aa r X i v : . [ m a t h . L O ] J a n INCOMPATIBLE BOUNDED CATEGORY FORCING AXIOMS
DAVID ASPER ´O AND MATTEO VIALE
Abstract.
We introduce bounded category forcing axioms for well-behaved class-es Γ. These are strong forms of bounded forcing axioms which completely decidethe theory of some initial segment of the universe H λ +Γ modulo forcing in Γ, forsome cardinal λ Γ naturally associated to Γ. These axioms naturally extend pro-jective absoluteness for arbitrary set-forcing—in this situation λ Γ = ω —to classesΓ with λ Γ > ω . Unlike projective absoluteness, these higher bounded categoryforcing axioms do not follow from large cardinal axioms, but can be forced undermild large cardinal assumptions on V . We also show the existence of many classesΓ with λ Γ = ω , and giving rise to pairwise incompatible theories for H ω . Contents
1. Introduction 21.1. Model companionship versus bounded category forcing axioms 61.2. Unbounded category forcing axioms 92. Forcing with forcings 102.1. Projective absoluteness 112.2.
BCFA (Γ) and how to get it 183. Forcing with forcings: definitions 203.1. The Factor Lemma for Γ. 203.2. Γ-iterability 213.3. Universality of (Γ , ≤ Γ ) and Γ-rigidity. 223.4. Well-behaved classes Γ 254. Forcing with forcings: proofs 264.1. Why Γ δ ∈ Γ? 264.2. Γ-freezeability versus Γ-rigidity 284.3. Proof of the Factor Lemma for a well-behaved Γ 305. Absolutely well-behaved classes 335.1. Forcing classes. 335.2. Incompatible bounded category forcing axioms 375.3. Proof of theorems 5.20 and 5.21. 385.4. Bounded category forcing axioms and stronger large cardinalassumptions 516. Appendix 526.1. Two-step iterations 526.2. Limit length iterations 54
Mathematics Subject Classification.
References 551.
Introduction
Forcing axioms are principles asserting the existence of sufficiently generic filtersfor all forcing notions in some reasonable class. In a general form they state, for agiven class Γ of forcing notions and a given cardinal κ , that for every P ∈
Γ andevery collection D of ≤ κ -many dense subsets of P there is a filter G of P such that G ∩ D = ∅ for each D ∈ D . We refer to this statement as FA (Γ) κ . These axiomsare successful in settling a wide range of problems undecidable on the basis of thecommonly accepted axioms for set theory. This is not surprising given that forcingaxioms can be often characterized, for reasonable classes Γ, as maximality principles with respect to generic extensions via forcing notions coming from Γ; specifically,the assertions that all statements in some reasonable class Σ that can be forced overthe universe by some forcing in Γ are in fact true.For certain classes Σ of statements, such a maximality principle amounts to as-serting generic absoluteness of the universe with respect to the relevant forcingextensions, i.e., the assertion that for every P ∈
Γ, every P -generic filter G over V ,and every σ ∈ Σ, V | = σ if and only if V [ G ] | = σ . This is the case, for example, if Σis the class of all Σ sentences with parameters in H λ , for a fixed cardinal λ , and allforcing notions in Γ preserve the cardinality of λ . Another example is given by theclass Σ of all projective sentences, i.e., all sentences of the form H ω | = σ , where σ isany sentence with parameters in H ω . It is a remarkable fact, due to Woodin, thatthe mere presence in V of sufficiently strong large cardinals—for example a properclass of Woodin cardinals, or a supercompact cardinal—outright implies this princi-ple of generic absoluteness, known as Projective Absoluteness . Generic absolutenessis a naturally attractive possible feature of the universe, at least for the set-theoristwith realist inclinations, in that it manages to neutralize, to some extent, the effectsof forcing on the universe (forcing being our prime method for proving independenceover our base set theory).It turns out that if we avail ourselves of the expressive power provided by secondorder set theory, we can make sense of hypotheses which in terms of consistencystrength lie way below the existence of Woodin cardinals and which neverthelesssuffice to produce models of Projective Absoluteness. To be specific, if we work in theextension of Morse-Kelley set theory ( MK ) with the axiom saying that the class Ordof all ordinals is Mahlo (i.e., every class of ordinals which is closed and unbounded This is equivalent to the axiom obtained by letting D consist of maximal antichains of P . There are of course interesting classes Σ for which the relevant notion of maximality doesnot imply the corresponding notion of generic absoluteness. Take for example the class Σ of allsentences of the form H ω | = σ , where σ is a Π sentence. Maximality for this class Σ relative toall set-forcing extensions is consistent, as it follows from Woodin’s P max axiom ( ∗ ). On the otherhand, generic absoluteness for Σ is simply false as CH can be expressed by a sentence in Σ andboth CH and ¬ CH are forcible. When λ is an infinite cardinal and Γ is a class of forcing axioms preserving the cardinality of allcardinals µ ≤ λ , the bounded forcing axiom BFA κ (Γ) can often be in fact characterized as preciselythis form of generic absoluteness for κ = λ + . BFA κ (Γ) is the axiom saying that for every P ∈
Γand every collection A of maximal antichains of P , if |A| ≤ κ and | A | ≤ κ for every A ∈ A , thenthere is a filter G ⊆ P such that G ∩ A = ∅ for each A ∈ A . ncompatible bounded category forcing axioms 3 contains an inaccessible cardinal), then we may find a set-forcing extension of V inwhich Projective Absoluteness holds. In fact, for every set-generic extension V [ G ]of V there is a set-generic extension of V [ G ] satisfying Projective Absoluteness. Ofcourse now we do not obtain Projective Absoluteness in V but only prove that itis forcible. In order to even be able to state our starting hypothesis we need totranscend first order logic—which of course was enough to express the existenceof a proper class of Woodin cardinals—and make use of the additional expressivepower of second order logic. This is arguably a drawback of the present situation.Nevertheless, we will make free use of second order logic in this paper as doing so willallow us to make sense of situations for which this would not be possible otherwise.The construction referred to above originates in the classical argument for deriv-ing Projective Absoluteness from large cardinals. The main observation is that ifOrd is Mahlo, then there are unboundedly many inaccessible cardinals δ with theproperty that if G is Coll( ω, <δ )-generic over V , then H V [ G ] ω (= V δ [ G ]) is an ele-mentary substructure of V [ H ] for some generic filter H over V for the class-forcingColl( ω, < Ord) (which of course is the same as Coll( ω, <
Ord) as computed in V [ G ]).Using the fact that every set-forcing extension can be absorbed in a forcing extensionvia Coll( ω, κ ), for some high enough cardinal κ , it follows that Projective Absolute-ness holds in V [ G ] in the above situation. This analysis shows in fact the following:If Ord is Mahlo, then Projective Absoluteness holds if and only if the identity on H ω is an elementary embedding from this structure into V Coll( ω,<δ ) . This justifiesdefining Projective Absoluteness to be precisely this axiom (which we do). We alsowrite PA for Projective Absoluteness.This paper can be naturally split in two parts. The main goal of the first part is toextend the above observation concerning PA and how to obtain it to fragments H κ of the universe beyond H ω . Let us assume that κ is a successor cardinal, κ = λ + . Our guiding idea for obtaining analogues of PA applying to H κ is to focus on someclass Γ of forcing notions preserving all cardinals µ ≤ λ and try to extend themethods in the argument for the forcibility of Projective Absoluteness in a suitableway to apply to forcings in Γ. We shall be dealing with fairly big classes Γ forwhich there is an iterability theorem, for example the class of all proper forcings, orthe class of all semiproper forcings. For these classes one cannot possibly hope toobtain generic absoluteness at the level of H ω relative to all extensions by membersof Γ. For example there is always a proper poset forcing CH and there is alwaysone forcing ¬ CH and, as already mentioned, CH is expressible over H ω . We willinstead aim at obtaining generic absoluteness relative to all forcing notions in asuitable class Γ which, moreover, force the second order axiom corresponding to theaxiom, in the PA situation, asserting that H ω is an elementary substructure of theColl( ω, < Ord)-extension of V .For technical reasons it will be convenient to deal with classes consisting ofcomplete Boolean algebras. The right analogue, in this context, of the collapseColl( ω, <
Ord) in the PA argument turns out to be the class-forcing whose condi-tions are all algebras in Γ, and where C ∈ Γ is stronger than B ∈ Γ, which we will As we will see, our methods naturally pertain to the theory of structures H κ for κ being asuccessor cardinal. See for example [28] for more details. There will be no real loss of generality in restricting thediscussion to classes of complete Boolean algebras thanks to the fact that all classes we will benaturally interested in will be closed under taking regular open completions.
