methods show that when CH fails, SCFA does notsay much about the reals or combinatorics on ω . One question is how far this canbe pushed. Question 5.1.
Does
SCFA have any implications on the reals or the structure of H ω ? Does it imply that there are no Kurepa trees? Does it imply that ♦ ∗ fails?As with other forcing axioms, one can strengthen SCFA to SCFA + , resultingin the axiom that says that whenever P is a subcomplete notion of forcing, D isa collection of dense subsets of P of size ω , and ˙ S is a P -name for a stationarysubset of ω , then there is a filter F ⊆ P that meets each set in D and has theproperty that ˙ S F is stationary. It is well-known that under MA + ω ( σ − closed ), thereare no Kurepa trees, and ♦ ∗ ω fails, so the same is true under SCFA + . Note that byconstruction, the “natural models” in the previous theorems satisfy the “+” versionof the relevant forcing axiom.We can ask similar questions about ω , and at that level, much more remainsopen. For instance, we do not even know whether the continuum can be larger than ℵ under SCFA . Question 5.2.
Is 2 ℵ > ℵ consistent with SCFA ?Another line of questioning concerns the utility and uniqueness of ∞ -subproperand ∞ -subcomplete forcing notions. Question 5.3.
Is every ∞ -subcomplete ( ∞ -subproper) forcing notion subcomplete(subproper)? What about just up to forcing equivalence? Are the forcing axiomsfor these classes equivalent?On another note, we ask about the relationship between RCS and nice itera-tions. Note that proper, semiproper, subproper and subcomplete forcing notionsare all iterable by both types of limits. It’s worth asking if this is simply a ZFC -phenomenon.
Question 5.4.
Suppose Γ is a definable class of forcing notions. If Γ is iterablewith RCS, is it iterable by nice iterations? What about the converse? There is a gap in the proof of [9, Cor. 9.1], as discovered by Sean Cox.
TERATION THEOREMS FOR SUBVERSIONS OF FORCING CLASSES 45
References [1] U. Abraham. Proper Forcing. In M. Foreman and A. Kanamori, editors,
Handbook of SetTheory . Springer, 2010.[2] G. Fuchs. Closure properties of parametric subcompleteness.
Archive for Mathematical Logic ,57(7-8):829–852, 2018.[3] G. Fuchs. Hierarchies of forcing axioms, the continuum hypothesis and square principles.
Journal of Symbolic Logic , 83(1):256–282, 2018.[4] G. Fuchs. Diagonal reflections on squares.
Archive for Mathematical Logic , 58(1):1–26, 2019.[5] G. Fuchs and K. Minden. Subcomplete forcing, trees and generic absoluteness.
Journal ofSymbolic Logic , 83(3):1282–1305, 2018.[6] M. Goldstern and S. Shelah. The bounded proper forcing axiom.
Journal of Symbolic Logic L -forcing. In C. Chong, Q. Feng, T. A. Slaman, W. H.Woodin, and Y. Yang, editors, E-recursion, forcing and C ∗ -algebras , volume 27 of LectureNotes Series, Institute for Mathematical Sciences, National University of Singapore ω ω -bounding and semiproper preorders. Kyoto UniversityResearch Repository (Axiomatic Set Theory) , pages 83–99, 2001.[14] T. Miyamoto. On iterating semiproper preorders.
Journal of Symbolic Logic , 67(4), 2002.[15] T. Miyamoto. A class of preorders iterated under a type of RCS. Unpublished, 2011.[16] M. Viale, G. Audrito, and S. Steila. A boolean algebraic approach to semiproper iterations.2014. Preprint: arXiv:1402.1714 [math.LO].(G. Fuchs)
Mathematics, The Graduate Center of The City University of New York,365 Fifth Avenue, New York, NY 10016 & Mathematics, College of Staten Island ofCUNY, Staten Island, NY 10314
E-mail address : [email protected]
URL : (C.B. Switzer) Mathematics, The Graduate Center of The City University of NewYork, 365 Fifth Avenue, New York, NY 10016
E-mail address ::