D. ASPER ´O AND M. VIALE denote by C ≤ Γ B , if and only if there is a complete Boolean algebra homomorphism i : B −→ C such that B forces the quotient algebra C /i [ ˙ G B ] to be in the class Γ asinterpreted in V B . Thus we are naturally seeing the category whose objects are allalgebras in Γ, and whose arrows are homomorphisms i : B −→ C of the above form,as a class-sized forcing notion. Given a class Γ of complete Boolean algebras, we will associate to Γ a certaincardinal λ Γ . This will be the supremum of the class of all cardinals preserved byall members of Γ. In all classes Γ we will consider in the second part of the paper, λ Γ = ω (and so H λ +Γ will be H ω ). If Γ has suitable nice properties in all extensionsby members of Γ, then forcing with (Γ , ≤ Γ ) preserves all cardinals µ ≤ λ Γ and makesOrd equal to λ +Γ (i.e., it forces V = H λ +Γ ), and every set in the extension V [ H ] isalready in V [ H ∩ V δ ] for some ordinal δ such that (the regular open completion of)Γ ∩ V δ is in Γ and H ∩ V δ is V -generic for Γ ∩ V δ . Furthermore, under suitable mildlarge cardinal assumptions—typically the same hypothesis we had for PA , namelythat Ord is Mahlo, suffices—we have that for every B ∈ Γ there is a B -name ˙ Q foran algebra in V B ’s version of Γ such that C = B ∗ ˙ Q is in Γ and is such that if G is C -generic over V , then H V [ G ] λ +Γ is an elementary substructure of V [ H ] for everygeneric filter H over V for the category forcing (Γ , ≤ Γ ) as computed in V [ G ]. Wewill call classes Γ satisfying all the relevant nice properties in all generic extensionsby members of Γ absolutely well-behaved . By extending the PA argument we willprove, in addition, that if Ord is Mahlo, Γ is absolutely well-behaved, and H λ +Γ is anelementary substructure of V [ H ] for every generic filter H over V for (Γ , ≤ Γ ), thenthe following is the case.(1) A strong form of the bounded forcing axiom BFA λ Γ (Γ) holds. (2) If G is a V -generic filter for some algebra in Γ and H V [ G ] λ +Γ is an elementarysubmodel of V [ H ] for some generic filter H over V [ G ] for the category forcing(Γ , ≤ Γ ) as computed in V [ G ], then H Vλ +Γ and H V [ G ] λ +Γ have the same theory. It follows, from (1) and (2) above, together with the discussion before (1), thatthe second order axiom saying that H λ +Γ is an elementary substructure of V [ H ] forsome V -generic filter H over (Γ , ≤ Γ ) is a strong form of the bounded forcing axiom BFA λ Γ (Γ) which achieves our goal (and which can actually be forced). We thereforecall this second order axiom the bounded category forcing axiom for Γ, and denoteit by
BCFA (Γ). Our results in the first part of the present paper turn forcing from a tool useful toprove undecidability results into a tool useful to prove theorems: In order to showthat
BCFA ( λ Γ ), together with the ambient set theory, implies H λ +Γ | = σ for a given We will in fact be working with definable classes Γ. In a statement like the one above we areof course really referring to some official definition of Γ. As we will soon mention, we will call our axioms corresponding to these categories Γ bounded category forcing axioms. λ Γ could sometimes be all of Ord (for example if Γ is the class of forcings with the countablechain condition), but in all classes we will consider λ Γ will be an actual cardinal (in this case it isof course the maximum cardinal preserved by all members of Γ). This part is easy. This is considerably more involved. There is no need to specify λ Γ as this cardinal can be read off from Γ. ncompatible bounded category forcing axioms 5 sentence σ , it suffices to show that BCFA (Γ), together with the ambient set theory,implies the existence of a forcing notion B in Γ which forces both BCFA (Γ) and H λ +Γ | = σ .In the second part of the paper we isolate ℵ -many absolutely well-behaved classesΓ of complete Boolean algebras, all of them with λ Γ = ω . Among these we havefor example the class of all complete Boolean algebras which are proper forcingnotions, the class of all complete Boolean algebras which are semiproper forcingnotions, the class of all complete Boolean algebras which are proper forcing notionspreserving Suslin trees, etc. The main point in this second part is to prove that all thecorresponding bounded category forcing axioms are pairwise provably incompatible,sometimes in the presence of extra mild large cardinal assumptions, regardless ofthe fact that we have Γ ⊆ Γ for many choices of Γ and Γ . For example, if Γ and Γ are, respectively, the class of semiproper forcing notions and the class ofproper forcing notions, we have that BCFA (Γ ) and BCFA (Γ ) are incompatible—assuming there is, for example, a measurable cardinal and an inaccessible cardinal δ such that V δ ≺ V —despite the fact that Γ ⊆ Γ and therefore BFA ℵ (Γ ) (whichis implied by BCFA (Γ )) implies BFA ℵ (Γ ) (which is implied by BCFA (Γ )). Indeed,for this choice of Γ and Γ we have that if BCFA (Γ ) holds and there is a measurablecardinal, then Club Bounding holds, whereas if BCFA (Γ ) holds and there is aninaccessible cardinal δ such that V δ ≺ V , then Club Bounding fails.Bounded category forcing axioms are to be seen as strong forms of bounded forcingaxioms providing a picture of the universe as being saturated by only forcing comingfrom the relevant class Γ. For classes Γ ⊆ Γ , even if it is of course true that BFA λ Γ (Γ ) implies BFA λ Γ (Γ ), there will typically be statements σ about H λ +Γ suchthat BFA λ Γ (Γ ) implies σ , whereas ¬ σ can be forced by a forcing in Γ and, onceforced, will be preserved by subsequent forcing in Γ . This explains why σ will followfrom BCFA (Γ ) whereas it will fail in the BCFA (Γ ) model.One lesson to be learned from these incompatibility results is that natural formsof (bounded) forcing axioms do not, by themselves, favour a universist conception ofset theory. There is unavoidable branching at the level of these axioms; in particular,the set-theorist with a universist mindset will need additional criteria—beyond the‘naive’ view on maximality provided by looking at the containment relation betweenthe classes Γ under consideration—to favour one of these axioms over the others.This should be compared with the fact that, as an empirical fact, consistent largecardinal axioms seem to be orderable under implication. A reasonable additionalcriterion available to the universist when assessing bounded category forcing axiom—consistent with the view that these axioms are indeed strong forms of boundedforcing axioms—could be to focus on maximizing the class of Π sentences over H λ +Γ Club Bounding is the statement that for every function f : ω −→ ω there is some α < ω such that every canonical function for α bounds f on a club. Our results in the second part of the paper provide some evidence that this will be the casefor all choices of Γ and Γ . At least upwards directed, in the sense that for any two large cardinal axioms A and A therealways seems to be a large cardinal axiom A subsuming both A and A . This seems to applyboth to large cardinal axioms over ZFC and to the family of large cardinal axioms in the wider ZF context. D. ASPER ´O AND M. VIALE implied by the axiom. In this respect, when looking at classes Γ with λ Γ = ω ,the class that fares the best is of course the class of all forcing notions preservingstationary subsets of ω (we will denote this class by SSP ). This move of courseamounts to ignoring the completeness granted by the axioms and instead focusingon the forcing axiom side of the axiom. When making it, we are probably taking theΠ maximality for H λ +Γ secured by the axiom as the main object of interest of strong(bounded) forcing axioms, and regard any possible completeness for the theory of H λ +Γ modulo forcing as a welcome extra feature, of foundational interest, that theseaxioms may have if they are strong enough.1.1. Model companionship versus bounded category forcing axioms.
Wenow point out the relevance of our results to the notions of model completenessand model companionship introduced by Robinson. There is a growing body ofevidence relating forcing axioms, generic absoluteness results, and the generic multi-verse to these model-theoretic notions. These notions describe, in a model-theoreticterminology applicable to an arbitrary first order theory T , the closure properties ofalgebraically closed fields and the way the elementary class given by such fields sitsinside the elementary class given by arbitrary fields. Applied to the set-theoreticrealm, the basic idea is that H ω plays with respect to the generic multiverse therole C does for arbitrary fields, while (assuming forcing axioms) H ω plays this samerole with respect to the generic multiverse given by forcings in an appropriate class(e.g. proper, SSP , etc).A first key notion in this context is that of existentially closed structures in asignature τ . A τ -structure M is existentially closed in a superstructure N if M ≺ N . For a τ -structure M with domain M , let us write ( M, τ M ) as a shorthand for( M, R M : R ∈ τ ).Consider a signature τ ST in which one adds a predicate symbol R φ of arity n for any ∆ formula φ ( x , . . . , x n ) and interprets in the models of set theory thesepredicate symbols R φ as the extension of the formula φ . In this set-up, one of thekey consequences of Shoenfield’s absoluteness is that ( H Vω , τ V ST ) ≺ ( H V [ G ] κ , τ V [ G ] ST )whenever G is a forcing extension of V and κ is an uncountable cardinal in V [ G ];actually Shoenfield’s absoluteness states that H Vω is existentially closed in all of itswell-founded τ ST -superstructures which model ZFC − (i.e. ZFC minus the power-setaxiom).Consider now bounded forcing axioms; these axioms state that ( H Vω , τ V ST ) ≺ ( H V [ G ] κ , τ V [ G ] ST ) whenever G is V -generic for a forcing notion P in a given class offorcings Γ (stationary set preserving, proper, semiproper, etc) and κ ≥ ω V [ G ]2 ; onceagain, these axioms assert that H Vω is existentially closed in its τ ST -superstructureswhich model ZFC − and are obtained by certain types of forcings.Let us explore briefly the notion of T -existentially closed structure and show howsuch structures are produced in model theory. Given a first order theory T for asignature τ , a τ -structure M is T -e.c. if M ≺ N whenever N is a superstructureof M which realizes T ∀ (the universal fragment of T ). And, after all, the mathematical applicability of bounded forcing axioms is correlated to theirΠ consequences for the theory of H λ +Γ . ncompatible bounded category forcing axioms 7 Note that neither M nor N may be models of T , the only sure thing is thatthey model T ∀ . A key (and not so trivial) fact which will play an essential role inour arguments is that whenever M is T -e.c., so is N if N ≺ M . The standardexample of a T -e.c. structure is an algebraically closed field, where T is the theoryof fields in signature { + , · , , } : if K is algebraically closed, for any L ⊇ K whichmodels T ∀ , any Σ formula with parameters in K (i.e. statements of the form ∃ ~y hV ni =1 p i ( ~y ) = 0 ∧ V mj =1 q j ( ~y ) = 0 i with p i , q j polynomials with coefficients in K )realized in L is already true in K .How does one construct a T -e.c. structure? The simplest way is to start with amodel M of T ∀ and construct using some book-keeping device a chain ( M α : α < κ )of models of T ∀ in which at each stage α one tries to make true in M α +1 someexistential formula with parameters in M α . Note that there is tension betweenthe constraint given by M α | = T ∀ and the requirement that M α +1 realizes some Σ formula with parameters in M α (for example this formula cannot be the negationof some universal axiom of T ).A key point is that an increasing chain of models of T ∀ is still a model of T ∀ , hencethe construction does not stop at limit levels; now if κ is large enough, at stage κ all existential formulae with parameters in some M α for α < κ have been realized(if possible) in some M β with β < κ , hence M κ is T -e.c.Compare this procedure with the standard proof of the consistency of boundedforcing axioms: for example to establish the consistency of BPFA one does exactly thesame, but now one starts from M = ( V, ∈ ) and in passing from M α = ( V [ G α ] , ∈ )to M α +1 = ( V [ G α +1 ] , ∈ ), one can use only forcings which are proper in V [ G α ];moreover, to catch one’s tail one must continue the iteration for a κ which is areflecting cardinal. Also, in this case there is tension between realizing some new Σ formula with parameters in H V [ G α ] ω , and doing so by using a proper forcing (whichpreserves the Π formula, for τ ST in parameter S ⊆ ω , S is stationary ).Now one of the guiding ideas of this paper is: what happens if we continuethis iterated construction along all the ordinals? If the ordinals are long enough,one should end up with a structure where all existential formulae which can beconsistently made true by a proper forcing have been made true. Note, in addition,that V [ G ] should also be a structure which resembles H ω : on the one hand, everyordinal above ω V will have been collapsed to ω V at some point in the iteration; onthe other hand, we should also expect that the regularity of Ord is preserved as alliterands are set-sized and, moreover, ω is preserved as the iteration is proper. Hence,the Ord-length iteration is < Ord-CC, ω -preserving, and collapsing all uncountableordinals to size ℵ , if things are properly organized. For example T is the theory of fields elementarily equivalent to Q in signature { + , · , , } , K is algebraically closed, L ⊇ K is a ring which is not a field. Then K is T -e.c., K ≺ L , both aremodels of T ∀ , but neither of them models T . Note that L may not even be a ring, it is just a structure with no-zero divisors, where + , · satisfy the commutativity, associatitvity, and distributivity laws, and 0 and 1 are the neutralelements of + , · . L is a ring if it satisfies the Π -sentence stating that ( L, + ,
0) is a group. Much in the same way one builds the algebraic closure of a field L by passing from L = L to L the field of fractions of L [ x ] /p ( x ) (with p an irreducible polynomial with coefficients in L ),and then inductively from L n to the field of fractions of L n [ x ] /p n ( x ) with the p n s given by somebook-keeping device. D. ASPER ´O AND M. VIALE
In the above situation, we can look at whether there are stages α for which H V [ G α ] ω is an actual elementary substructure of V [ G ]. And if the ordinals are long enough(e.g., Ord is Mahlo), this will in fact be the case. We can pursue this approach notonly for proper forcings but for a variety of classes Γ including the class of properforcing notions, semiproper forcing notions, SSP , etc.It turns out that if Γ is a sufficiently well-behaved class, then the correspondingOrd-length iteration is equivalent to the category forcing (Γ , ≤ Γ ). We can thendefine the bounded category forcing axiom for Γ, BCFA (Γ), as the assertion that H Vω is an elementary substructure of V [ H ], for some generic extension H of V by(Γ , ≤ Γ ). Working in G¨odel-Bernays or in Morse-Kelley set theory, this is a perfectlymeaningful statement. Using again the fact that Γ is well-behaved, we can thenprove, in the presence of our (mild) large cardinal assumption, that for every forcingextension V [ G ] via a member from Γ there is a further forcing extension V [ G ][ g ] viaa member from Γ V [ G ] satisfying BCFA (Γ); and furthermore, if V | = BCFA (Γ) and G and g are as above, then H Vω and H V [ G ][ g ] ω have the same theory.These bounded category for axiom for Γ not only imply the corresponding boundedforcing axioms but also strengthen the resurrection axioms introduced by Johnstoneand Hamkins [11] and their iterated version introduced by Audrito and the secondauthor [5]. We are now in a position to briefly outline how these results compare tothe notion of model completeness and model companionship.A first order theory T is model complete if and only if M ≺ N whenever N is asuperstructure of M and they are both models of T ; equivalently if and only if anymodel of T is T ∀ -e.c.. T is the model companion of S if T is model complete and ifevery model of T embeds into a model of S and vice versa; equivalently, if T ∀ = S ∀ and any model of T is S -e.c.A standard example is obtained by taking S to be the theory of rings with nozero divisors in signature { + , · , , } and T the theory of algebraically closed fieldsin the same signature.Now let us look at the Γ-generic multiverse (for a sufficiently well-behaved classΓ): n H V [ G ] κ : κ ≥ ω V [ G ]2 , G is V -generic for a forcing in Γ o . The above considerations show that the class (cid:8) H V [ G ] ω : V [ G ] | = BCFA (Γ) , G is V -generic for a forcing in Γ (cid:9) sits inside the Γ-generic multiverse much in the same way the elementary class ofmodels of T sits inside the elementary class of models of S , whenever T is the modelcompanion of S .These considerations do not establish that the theory of H ω is model complete if V | = BCFA (Γ). However, much stronger connections between generic absoluteness,forcing axioms, and model companionship can be obtained if one works with richersignatures. For instance, [24] shows that the theory of H Vω is the model companion ofthe theory of V in the signature τ UB = τ ST ∪ UB V where UB V is the class of universallyBaire sets, and each B ∈ UB V defines a predicate symbol for τ UB . Using the resultof the first author and Schindler establishing that Woodin’s axiom (*) follows from MM ++ [2], the second author [25] has shown that in models of MM ++ where UB ♯ All of the above class are. We are of course identifying Γ with some reasonable definition of it. ncompatible bounded category forcing axioms 9 is invariant across forcing extensions, the theory of H ω is the model companionof the theory of V in signature τ NS ω , UB = τ UB ∪ { NS ω } (where NS ω is the non-stationary ideal on ω , and is the canonical interpretation of its associated unarypredicate symbol in τ NS ω , UB ). Note that assuming large cardinals, the τ UB -theory of H ω is generically invariant, while in [25] it is also shown that the universal fragmentof the τ NS ω , UB -theory of V is invariant across the generic multiverse. Also, the modelcompleteness results of [25] entail that the τ NS ω , UB -theory of H ω is invariant acrossforcing extension of V which model MM ++ . In particular, the results of the presentpaper are weaker than those obtained in [25] when predicated about the class Γ ofSSP-forcings, but as we will see below, they can also be asserted for a variety ofclasses Γ other than SSP (for example proper, semiproper, etc); finally the presentpaper generalizes and improves results appearing in [11, 26, 27, 6], where appropriatestrengthenings of forcing axioms are introduced with the aim of achieving similargoals.1.2.
Unbounded category forcing axioms.
It is worth pointing out that boundedcategory forcing axioms are natural bounded forms of much stronger axioms, whichwe call category forcing axioms . Given an absolutely well-behaved class Γ of com-plete Boolean algebras, the category forcing axiom for Γ,
CFA (Γ), asserts that theclass of certain pre-saturated towers of ideals whose regular open completion is in Γand which have certain ‘rigidity’ properties is dense in the category forcing (Γ , ≤ Γ ). The theory of category forcing axioms, developed by the authors of the present paper,generalizes the theory of the strong form of Martin’s Maximum know as MM +++ .The axiom MM +++ , isolated by the second author, is in fact CFA ( SSP ).In the presence of sufficiently strong large cardinals, we obtain the following.(1) CFA (Γ) can always be forced by a forcing in Γ, and in fact by the intersectionof (Γ , ≤ Γ ) with V δ for some supercompact cardinal δ .(2) CFA (Γ) implies a strong form of the forcing axiom FA λ Γ (Γ).(3) If B ∈ Γ, G is V -generic for B , and V [ G ] | = CFA (Γ), then C Vλ +Γ , the λ +Γ -Changmodel as computed in V , is an elementary substructure of C V [ G ] λ +Γ .In (3) above, given an infinite cardinal λ , the λ -Chang model , denoted by C λ ,is the ⊆ -minimal transitive model of ZF containing all ordinals and closed under λ -sequences. It can be construed as C λ = L (Ord λ ). In particular, L ( P ( λ )) is adefinable inner model of C λ , and H λ + ⊆ C λ is definable in C λ from λ . As is well-known, C λ need not satisfy AC . For instance, by a result of Kunen [14, Thm. 1.1.6and Rmk. 1.1.28], if there are λ + -many measurable cardinals, then C λ | = ¬ AC .It is not difficult to see that if Γ is an absolutely well-behaved class, then CFA (Γ)implies
BCFA (Γ) (again, assuming sufficiently strong large cardinals). In particular, By ‘tower of ideals’ we mean the forcing notion resulting from naturally relativizing the defi-nition of the stationary tower forcing to some given family of ideals of sets with suitable properties.Forcing with such a forcing gives rise to an elementary embedding j : V −→ M , for a certainsubclass M of the generic extension. In the current situation, j has critical point λ +Γ . The relevant large cardinals—typically in the region of the existence of a proper class ofsupercompact cardinals together and one almost super-huge cardinal—are now much strongerthan the ones we need for our analysis of bounded category forcing axioms. Where L (Ord λ ) = S α ∈ Ord L (Ord λ ∩ V α ) = S α ∈ Ord L [Ord λ ∩ V α ]. all axioms CFA (Γ) obtained from considering the classes Γ from the second part ofthe present paper are provably pairwise incompatible.We should point out that, unlike the bounded category forcing axioms we will bestudying in the present paper, the stronger category forcing axioms we are referringto here admit a first order definition. In fact, given that the relevant classes Γ admita ∆ definition, possibly with some parameter, it is easy to see that CFA (Γ) can beexpressed as a Π sentence (in the same parameter).We should clarify that, despite the more attractive aspects of category forcing ax-ioms, compared to their bounded forms, we have opted to present in this paper onlythe theory of the latter. The reason for this is that the theory of their unboundedcounterparts is much more involved and would require a longer article. This workwill appear elsewhere. 2. Forcing with forcings
The aim of this section is to develop a general theory of category forcings i.e. classforcings (Γ , ≤ Γ ) with Γ a definable class of forcings and P ≤ Γ Q if whenever G is V -generic for P , in V [ G ] there is H V -generic for Q such that V [ G ] is a genericextension of V [ H ] by a forcing in Γ V [ H ] . It will become apparent that such ananalysis can be systematically carried out if one focuses on the subclass of Γ givenby the complete boolean algebras in Γ. First of all this is no loss of information sinceour analysis will identify properties of elements of Γ which are invariant with respectto boolean completions. For example “whenever G is V -generic for P , in V [ G ] thereis H V -generic for Q ” can be formulated in algebraic terms as “there is a (possiblynon-injective) complete homomorphism i : RO ( Q ) → RO ( P )”. We hope that thisexample makes transparent that, by focusing on cbas rather than posets, we willbe able to leverage the algebraic structure of complete boolean algebra to greatlysimplify the formulation of certain concepts, as well as many proofs. However mostof the forcings in Γ we will consider are not naturally presented as cbas; for exampleone of our main result will be to show that (Γ ∩ V δ , ≤ Γ ∩ V δ ), which is not evenseparative, is in Γ for most inaccessible cardinals δ . So we will feel free to decidedepending on the issue under examination whether to focus on Γ as a class of cbasor rather as a class of partial orders, keeping in mind that our results applie equallywell in both set-ups. Notation 2.1.
Given a cba B ∈ V and a family A of elements of V B , A ◦ = {h τ, B i : τ ∈ A } . For a partial order P and p ∈ P , P ↾ p = { q ∈ P : q ≤ p } . Definition 2.2.
Let B be an infinite complete boolean algebra and ˙ κ ∈ V B be suchthat J ˙ κ is a regular cardinal K B = 1 B . We define H B ˙ κ = (cid:8) τ : τ ∈ V B , J trcl( τ ) is hereditarily of size less than ˙ κ K B = 1 B (cid:9) ( H B ˙ κ ) ◦ ∈ V B is a canonical name for H V [ G ]˙ κ G whenever G is V -generic for B . Moreprecisely: This very convenient notational trick is due to Asaf Karagila. ncompatible bounded category forcing axioms 11
Assume G is V -generic for B . Then( H B ˙ κ ) ◦ G = (cid:8) τ G : τ ∈ H B ˙ κ (cid:9) = H V [ G ]˙ κ G . On the face of its definition H B ˙ κ is a proper class. To avoid unnecessary complica-tions we can use Scott’s trick and consider just its elements whose rank is boundedby some sufficiently large fixed α . As we will see, in many cases of interest it sufficesto take α = | B | . Definition 2.3.
Given a complete boolean algebra B , we denote by ˙ ω a canonicallychosen member of V B such that J ˙ ω is the first uncountable cardinal K B = 1 B . There is of course a dependence on B in the above definition of ˙ ω . However, wewill not need to make this dependence explicit in the notation.2.1. Projective absoluteness.
We want to sketch the reason why Solovay’s modelfor set theory obtained by forcing with Coll( ω, <δ ) gives the “correct” theory ofprojective sets. While doing so we outline how these results could be generalizedto larger fragment of the universe sets. The following is a weakening of Woodin’sgeneric absoluteness results for the ω -Chang model: Theorem 2.4. [14, Cor. 3.1.7] (Woodin) . Assume V is a ZFC model in whichthere are class-many Woodin cardinals. Let φ ( x , . . . , x n ) be any formula for thesignature {∈} and a , . . . , a n ∈ H ω . TFAE:(1) H Vω | = φ ( a , . . . , a n ) .(2) V | = J H ˙ ω | = φ (ˇ a , . . . , ˇ a n ) K B = 1 B for some cba B .(3) V | = J H ˙ ω | = φ (ˇ a , . . . , ˇ a n ) K B = 1 B for all cbas B . In particular, the second order theory of the natural numbers is provably invariantunder set-sized forcing.We will prove a weak form of the above theorem (considerably weakening thelarge cardinal assumptions); this will be a first approximation to the type of axiomwe will introduce to get generic absoluteness for the theory of H λ + with respect toarbitrary infinite cardinals λ . Notation 2.5.
For δ an ordinal • p ∈ Coll( ω, <δ ) if and only if p : ω × δ → δ is such that dom( p ) is finite and p ( n, α ) ∈ α for all α ∈ dom( p ). • B δ denotes the boolean completion of the forcing notion Coll( ω, <δ ). • In models ( V, V ) of MK , Coll( ω, < Ord) , B Ord ∈ V are the class forcings ob-tained replacing δ with Ord. B Ord is a well defined element of V because Coll( ω, < Ord) is < Ord-CC, hence thesuprema needed to define B Ord are suprema of set-sized antichains, and therefore B Ord is a proper class.Generically Coll( ω, <δ ) adds bijections between ω and all ordinals less than δ ,while making δ the first uncountable cardinal; Coll( ω, < Ord) therefore makes allsets existing in the generic extension hereditarily countable.Solovay realized that these forcings produce natural models of the theory of pro-jective sets (equivalently of the first order theory of ( H ω , ∈ )).The following are well-known properties of B δ (which hold also for δ = Ord by astraightforward generalization of the proofs). Theorem 2.6.
Let ( V, V ) be a model of MK . Assume δ is an inaccessible cardinalor δ = Ord . Then: B δ is <δ -CC: [12, Theorem 15.17(iii)] Hence H B δ ˙ ω = V δ ∩ V B δ . Universality: [12, Lemma 26.9] B δ contains as a complete subalgebra an iso-morphic copy of any cba of size smaller than δ . Factor Lemma: [12, Cor. 26.11]
For all cbas B ∈ V δ , B ∗ ˙Coll(ˇ ω, < ˇ δ ) ∼ = Coll( ω, <δ ) . Logical Completeness: B δ is homogeneous [12, Thm 26.12] .Hence T δ = { φ ( a , . . . , a n ) : a , . . . , a n ∈ H ω , J H ˙ ω | = φ (ˇ a , . . . , ˇ a n ) K B δ = 1 B δ } is a complete first order theory. Definition 2.7. PA (Projective Absoluteness) H ω | = T Ord . PA states that for all formulae φ ( x , . . . , x n ) and a , . . . , a n ∈ H ω we have that H ω | = φ ( a , . . . , a n ) if and only if J φ (ˇ a , . . . , ˇ a n ) K B Ord = 1 B Ord . A complete boolean algebra C ∈ V satisfies PA if and only if (viewing it as acomplete subalgebra of B Ord ) for all τ , . . . , τ n ∈ H C ˙ ω and formulae φ ( x , . . . , x n )without class quantifiers, we have that J φ ( τ , . . . , τ n ) K B Ord = q ( H C ˙ ω ) ◦ | = φ ( τ , . . . , τ n ) y C . Theorem 2.8.
Assume ( V, V ) | = MK + Ord is Mahlo . (i.e. every club subclass of the ordinals in V contains a regular cardinal) .Let (Ω , → Ω ) be the category given by the class of cbas as objects and the class ofcomplete homomorphisms between them as arrows. Set C ≤ Ω B if there is i : B → C in → Ω .Then D PA = { B ∈ Ω : B satisfies PA } is dense in the class partial order (Ω , ≤ Ω ) . In particular PA is a consistent axiom. Proof.
Since B ≥ Ω B δ for any inaccessible δ > | B | , it suffices to show that there arestationarily many δ such that B δ satisfies PA .We need to put together three separate observations on the properties of B γ :(1) For any ordinal γ Coll( ω, <γ ) ⊑ Coll( ω, <
Ord) , (i.e. the inclusion map is a complete embedding of partial orders). Note that the homomorphism may not be injective. The key point is that the preimage i − [ G ]by a complete (possibly non-injective) homomorphism i : B → C of a V -generic filter G for C is a V -generic filter for B . ncompatible bounded category forcing axioms 13 (2) Assume δ is an inaccessible cardinal . Since B δ is <δ -CC and has size δ , itis not hard to check that( H δ ˙ ω ) ◦ G = (cid:8) τ G : τ ∈ V δ ∩ V B , J trcl( τ ) is countable K B δ = 1 B (cid:9) = H V [ G ] ω = V δ [ G ]whenever G is V -generic for B δ .Observe also that Coll( ω, <δ ) is a definable class forcing in the ZFC -model( V δ , ∈ ), which the MK -model ( V δ , V δ +1 , ∈ ) proves to be <δ -CC.By the forcing theorem applied in V δ to the definable class forcing Coll( ω, <δ ),we get that for all V -generic filters G for B δ , all formulae φ ( x , . . . , x n ), andall τ , . . . , τ n ∈ V δ ∩ V B ,( H δ ˙ ω ) ◦ G = V δ [ G ] | = φ ( τ , . . . , τ n )if and only if for some p ∈ G ∩ Coll( ω, <δ ), V δ | = p (cid:13) φ ( τ , . . . , τ n ) . (3) Since h V, Vi is a model of MK , the class C = { γ : ( V γ , ∈ ) ≺ ( V, ∈ ) } is a club subset of Ord.Now for each γ ∈ C , each formula φ ( x , . . . , x n ) without class quantifiers,and each τ , . . . , τ n ∈ V γ ∩ V B , V γ | = p (cid:13) Coll( ω,<γ ) φ ( τ , . . . , τ n )if and only if V | = p (cid:13) Coll( ω,<
Ord) φ ( τ , . . . , τ n ) . This is the case since the classes involved in the definition of p (cid:13) Coll( ω,<
Ord) φ ( τ , . . . , τ n )are all definable in V using parameters in V γ .Since Ord is Mahlo in h V, Vi , there are stationarily many inaccessible cardinals δ in C , and for any inaccessible δ ∈ C the three properties outlined above for B δ holdsimultaneously.The following claim suffices to complete the proof of the theorem. Claim 1.
Assume δ ∈ C is inaccessible. Then for all formulae φ ( x , . . . , x n ) withoutclass quantifiers and τ , . . . , τ n ∈ H B δ ˙ ω = V δ ∩ V B δ , J H ˙ ω | = φ ( τ , . . . , τ n ) K B δ = J φ ( τ , . . . , τ n ) K B Ord . The forcing theorem applies in V δ to the class forcing (relative to V δ ) Coll( ω, <δ ) becauseColl( ω, <δ ) is definably <δ -CC, in the sense that any definable antichain over ( V δ , ∈ ) belongs to V δ . Proof.
Since δ ∈ C is inaccessible, φ does not have class quantifiers, and Coll( ω, <δ ) ⊑ Coll( ω, <
Ord), we have that: J H ˙ ω | = φ ( τ , . . . , τ n ) K B δ == _ Coll( ω,<δ ) (cid:8) p ∈ Coll( ω, <δ ) : h V δ , ∈i | = p (cid:13) Coll( ω,<δ ) φ ( τ , . . . , τ n ) (cid:9) == _ Coll( ω,<δ ) (cid:8) p ∈ Coll( ω, <
Ord) : h V, ∈i | = p (cid:13) Coll( ω,<
Ord) φ ( τ , . . . , τ n ) (cid:9) == _ Coll( ω,<
Ord) (cid:8) p ∈ Coll( ω, <
Ord) : h V, ∈i | = p (cid:13) Coll( ω,<
Ord) φ ( τ , . . . , τ n ) (cid:9) == J φ ( τ , . . . , τ n ) K B Ord . The first and last equalities hold by definition (for the first observe that ˙ H δω = V δ ∩ V B δ is a canonical B δ -name for H V [ G ] ω for G V -generic for B δ ); the second equalityholds because ( V δ , ∈ ) ≺ ( V, ∈ ); the third equality holds because Coll( ω, <δ ) ⊑ Coll( ω, <
Ord). (cid:3)
The theorem is proved. (cid:3)
Remark . If γ ∈ C is not inaccessible, then Coll( ω, <γ ) is not <γ -cc and collapses γ to become countable. Hence H B γ ˙ ω = V γ ∩ V B γ . This gives that PA may not besatisfied by B γ in this case, since the boolean value of J φ ( τ , . . . , τ n ) K B Ord may requiresome p Coll( ω, <γ ) to be computed if there is some τ i which is not in V γ . Corollary 2.10.
Assume ( V, V ) | = MK and i : B → C is a complete homomorphism.Assume B and C both satisfy PA . Then for any G V -generic for C , letting H = i − [ G ] , we get that H B ˙ ω [ H ] = H V [ H ] ω ≺ H V [ G ] ω = H C ˙ ω [ G ] . Proof.
The Corollary is an immediate consequence of the following basic model-theoretic observation:
Fact 2.11.
Assume M , M and N are L -structures such that the following diagramis realized: M NM ω Σ ω ⊑ Then M ≺ M . Notice that if K is V -generic for B Ord , G ∈ V [ K ] is V -generic for C , and H ∈ V [ G ]is V -generic for B , we obtain the above configuration: H V [ H ] ω H V [ K ] ω H V [ G ] ω Σ ω Σ ω ⊑ (cid:3) ncompatible bounded category forcing axioms 15 Corollary 2.12.
Assume ( V, V ) | = MK + PA + Ord is Mahlo . TFAE:(1) H ω | = φ ( a , . . . , a n ) .(2) q H B ω | = φ (ˇ a , . . . , ˇ a n ) y B = 1 B for some cba B which satisfies PA .(3) q H B ω | = φ (ˇ a , . . . , ˇ a n ) y B = 1 B for all cbas B which satisfy PA . Woodin’s generic absoluteness results for projective sets provide significant strength-enings of the conclusion of the theorem above in the presence of stronger hypotheses.The following is a weakening of [14, Theorem 3.1.2] which has the same flavor ofwhat we have been showing so far:
Theorem 2.13 (Woodin) . Assume δ is a Woodin cardinal which is a limit of Woodincardinals and B ∈ V δ is a cba.Then H B ˙ ω ≺ H B δ ˙ ω . Now observe that the fact whether or not δ is Woodin is detected at stage V δ +1 : V | = δ is a Woodin cardinal ⇐⇒ V δ +1 | = δ is a Woodin cardinal . In particular ‘Ord is Woodin ’ is expressible in any model of MK . Corollary 2.14.
Assume ( V, V ) | = MK + Ord is Woodin + there are class many Woodin cardinals . Then every B in Ω satisfies PA .Hence TFAE:(1) H ω | = φ ( a , . . . , a n ) .(2) q H B ω | = φ (ˇ a , . . . , ˇ a n ) y B = 1 B for some cba B .(3) q H B ω | = φ (ˇ a , . . . , ˇ a n ) y B = 1 B for all cbas B . We observe the following:
Remark . The key steps in the proof of the density of D PA = (cid:8) B : H B ˙ ω ≺ V Coll( ω,<
Ord) (cid:9) for the preorder given by the class category (Ω , → Ω ) in a model of MK + Ord is Mahlorely on the following crucial properties: • Coll( ω, <δ ) preserves the regularity of δ for any regular cardinal δ and makes δ the first uncountable cardinal; • Coll( ω, <δ ) ⊑ Coll( ω, <
Ord) for all regular cardinals δ ; • any cba B ∈ V δ embeds as a complete suborder in Coll( ω, <δ ) for any inac-cessible δ ;We want to replicate the above proof pattern for arbitrary classes of forcings Γclosed under two-step iterations. The guiding idea will be that projective absolute-ness is just BCFA (Γ) for Γ = Ω, the class of all forcing notions. We will show that for a variety of definable classes of forcings Γ there is a (uniformly) definable cardinal λ Γ associated to Γ such that D Γ = n B ∈ Γ : H B ˙ λ +Γ ≺ V Γ ↾ B o is dense in the class forcing induced by the category (Γ , → Γ ).Our analysis of projective absoluteness in the previous section shows that λ Ω = ω .We will focus in section 5 on the analysis of classes Γ for which λ Γ = ω . However,the machinery we will present below is modular and shows that for any class offorcings Γ satisfying certain reasonable properties, λ Γ is well defined and D Γ isdense in the class forcing induced by the category (Γ , → Γ ).First of all we need to introduce the terminology and notation required to formu-late precisely what is meant by the class forcing induced by the category (Γ , → Γ ),by BCFA (Γ), etc.
Notation 2.16.
Let i : B → C be a complete homomorphism of cbas.ker( i ) = _ { b ∈ B : i ( b ) = 0 C } , coker( i ) = ¬ ker( i ) ,i is a regular embedding if it is injective and complete.For any b ∈ B k b : B → B ↾ bc c ∧ b. ˙ G B = (cid:8) h ˇ b, b i : b ∈ B (cid:9) is the canonical B -name for the V -generic filter. Remark . All complete homomorphisms i : B → C are of the form i ◦ k b with i : B ↾ b → C a regular embedding and b = coker( i ). Notation 2.18.
Let i : B → C a complete homomorphism and ˙ κ ∈ V B , ˙ δ ∈ V C besuch that J ˙ κ is a regular uncountable cardinal K B = 1 B and r ˙ δ is a regular uncountable cardinal z C = 1 C , where r ˆ i ( ˙ κ ) ≤ ˙ δ z = 1 C . We say that H B ˙ κ ≺ H C ˙ δ if and only if for all τ , . . . , τ n ∈ H B ˙ κ and formulae φ ( x , . . . , x n ) without class quan-tifiers, we have that i ( J H ˙ κ | = φ ( τ , . . . , τ n ) K B ) = r H ˙ δ | = φ (ˆ i ( τ ) , . . . , ˆ i ( τ n )) z C . Here, ˆ i : V B → V C τ n h ˆ i ( σ ) , i ( b ) i : h σ, b i ∈ τ o ncompatible bounded category forcing axioms 17 Our analysis of projective absoluteness covers the situation in which i : B = RO (Coll( ω, <δ )) → C = RO (Coll( ω, <γ ))is the inclusion, δ < γ are both in C (hence they both force PA ), and ˙ κ = ˙ ω B and˙ δ = ˙ ω C are the canonical names for the first uncountable cardinal, for B and C ,respectively. Notation 2.19.
Given the standard model ( V, V ) of MK , a definable class Γ denotesan element of V defined by a first order formula φ Γ ( x, a Γ ) with a Γ ∈ V and φ Γ ( x, y )a formula without class quantifiers.Note that we allow the free variable x to also take values in V , since one of ourobjective will be to infer that φ Γ (Γ , a ) holds in ( V, V , ∈ ) for a wide range of classesΓ. Definition 2.20.
Let Γ be a definable class of pre-orders. • (Γ , → Γ ) is the category whose objects are cbas in Γ, and whose arrows arethe Γ-correct homomorphisms (not necessarily injective). • For cbas B , C , we write – B ≥ Γ C if there is a Γ-correct i : B → C , and – B ≥ ∗ Γ C if there is a Γ-correct i : B → C which is also injective (i.e. i isregular).For any ordinal δ , Γ δ = Γ ∩ V δ . Definition 2.21.
Let Γ be a definable class of pre-orders. • Γ is stable under forcing if for any P ∈ Γ, any pre-order Q such that RO ( P ) ∼ = RO ( Q ) is also in Γ. • A complete (not necessarily injective) homomorphism of cbas i : B → C isΓ -correct if r φ Γ ( C / i [ ˙ G B ] , ˇ a Γ ) z B = r C / i [ ˙ G B ] ∈ ˙Γ z B ≥ coker( i ) . • Γ is closed under two-step iterations if C ∈ Γ whenever B ∈ Γ and i : B → C is Γ-correct. • (Γ , → Γ ) is the category whose objects are cbas in Γ, and whose arrows arethe Γ-correct homomorphisms (not necessarily injective). • For cbas B , C , we write – B ≥ Γ C if there is a Γ-correct i : B → C , and – B ≥ ∗ Γ C if there is a Γ-correct i : B → C which is also injective (i.e. i isregular).For any ordinal δ , Γ δ = Γ ∩ V δ . Remark . Stability under forcing in Γ asserts that Γ is closed with respect tothe equivalence relation on preorders given by P ∼ Q iff RO ( P ) ∼ = RO ( Q ). All theproperties of Γ we are interested in are invariant under this equivalence relation,and each equivalence class according to ∼ has a canonical representative given bythe unique (modulo isomorphism) cba B belonging to it. In particular, we canrestrict our analysis of Γ concentrating just on its subclass given by the cbas init. It will be however sometimes convenient to allow as elements of Γ also partial This set-up provides an algebraic expression of the condition that whenever G is V -genericfor C with i (coker( i )) ∈ G and H = i − [ G ], C / i [ H ] is a cba in V [ H ] which is in Γ V [ H ] . orders P ∈ V which are not cbas but are such that φ Γ ( P, a γ ) holds: specifically,we will need to infer that for well-behaved classes Γ, Γ δ ∈ Γ for many inaccessiblecardinals δ ; (Γ δ , ≤ Γ ) is a pre-order but not a cba, and it is much simpler to describethe combinatorial properties of Γ δ than those of its boolean completion. • If (Γ , → Γ ) has lower bounds for its finite subsets, then ≤ Γ is a trivial forcingnotion since all conditions are compatible. • For any cba B there is a regular embedding i : B → RO (Coll( ω, δ )) for anylarge enough δ ; consequently the class (Ω , → Ω ) of all cbas and all completehomomorphisms between them has lower bounds for its finite subsets, hence ≤ ∗ Ω and ≤ Ω are trivial forcing notions. Definition 2.23.
A partial order P is in SSP if for any stationary S ⊆ ω , P (cid:13) ˇ S is stationary . Fact 2.24. ( SSP , ≤ Ω ) (hence also ( SSP , ≤ SSP ) ) is an atomless partial order.Proof. Assume • P is Namba forcing on ℵ , • Q is Coll( ω , ω ).Then RO ( P ) and RO ( Q ) are incompatible conditions in ( SSP , ≤ Ω ) (and therefore also in( SSP , ≤ SSP )): Assume R ≤ SSP RO ( P ) , RO ( Q ), and H is V -generic for R . Then • ω V [ H ]1 = ω , • there are G, K ∈ V [ H ] V -generic filters for P and Q respectively (since R ≤ Ω RO ( P ) , RO ( Q )). G gives in V [ H ] a sequence cofinal in ω V of type ω . K gives in V [ H ] a sequence cofinal in ω V of type ω V .Contradiction with the preservation of ω in V [ H ] (which holds since R ∈ SSP ).This argument can be repeated in V B for any B ∈ SSP . (cid:3) Remark . Similar arguments show that ≤ Γ defines an atomless partial order for avariety of Γ ⊆ SSP (for example for Γ being the class of proper posets, or the class ofsemiproper posets).2.2.
BCFA (Γ) and how to get it.
First of all we define the cardinal λ Γ we attach to agiven class Γ of forcing notions. λ Γ could actually be Ord. However, in all cases of interest λ Γ will in fact exist as a cardinal. It will be needed to formulate BCFA (Γ) properly (it willalso be a key parameter in the proper formulation of the iteration theorem we will laterrequire to hold for forcings in Γ).
Definition 2.26.
Given a definable class of forcings Γ, λ Γ is the supremum of all cardinals η ∈ V such that all forcings in Γ preserve η . Remark . λ Ω = ω , λ SSP = ω , and the same holds for all Γ ⊆ SSP such that everycountably closed forcing is in Γ (e.g. the class of proper forcings and the class of semiproperforcings). It is easy to see that λ Γ is either Ord or the maximum of the set of cardinalsof which it is a supremum. We will be interested just in the case in which λ Γ is anuncountable regular cardinal (and the reader can safely assume throughout the paper that λ Γ = ω ). Definition 2.28.
Let ( V, V ) be the standard model of MK and Γ ∈ V be a definable classof forcings closed under two-step iterations and such that λ Γ ∈ Ord. We say that the
Bounded Category Forcing Axiom for Γ, BCFA (Γ), holds if H λ +Γ ≺ V Γ , ncompatible bounded category forcing axioms 19 and D Γ = n B ∈ Γ : H B ˙ λ +Γ ≺ V Γ ↾ B o is dense in (Γ , ≤ Γ ). As we will see, the density of D Γ can be proved right away in MK for many classes offorcings Γ, hence BCFA (Γ) holds once we force with some element of D Γ .Assume we have a definable class of forcings (Γ , → Γ ) with the following properties. • Pretameness: V Γ | = Ord = λ +Γ . • Factor Lemma:
Γ is closed under two-step iterations; therefore B ∗ ˙Γ ∼ = Γ ↾ B holds for all cbas B ∈ Γ; • Self-similarity:
For stationarily many inaccessible cardinals α , – Γ ∩ V α = Γ α is such that (Γ α , ≤ Γ ) ∈ Γ, and (as in the case α = Ord) (Γ α , ≤ Γ )preserves the regularity of α making it the successor of λ Γ , and – the map B Γ α ↾ B regularly embeds Γ α into Γ ↾ Γ α . • Universality:
For stationarily many inaccessible cardinals α , every B ∈ Γ α regu-larly embeds into Γ α ↾ C for some C ≤ Γ B in Γ α .Then we would be able to replicate the same proof pattern we used to establish theconsistency of projective absoluteness replacing Coll( ω, < Ord) by Γ and Coll( ω, <α ) byΓ α to infer that D Γ = n B ∈ Γ : H B ˙ λ +Γ ≺ V Γ o is dense in (Γ , ≤ Γ ), as witnessed by the forcings Γ α ↾ C for α inaccessible with Γ α ∈ Γ, V α ≺ V , and C ∈ Γ α .We also want to observe some other basic facts concerning the above conditions. Fact 2.29.
Assume Γ is pretame , satisfies the Factor Lemma , the
Self-similarity condition, and the
Universality condition. Then whenever K is V -generic for Γ and C ∈ K , in V [ K ] there is a V -generic filter for C .Proof. This is easily granted if C = Γ α ∈ K ( K ∩ V α is V -generic for Γ α by self-similarity ).Now assume B ≥ Γ Γ α ↾ C for some Γ α ↾ C ∈ K . Let i : B → Γ α ↾ C witness B ≥ Γ Γ α ↾ C ( i exists by universality ). Then i − [ K ] ∈ V [ K ] is V -generic for B . (cid:3) The following observation is a first indication that these properties are closely relatedto generic absoluteness results and to forcing axioms.
Fact 2.30.
Assume Γ is pretame , satisfies the Factor Lemma , the
Self-similarity condition, and the
Universality condition. Let B ∈ D Γ , and let H be V -generic for B .Then H V [ H ] λ +Γ ≺ Σ V [ H ] C for all C ∈ Γ V [ H ] . H λ +Γ ≺ V Γ means that H Vλ +Γ ≺ V [ G ] for every Γ-generic filter G over V , whereas H B ˙ λ +Γ ≺ V Γ ↾ B means that whenever G is V -generic for Γ ↾ B , in V [ G ] there is H , V -generic for B , such that H V [ H ]˙ λ +Γ ≺ V [ G ] . While parsing through these items, the reader should keep in mind the case Γ being
SSP orthe class of (semi)proper forcings with λ Γ = ω . As we will see below, it will be rather delicate to define ˙Γ and to infer that B ∗ ˙Γ and Γ ↾ B define equivalent class forcings in V ; this despite the intuition that closure of Γ under two-stepiterations should grant it. Proof.
Notice that if C ≤ Γ B , K is V -generic for Γ, with H ∈ V [ K ] V -generic for C , and G ∈ V [ H ] is V -generic for B , we obtain the configuration: H V [ G ] λ Γ V [ K ] V [ H ] Σ ω ⊑⊑ This gives that any Σ -property with parameters in H V [ G ] λ Γ true in V [ H ] remains true in V [ K ] and thus reflects to H V [ G ] λ Γ . (cid:3) In particular, the elements of D Γ are forcing strong versions of the correspondingbounded forcing axiom for Γ.We now give rigorous definitions and outline how to infer the above properties of (Γ , → Γ )for a wide family of classes of forcings which includes the classes of proper, semiproper,and SSP forcings. 3.
Forcing with forcings: definitions
The Factor Lemma for Γ . In Subsection 3.4 we will define the notion of absolutelywell-behaved class. One of the things we will prove about such classes is the following.
Lemma 3.1 (The Factor Lemma for Γ) . Let h V, Vi | = MK , and let Γ ∈ V be a definableabsolutely well-behaved class of forcings. Let B ∈ Γ and Γ B = n ˙ C ∈ V B : r φ Γ ( ˙ C , ˇ a Γ ) z B = 1 B o , → Γ B = n ˙ k ∈ V B : r ˙ k : ˙ C → ˙ D is Γ -correct z B = 1 B o . Then whenever G is V -generic for B , we have that (Γ B ) ◦ G = Γ V [ G ] and ( → Γ B ) ◦ G = ( → Γ ) V [ G ] .Moreover, for any C ≤ Γ B fix in V i C : B → C witnessing this. Then the map: Θ B :Γ ↾ B → B ∗ Γ ◦ B C B ∗ ( C / i C [ ˙ G B ] ) defines a dense embedding of (Γ ↾ B , ≤ Γ ) into the class partial order B ∗ Γ ◦ B with ordergiven by ( d, ˙ D ) ≤ ( c, ˙ C ) if and only if d ≤ B c and r ∃ ˙ k : ˙ C → ˙ D Γ -correct z B ≥ d. We defer the proof to Section 4.3. We don’t need the Factor Lemma for any of theproofs occurring in Sections 3.2, 3.3, 4.1, 4.2. On the other hand, this lemma is needed inthe proof of Theorem 3.14, which will be an easy corollary of all the results obtained in theabove sections combined with the Factor Lemma, and it is also needed in the derivationof Corollary 4.10.Despite the apparent self-evidence of the Factor Lemma, its proof requires a carefulformulation and unfolding of the relation existing between elements of Γ V [ G ] and → Γ V [ G ] and their corresponding B -names Γ B , → Γ B . To appreciate the difficulties one may encounterwhen proving this lemma, observe that the following set of equalities holds true for all Given a B -name ˙ C ∈ V B for a forcing notion, i ˙ C : B → RO ( B ∗ ˙ C ), b
7→ h b, ˙ C i denotes thecanonical embedding of B in the boolean completion of the two-step iteration B ∗ ˙ C . ncompatible bounded category forcing axioms 21 V -generic filters G for B :(Γ B ) ◦ G = n ˙ C G : ˙ C ∈ Γ B o == n ˙ C G : r ˙ C ∈ (Γ B ) ◦ z B = 1 B o == n ˙ C G : r ˙ C ∈ (Γ B ) ◦ z B ∈ G o == n C / i ˙ C [ G ] : C = RO ( B ∗ ˙ C ) and r φ Γ ( ˙ C , ˇ a Γ ) z B = 1 B o == (cid:8) C / i [ G ] : C ∈ Γ ↾ B , i : B → C is Γ-correct (cid:9) . Nonetheless, the equality between the terms in the second and third lines is a bit delicateto prove, and requires a careful reformulation of Cohen’s forcing theorem.It will be even more delicate to prove the corresponding set of equalities for the canonical B -name ( → Γ B ) ◦ for → Γ V [ G ] , and to infer the other desired properties of the B -names (Γ B ) ◦ ,( → Γ B ) ◦ .3.2. Γ -iterability. There are two further key properties of Γ we need to outline in order toinfer the nice properties for Γ needed to establish the consistency of
BCFA (Γ). Formulatedin categorial terms, we need to have that Γ is closed under set sized products and thatmany Γ-valued diagrams have a colimit. Formulated in the forcing terminology, we needthe closure of Γ under lottery sums, and an iteration theorem for Γ.Let’s first focus on the definition of the iterability property for forcings in Γ:
Definition 3.2.
Assume Γ is a definable class of pre-orders closed under two-step itera-tions, and λ Γ is a regular uncountable cardinal. • Γ has the
Baumgartner property if whenever δ is inaccessible and F = { i αβ : B α → B β : α ≤ β < δ } ⊆→ Γ ∩ V δ is an iteration system with B α = lim −→ F ↾ α ∈ Γ for stationarily many α < δ , thenlim −→ F ∈ Γ. • Γ is iterable if Γ has the Baumgartner property and Player II has a winningstrategy Σ(Γ) in the game G (Γ) of length Ord between players I and II definedas follows: – players I and II alternate playing Γ-correct injective homomorphism i α,α +1 : B α → B α +1 ; – player I plays at odd stages, player II at even stages (0 and all limit ordinalsare even); – at stage 0, II plays the identity on the trivial cba . – at limit stages η , II must play a B η ∈ Γ which admits for each α < η aΓ-correct i αη : B α → B η such that i β,η ◦ i α,β = i β,η for all α ≤ β < η ; moreover II must play lim −→{ B α : α < η } at stage η if: ∗ either cof( η ) = λ Γ , ∗ or η is inaccessible and { i αβ : B α → B β : α ≤ β < η } ⊆ Γ ∩ V η ; – II wins G (Γ) if she can play at all stages. Remark . The reader may keep in mind λ Γ = ω and Γ being the class of semiproper forcings whileparsing through the definition. lim −→{ B α : α < η } is the direct limit of the iteration system given by the maps i γβ : B γ → B β which are built along the play of G (Γ) (see appendix 6 for the definition of direct and inverse limitof an iteration system). • If Γ has the Baumgartner property, δ > λ Γ is inaccessible, and F = { i αβ : B α → B β : α ≤ β < δ } ⊆→ Γ ∩ V δ is a play of G (Γ), lim −→ F ∈ Γ is <δ -CC, since: – All B α are <δ -CC having size less than δ , and for all α < δ of cofinality λ Γ , B α is the direct limit of F ↾ α , and therefore Baumgartner’s theorem 6.10applies. – lim −→ F ∈ Γ by the Baumgartner property.In particular, if Γ is closed under two-step iterations but is not iterable, and Σ isa strategy for player II , a play F = { i αβ : B α → B β : α ≤ β < δ } of the game G (Γ) which II cannot win using Σ is such that either – cof( δ ) < δ , or – cf( δ ) = λ Γ and lim −→ ( F ) / ∈ Γ, or – cf( δ ) = λ Γ and there is no lower bound in Γ to the cbas played in the gamebefore stage δ , or – some B α has size bigger than δ .Moreover II can always play at successor stages of G (Γ). • The class Γ of proper forcings is iterable: II plays the identity at all non-limitstages, the full limit at limit stages of countable cofinality, and the direct limit atlimit stages of uncountable cofinality. • With slightly more refined strategies one can prove that also the class of semiproperforcings is iterable, and that so is the class of stationary set preserving forcingsassuming the existence of class many supercompact cardinals. We will addressthis issue with more care in Section 5.3.3.
Universality of (Γ , ≤ Γ ) and Γ -rigidity. What conditions grant that (Γ , ≤ Γ ) ab-sorbs as a complete suborder any set sized P ∈ Γ?The optimal case is that there is a complete embedding i B : B → Γ ↾ B for a dense set of B ∈ Γ.If this is the case, take Q ∈ Γ, find B ≤ Γ Q in the above dense set and i : Q → B in → Γ witnessing this.Then i B ◦ i : Q → Γ ↾ B will witness that Γ ↾ B absorbs Q as well.Now, given B ∈ Γ, we have a natural candidate for a complete embedding i B : B → Γ ↾ B : i B : B → Γ ↾ B b B ↾ b. • i B is order preserving: If b ≤ b , the map i b : B ↾ b → B ↾ b c c ∧ b is Γ-correct and witnesses that B ↾ b ≥ Γ B ↾ b . • i B preserves sups: If { a i : i ∈ I } ⊂ B + is a maximal antichain in B , the product algebra Y i ∈ I ( B ↾ a i )(the lottery sum of { B ↾ a i : i ∈ I } ) is isomorphic to B , the top element of Γ ↾ B in (Γ ↾ B , ≤ Γ ). ncompatible bounded category forcing axioms 23 • PROBLEM:
Does this map preserve incompatibility? In general NO!Assume B is homogeneous (for example B is the boolean completion of Cohen’sforcing 2 <ω ). Assume s, t are incompatible conditions in B ; by homogeneity, B ↾ s is isomorphic to B ↾ t ; therefore the incompatible s, t ∈ B are mapped to thecompatible conditions B ↾ s , B ↾ t in (Γ , ≤ Γ ).To overcome this problem we need to find densely many highly inhomogeneous B ∈ Γ.Suppose for the moment that the map i B : B → Γ ↾ B b B ↾ b. defines a complete embedding (i.e. preserves the incompatibility relation), and pick a V -generic H for Γ such that B ∈ H . Then G = i − B [ H ] is V -generic for B .Suppose now that { B : i B defines a complete embedding } is dense in (Γ , ≤ Γ). Assume H is V -generic for Γ. Pick Q ∈ H . By density there is B ∈ H refining Q and such that i B defines a complete embedding of B into Γ ↾ B . Let i : Q → B witness B ≤ Γ Q . Then K = ( i B ◦ i ) − [ H ] is V -generic for Q .In particular we get the following: Fact 3.4.
Assume E Γ = { B ∈ Γ : i B : b B ↾ b defines a complete embedding of B into Γ ↾ B } is dense in (Γ , ≤ Γ ) .Then any V -generic filter H for Γ adds a V -generic filter for any Q ∈ H . How do we get to the density of E Γ ? The key step is to reformulate properly thecondition that i B is a complete embedding. Definition 3.5.
Let Γ be a definable class of cbas closed under two-step iterations, andlet B ∈ Γ. B ∈ Γ is Γ-rigid if for i , i : B → Q in → Γ we have that i = i . Remark . Γ-rigid cbas B are absorbed by (Γ ↾ B , ≤ Γ ) using the map i B : b B ↾ b . Seethe Lemma below. Lemma 3.7.
The following are equivalent characterizations of Γ -rigidity for an algebra B ∈ Γ :(1) for all b , b ∈ B such that b ∧ B b = 0 B , B ↾ b is incompatible with B ↾ b in (Γ , ≤ Γ ) .(2) For every C ≤ Γ B and every V -generic filter H for C , there is just one Γ -correct V -generic filter G ∈ V [ H ] for B .(3) For all C ≤ Γ B in Γ there is only one Γ -correct homomorphism i : B → C .Proof. We prove these equivalences by contraposition as follows:
Assume 1 fails as witnessed by i j : B ↾ b j → Q for j = 0 , b incompatible with b in B . Pick H , a V -generic filter for Q . Then G j = i − j [ H ] ∈ V [ H ] (for j = 0, 1) are distinct and Γ-correct V -generic filters for B in V [ H ], since b j ∈ G j \ G − j . Assume 3 fails for B as witnessed by i = i : B → C . Let b be suchthat i ( b ) = i ( b ). W.l.o.g. we can suppose that r = i ( b ) ∧ i ( ¬ b ) > C . Then j : B ↾ b → C ↾ r and j : B ↾ ¬ b → C ↾ r given by j k ( a ) = i k ( a ) ∧ r for k = 0 , a in the appropriate domain witness that B ↾ ¬ b and B ↾ b are compatible in(Γ , ≤ Γ ), i.e. that 1 fails. Assume 2 fails for B as witnessed by some C ≤ Γ B , a V -generic filter H for C , and Γ-correct V -generic filters G = G ∈ V [ H ] for B . Let ˙ G , ˙ G ∈ V C be such that ( ˙ G ) H = G = ( ˙ G ) H = G are Γ-correct V -generic filters for B in V [ H ] for both j = 1 ,
2. Find q ∈ G forcing that b ∈ ˙ G \ ˙ G for some fixed b ∈ B . Then for some r ∈ H refining q , we have that both homomorphisms i j = i ˙ G j ,r : B → C ↾ r defined by a r ˇ a ∈ ˙ G j z C ∧ r are Γ-correct. However i ( b ) = r = i ( ¬ b ), and hence i = i witness that 3 fails for B and C ↾ r . (cid:3) Fact 3.8.
The class of Γ -rigid cbas is closed under set-sized products (i.e. lottery sums),and the restriction operation B B ↾ b for b ∈ B + .Proof. We leave to the reader to check that for all b ∈ B + , B ↾ b is Γ-rigid if so is B .Assume now that { B i : i ∈ I } is a family of Γ-rigid cbas.Towards a contradiction, assume k j : ( Q i ∈ I B i ) → C are distinct and Γ-correct for j = 0 ,
1. if k ↾ B i = k ↾ B i for all i ∈ I , we get that k = k , which is a contradiction.Hence for some i ∈ I , k ↾ B i = k ↾ B i .Then for some a ∈ B i , k ( a ) = k ( a ), therefore k ( a )∆ k ( a ) > C . So either k ( a ) ∧ k ( ¬ a ) = c > C or k ( a ) ∧ k ( ¬ a ) = c > C . In either cases we get that B i is not Γ-rigidas witnessed by the distinct Γ-correct maps (for i = 0 , k ∗ i : b k i ( b ) ∧ c with domain B i and range C ↾ c . (cid:3) Lemma 3.9.
Let Γ be closed under set-sized products (i.e. set-sized lottery sums) andsuch that the Γ -rigid forcings are dense in (Γ , ≤ Γ ) . Assume D is a dense open class of (Γ , ≤ Γ ) .Then for all B ∈ Γ , there is a Γ -rigid C ≤ ∗ Γ B and a maximal antichain A of C suchthat k C [ A ] ⊆ D .Proof. Given B ∈ Γ, find a Γ-rigid C ≤ Γ B with C ∈ D , with k : B → C a witness that C ≤ Γ B and b = coker( k ) . Then k ↾ b : B ↾ b → C b k ( b )is injective and Γ-correct and witnesses B ↾ b ≥ ∗ Γ C .Now find C ≤ Γ B ↾ ¬ b Γ-rigid and in D with k : B ↾ ¬ b → C a witness that C ≤ Γ B and b = coker( k ) ≤ ¬ B b . Then k ↾ b : B ↾ b → C is injective.Continuing this way we construct by induction a maximal antichain E = { b α : α < γ } of B such that for all α < γ there is b α ≤ ¬ W β<α b β , C α ∈ D , and k α : B ↾ b α → C α witnessing C α ≤ ∗ Γ B ↾ b α .Let k : B → Y α<γ C α b
7→ h k α ( b ∧ b α ) : α < γ i . Then C ≤ ∗ Γ B as witnessed by k , and A = { c α = k ( b α ) : α < γ } = k [ E ] is a maximalantichain of C such that for all α < γ : • k ( b α ) = h C , . . . , C η , . . . , C α , C α +1 , . . . , C ξ , . . . . . . i ∈ A ; • C α ∼ = C ↾ k ( b α ) = k C ( c α ) ∈ D . ncompatible bounded category forcing axioms 25 Finally notice that C is Γ-rigid, being the lottery sum of Γ-rigid forcings. (cid:3) It can also be shown that Γ-rigidity is preserved by passing to generic quotients; i.e.
Assume C ∈ V is Γ -rigid, G is V -generic for B , and i : B → C is a Γ -correct homomorphism. Then C / i [ G ] is Γ V [ G ] -rigid in V [ G ] . But this fact (which we don’t need) has a very convoluted proof (it essentially amountsto a different proof of the Factor Lemma), so we omit it.3.4.
Well-behaved classes Γ . We can now give the key definitions and state the maintheorem which will be repeatedly used in Section 5.
Definition 3.10.
Let ( V, V ) be the standard model of MK . Given a definable class ofcbas Γ ∈ V closed under two-step iterations: • (Γ , ≤ Γ ) is strategically < Ord -closed if λ Γ is a regular uncountable cardinal, and Γis iterable with an iteration strategy Σ(Γ) definable in the ZFC -model ( V, ∈ ). • Γ is closed under lottery sums if any set-sized product of cbas in Γ is in Γ. • Γ is closed under isomorphisms if C ∈ Γ whenever B ∈ Γ and C ∼ = B . • Γ is closed under restrictions and complete subalgebras if for every B ∈ Γ: – for every b ∈ B , the map k b : B → B ↾ bc c ∧ b is Γ-correct, and – any complete subalgebra of B is in Γ.The following is the key definition of the paper. Definition 3.11.
Assume ( V, V ) is a model of MK + Ord is Mahlo .A definable class Γ is well-behaved in V if:(1) λ Γ is a regular uncountable cardinal.(2) Γ is closed under isomorphisms, two-step iterations, lottery sums, restrictions, andcomplete subalgebras.(3) For all B ∈ Γ and
G V -generic for B , V [ G ] models that Γ V [ G ] is strategically < Ord-closed.(4) Γ contains as elements all < λ Γ -closed forcings.(5) For all inaccessible cardinals δ > | a Γ | and for all B ∈ V δ , V δ | = φ Γ ( B , a Γ ) if and only if V | = φ Γ ( B , a Γ ) . (6) The Γ-rigid cbas are dense in (Γ , ≤ Γ ).Γ is absolutely well-behaved if Γ V [ G ] is well-behaved in V [ G ] for any V -generic filter G for some B ∈ Γ. Remark . In all classes Γ we will consider, clause 5 in Definition 3.11 will be provedby showing that Γ can be defined both by a Σ formula and by a Π formula, possiblywith parameters. Remark . • We will show that
SSP , proper, semiproper and many other well-known classesof forcings contained in
SSP are absolutely well-behaved (in some cases assuminglarge cardinals in V ).The key point is that being well-behaved for all these Γ is provable in ZFC (+large cardinals). The only property of well-behavedness not covered elsewhere inthe literature for these classes of forcings is the density of Γ-rigid forcings. • The class Γ given by CCC forcings is not well-behaved; for example it is not closedunder lottery sums (easy), and it does not have Γ-rigid elements (less straightfor-ward).The following is one of the main results of this paper.
Theorem 3.14.
Assume h V, Vi satisfies • MK , • λ Γ is a regular uncountable cardinal, and • Γ is absolutely well-behaved.Then V Γ models that Ord is the successor of λ Γ .Moreover, for all inaccessible δ such that V δ ≺ V :(1) Γ δ ∈ Γ is Γ -rigid, and is such that for all B ∈ Γ δ there is C ∈ Γ δ such that Γ δ ↾ C ≤ Γ B .Hence the class of Γ -rigid forcings is pre-dense as witnessed by the forcings Γ δ as δ ranges on the inaccessible cardinals with V δ ≺ V .(2) Γ δ preserves the regularity of δ and forces it to become the successor of λ Γ .(3) If λ Γ = ω , Γ δ forces BFA (Γ) , and if δ is supercompact, it forces also FA ω (Γ) .(4) For all G V -generic for Γ with Γ δ ∈ V [ G ] , letting G δ = G ∩ V δ we have that H V [ G δ ]( λ Γ ) + = V δ [ G δ ] ≺ V [ G ] = H V [ G ]( λ Γ ) + . Corollary 3.15.
Assume h V, Vi satisfies • MK , • Ord is Mahlo, • λ Γ is a regular uncountable cardinal, and • Γ is absolutely well-behaved.Then D Γ = (cid:26) B ∈ Γ : H B ˙ λ +Γ ≺ H Γ ↾ B ˙ λ +Γ (cid:27) is dense as witnessed by Γ δ ↾ C as δ ranges among the inaccessible cardinals γ with ( V γ , ∈ ) ≺ ( V, ∈ ) and C among the elements of Γ δ .In particular, for every B ∈ Γ there is some C ≤ Γ B such that C forces BCFA (Γ) . The next section is devoted to the proof of Theorem 3.14. We will pay special attentionto giving detailed proofs of 3.14(1) and 3.14(2). The reader familiar with these proofs willbe able to fill in the details needed to prove the remaining assertions of Theorem 3.14.The key result is 3.14(1).
Notation 3.16.
Given a category forcing (Γ , ≤ Γ ) with Γ a definable class of completeboolean algebras and ≤ Γ the order induced on Γ by the Γ-correct homomorphisms betweenelements of Γ, we denote the incompatibility relation with respect to ≤ Γ by ⊥ Γ , and thesubclass of Γ given by its Γ-rigid elements by Rig Γ .4. Forcing with forcings: proofs
Why Γ δ ∈ Γ ?Notation 4.1. Given a Γ-rigid forcing B : • k B : b B ↾ b is the canonical embedding of B into Γ ↾ B . • if C ≤ Γ B , k BC denotes the unique Γ-correct homomorphism of B into C . ncompatible bounded category forcing axioms 27 Theorem 4.2.
Assume Γ is absolutely well-behaved, and δ is inaccessible and such that V δ ≺ V . Then Γ δ ∈ Γ preserves the regularity of δ making it the successor of λ Γ . Notice that Theorem 4.2 proves Theorem 3.14(1) (except for the assertion that for all B ∈ Γ δ there is C ∈ Γ δ such that Γ δ ↾ C ≤ Γ B ) and Theorem 3.14(2). Proof.
Let P δ Γ be the set of F = { i α,β : B α → B β : α ≤ β < η } subsets of → Γ ∩ V δ suchthat: • F ∈ V δ is a partial play of G δ (Γ) according to Σ δ (Γ); • all the moves of I in F are Γ-rigid forcings; • G ≤ F if G is an end-extension of F .Since Γ is iterable, it is immediate to check that P δ Γ is <δ -closed, hence in Γ, since δ ≥ λ Γ and Γ is well-behaved (by clause 3.11(4)).Moreover, let G = { B α : α < δ } be (the union of) a V -generic G filter for P δ Γ . Then G is an iteration system in V [ G ], and it is clear that V [ G ] = V [ G ].Since P δ Γ is <δ -closed, V V [ G ] δ = V δ ≺ V . In particular δ is inaccessible in V [ G ], henceΓ V [ G ] δ = Γ V [ G ] ∩ V V [ G ] δ = Γ ∩ V δ where the equalities hold because of clause 3.11(5) in the definition of well-behaved classapplied in V and in V [ G ].Now Γ V [ G ] is iterable in V [ G ] (by clause 3.11(3)), hence it has the Baumgartner propertyin V [ G ]. Observe that for all ξ < δ of cofinality λ Γ , B ξ is the direct limit of G ↾ ξ , and theset of such ξ is stationary in δ in V [ G ] since P δ Γ is <δ -closed.By the Baumgartner property of Γ V [ G ] and Theorem 6.10 applied in V [ G ], letting B G ∈ V [ G ] be the direct limit of the iteration system G , V [ G ] | = B G is in Γ V [ G ] and is <δ -CC . Therefore V | = P δ Γ ∗ B ˙ G is in Γ (by clause 3.11(2)) and preserves the regularity of δ ,being a two-step iteration of a <δ -closed forcing with a <δ -CC forcing. Claim 2.
Let H be V [ G ] -generic for B G . Then { B α ↾ f ( α ) : f ∈ H, α < δ } is V -generic for Γ δ . By the Claim we get that Γ δ ⊑ P δ Γ ∗ ˙ B , hence Γ δ is in Γ by clause 3.11(2).Therefore it suffices to prove the Claim to conclude that Γ δ is in Γ and preserves theregularity of δ . Proof.
We leave to the reader to check that { B α ↾ f ( α ) : f ∈ H, α < δ } is a filter on Γ δ . We need to prove that this filter is V -generic.The key to the proof is the following: Subclaim 1.
For any D dense open subset of Γ δ , the set of partial plays F = { B η : η < α } ∈ P δ Γ such that there is ξ < α and A maximal antichain of B ξ with { B ξ ↾ a : a ∈ A } = k B ξ [ A ] ⊆ D is dense open in P δ Γ . Assume the subclaim holds. Then for D dense open subset of Γ δ , there is F = { B η : η < α } ∈ G , ξ < α and A maximal antichain of B ξ such that { B ξ ↾ a : a ∈ A } ⊆ D. Now { f ( ξ ) : f ∈ H } is V [ G ]-generic for B ξ , and A is still a maximal antichain of B ξ in V [ G ]. Therefore forsome a ∈ A and f ∈ H , f ( ξ ) ≤ a . Hence B ξ ↾ a ∈ D ∩ H , proving the claim.We prove the subclaim: Proof.
Assume { B ξ : ξ < α } ∈ P δ Γ and D is dense open in Γ δ .Then { B ξ : ξ < α } is a play according to Σ(Γ). Notice that there is some freedom todecide what B ξ is only for odd ξ and for B , because the even stages are decided by thewinning strategy Σ(Γ) for player II . W.l.o.g. (by prolonging { B ξ : ξ < α } if necessary) wemay assume that α is odd so that it is I ’s turn to play. This gives that α = β + 1. Then(by Lemma 3.9) there is B β +1 ≤ ∗ Γ B β which is Γ-rigid and such that some A ⊆ B β +1 is amaximal antichain with k B β +1 [ A ] ⊆ D . By definition of P δ Γ , { B ξ : ξ ≤ β } ∪ { B β +1 } ∈ P δ Γ and (cid:8) B B β +1 ↾ a : a ∈ A (cid:9) = k B β +1 [ A ] ⊆ D. (cid:3) The Claim is proved. (cid:3)
To conclude the proof of the Theorem, we are left with proving the following.
Claim 3. Γ δ makes δ the successor of λ Γ .Proof. Since λ Γ is preserved by all forcings in Γ, we get that λ Γ is preserved by Γ δ . Wealso know that δ is a regular cardinal of V [ G ] whenever G is V -generic for Γ δ .We must show that δ is the successor of λ Γ in V [ G ].For any ordinal α ≥ λ Γ and B ∈ Γ, V models that B ∗ ˙Coll( λ Γ , α ) ∈ Γ (since Γ containsall λ Γ -closed forcings and is closed under two-step iterations); we easily get (since δ isinaccessible) that B ∗ ˙Coll( λ Γ , α ) ∈ Γ δ for all λ Γ ≤ α < δ .In particular, for any λ Γ ≤ α < δ , the set D α of C ∈ Γ δ which collapse α to have size λ Γ is dense open in Γ δ .Since V δ ≺ V , also the set Rig Γ ∩ D α is dense in Γ δ for all α < δ . By Remark 3.6 (andusing V δ ≺ V ) b B ↾ b is a complete embedding of B into Γ δ ↾ B for any B ∈ Rig Γ . Inparticular, if G is V -generic for Γ δ , in V [ G ] there is a V -generic filter H for some B ∈ D α for any α < δ ; any such generic filter H adds a surjection of λ Γ onto α existing in V [ G ].We are done. (cid:3) The theorem is proved. (cid:3) -freezeability versus Γ -rigidity. It will be convenient in order to establish thata certain class is well-behaved to prove that it satisfies a clause weaker than 3.11(6): theΓ-freezing property.
Definition 4.3.
Let Γ be a definable class of cbas closed under two-step iterations, and B ∈ Γ. A Γ-correct k : B → C is Γ -freezing if for all i , i : C → Q in → Γ we have that i ◦ k = i ◦ k (i.e. if the map b C ↾ k ( b ) is incompatibility preserving for ≤ Γ ).We can give the following characterizations of Γ-freezeability, the proof of which is alongthe same lines of the proof of Lemma 3.7 and is left to the reader. Lemma 4.4.
Let k : B → Q be a Γ -correct homomorphism. The following are equivalent:(1) For all b , b ∈ B such that b ∧ B b = 0 B we have that Q ↾ k ( b ) is incompatiblewith Q ↾ k ( b ) in (Γ , ≤ Γ ) . We do not as yet say that it is also a Γ-correct embedding; this is indeed the case but to inferit we need the Factor Lemma, whose proof is not as yet granted. ncompatible bounded category forcing axioms 29 (2) For every R ≤ Γ Q and every V -generic filter H for R , there is just one Γ -correct V -generic filter G ∈ V [ H ] for B such that G = k − [ K ] for all Γ -correct V -genericfilters K ∈ V [ H ] for Q .(3) For all R ≤ Γ Q in Γ and i , i : Q → R witnessing that R ≤ Γ Q we have that i ◦ k = i ◦ k . Theorem 4.5.
Assume Γ is a definable class of forcings sastisfying clauses 3.11(1),3.11(2), 3.11(3), 3.11(4), 3.11(5) of Def. 3.11. Assume further that for all B ∈ Γ there is i B : B → C injective and Γ -freezing B . Then the class of Γ -rigid partial orders is dense in (Γ , ≤ ∗ Γ ) .Proof. Fix B ∈ Γ and Σ λ Γ (Γ) be a winning strategy for player II in G λ Γ (Γ).Define F = { k αβ : B α → B β : α ≤ β < λ Γ } by recursion on λ Γ as follows: • at stage 0, II plays k : → B ; • at odd stages α , I plays k α,α +1 : B α → B α +1 Γ-correct and injective which freezes B α ; • at even stages α , II plays according to Σ λ Γ .By iterability of Γ, B λ Γ is the direct limit of F and belongs to Γ. Clearly k λ Γ : B → B λ Γ witnesses B λ Γ ≤ ∗ Γ B . It suffices to prove the following: Claim 4. B λ Γ is Γ -rigid.Proof. Assume B λ Γ ↾ f is compatible with B λ Γ ↾ g in (Γ , ≤ Γ ) for some threads f, g incom-patible in B λ Γ .Let R ≤ Γ B λ Γ ↾ f, B λ Γ ↾ g .Since B λ Γ is a direct limit, f , g are threads with support bounded by some β < λ Γ .Hence: f ( α ) and g ( α ) are incompatible in B α for all λ Γ ≥ α ≥ β .Now k ββ +2 freezes B β , hence B β +2 ↾ k ββ +2 ( f ( β )) is incompatible with B β +2 ↾ k ββ +2 ( g ( β )).But k ββ +2 ( f ( β )) = f ( β + 2) and k ββ +2 ( g ( β )) = g ( β + 2), since both threads f, g havesupport at most β . Hence B β +2 ↾ g ( β + 2) ⊥ Γ B β +2 ↾ f ( β + 2) . We reached a contradiction: • On the one hand, for all α < λ Γ : B α ↾ f ( α ) ≥ Γ B λ Γ ↾ f ≥ Γ R , and B α ↾ g ( α ) ≥ Γ B λ Γ ↾ g ≥ Γ R . • On the other hand, B β +2 ↾ g ( β + 2) ⊥ Γ B β +2 ↾ f ( β + 2) . (cid:3) The theorem is proved. (cid:3)
Notice the following:
Fact 4.6.
Assume Γ ⊆ ∆ are definable classes of forcings. Then ≤ Γ ⊆≤ ∆ and ⊥ ∆ ⊆ ⊥ Γ .Hence, if i : B → C is Γ -correct and ∆ -freezes B , we also have that i is ∆ -correct and Γ -freezes B . Here we use crucially use that B λ Γ is a direct limit! For inverse limits it is well possible thattwo incompatible threads f , g are such that f ( α ) and g ( α ) are compatible in B α for all α . This fact will be repeatedly used to show that various classes of forcings ∆ have the∆-freezeability property providing for some Γ ⊆ ∆ an i : B → C which is Γ-correct and∆-freezes B . As we will see, all our freezeability results proceed by proving the existence,given B ∈ Γ, of a B -name ˙ Q for a forcing in Γ such that C = B ∗ ˙ Q codes the generic filter˙ G B for B as a subset A ˙ G B of ω in some absolute manner, in the sense that in every outermodel M of V C preserving stationary subsets of ω , A ˙ G B is the unique subset of P ( ω )satisfying some given property. It will thus follow that C SSP -freezes B , which will be aninstance of the above since we will always have SSP ⊇ Γ for the Γ of interest to us.4.3.
Proof of the Factor Lemma for a well-behaved Γ .Notation 4.7. Given a well-behaved Γ, for each R ∈ Rig Γ let k R : R → Γ ↾ R be given by r R ↾ r . Then k R is an order and incompatibility preserving embeddingof R in the class forcing Γ ↾ R which maps maximal antichains to maximal antichains.Moreover, for every B ≥ Γ C with B ∈ Rig Γ , let i B , C : B → C denote the unique Γ-correct homomorphism from B into C .By the results of the previous sections, Rig Γ is a dense subclass of Γ and is a separativepartial order. Hence, in order to simplify our calculations slightly, we focus on Rig Γ ratherthan on Γ when analyzing this class forcing. Definition 4.8.
Given B ∈ Γ, fix k : B → B Γ-freezing B and such that B ∈ Rig Γ . Let i C = i B , C ◦ k and k = k B ◦ k : B → Γ ↾ B b B ↾ k ( b )Given G a V -generic filter for B , define in V [ G ] the class quotient forcing P B = (( Rig Γ ↾ B ) V / k [ G ] , ≤ Γ / k [ G ] )as follows: C ∈ P B if and only if C ∈ ( Rig Γ ↾ B ) V and, letting J be the dual ideal of G , we have that 1 C i C [ J ](or, equivalently, if and only if coker( i C ) ∈ G ).We let C ≤ Γ / k [ G ] R if C ≤ Γ R holds in V . Theorem 4.9.
Suppose Γ is absolutely well-behaved. Let B ∈ Γ , and let k : B → B bea Γ -freezing homomorphism for B with B ∈ Rig Γ . Set k = k B ◦ k and i C = i B , C ◦ k forall C ≤ Γ B in Γ . V B B Γ ↾ B k k B k Let G be V -generic for B . Then: ncompatible bounded category forcing axioms 31 (1) The class forcing P B = (( Rig Γ ↾ B ) V / k [ G ] , ≤ Γ / k [ G ] ) is in V [ G ] forcing equivalent to the class forcing Q B = (Γ V [ G ] ↾ ( B / k [ G ] ) , ≤ Γ V [ G ] ) via the map i ∗ : P B → Q B C C / i C [ G ] .V [ G ] B / G B / k [ G ] (Γ ↾ B ) / k [ G ] V [ G ] ↾ ( B / k [ G ] ) CC / i C [ G ] k ∼ = k B ∼ = (2) Moreover let δ > | B | be inaccessible and such that ( V δ , ∈ ) ≺ ( V, ∈ ) . Then:(a) Γ V [ G ] δ ↾ ( B / k [ G ] ) is forcing equivalent in V [ G ] to (Γ δ ↾ B ) V / k [ G ] via the samemap.(b) V models that k B : B → Γ δ ↾ B is Γ -correct. Notice that this theorem proves the missing part of Theorem 3.14(1), i.e. the assertion:
For all B ∈ Γ δ there is C ∈ Γ δ such that Γ δ ↾ C ≤ Γ B .In particular this theorem and Theorem 4.2 give a completely self-contained and detailedproof of Theorem 3.14(1) and Theorem 3.14(2). Proof.
Part 2a of the theorem follows immediately from its part 1 relativizing every as-sumption in part 1 to V δ +1 . To prove part 2b, first observe that if B = B , k is necessarilythe identity map, G is V -generic for B , and this gives that ( B / k [ G ] ) is the trivial completeboolean algebra 2 = { , } , i.e.: Γ V [ G ] δ ↾ ( B / k [ G ] ) = Γ V [ G ] δ . Now • δ is inaccessible in V [ G ]; • V δ ≺ V grants that V δ [ G ] ≺ V [ G ], since G ∈ V δ [ G ];hence the set of Γ V [ G ] -rigid forcings is dense in Γ V [ G ] ∩ V δ [ G ], since it is dense in Γ V [ G ] ,being Γ V [ G ] well-behaved in V [ G ]. By Theorem 4.2 applied in V [ G ], (Γ δ ) V [ G ] ∈ Γ V [ G ] .By part 2a (applied in V δ +1 [ G ]) for the case B = B (so that k = k B ), we get that(Γ δ ↾ B ) / k B [ G ] ∼ = (Γ δ ) V [ G ] holds in V [ G ] for all G V -generic for B . This concludes the proofof 2b in case B = B . The desired conclusion 2b for an arbitrary B ∈ Γ δ follows using thefact that the set of B ≤ Γ B in Rig Γ is dense in Γ δ and applying 2b to all such B .We are left with proving part 1: Following the notation introduced in 4.8, we let i R denote the Γ-correct homomorphism i B , R ◦ k for any R ≤ Γ B , and we let k denote themap k B ◦ k : B → Γ ↾ B given by b B ↾ k ( b ). Let G be V -generic for B and J denote its dual prime ideal. We first observe that in V [ G ], ↓ k [ J ] = { R ∈ Γ V : ∃ q ∈ J R ≤ V Γ B ↾ k ( q ) } . We show that in V [ G ] the map i ∗ is total, order and incompatibility preserving, andwith a dense target. This suffices to prove this part of the theorem. i ∗ is total and with a dense target: By Theorem 6.2, any Q ∈ Q B is isomorphicto C / i C [ G ] for some C ∈ (Γ ↾ B ) V such that 1 C / ∈↓ i C [ J ], since Q is a non-trivialcomplete boolean algebra in V [ G ]. Let in V R ∈ Rig Γ refine C in the ≤ ∗ Γ -order.We claim that R / i R [ G ] refines Q in Q B .Assume towards a contradiction that R / i R [ G ] Q B . Then we would get that1 R ∈ i R [ J ]. Therefore for any Γ-correct injective u : C → R witnessing that R ≤ ∗ Γ C , we would have that i C [ J ] = u − [ i R [ J ]]. This gives that 1 C ∈ i C [ J ], andcontradicts our assumption that 1 C / ∈↓ i C [ J ].Therefore 1 R / ∈ i R [ J ], and u/ J : C / i C [ G ] → R / i R [ G ] witnesses that i ∗ ( R ) refines Q in Q B . Hence i ∗ has a dense image.Moreover for any R ∈ P B , 1 R i R [ J ], hence R / i R [ G ] is a non-trivial completeboolean algebra in Γ V [ G ] . Thus i ∗ is also well defined on all of ( Rig Γ ↾ B ) V / k [ G ] . i ∗ is order and compatibility preserving: Let i Q Q : Q → Q be a Γ-correctcomplete homomorphism in V with Q , Q ∈ P B witnessing that Q ≤ Γ / k [ G ] Q .This occurs only if 1 Q / ∈ i Q [ J ]. By Lemma 6.5, i Q Q / J : Q / i Q [ J ] → Q / i Q [ J ] isΓ V [ G ] -correct and witnesses that Q / i Q [ J ] ≥ Γ Q / i Q [ J ] holds in V [ G ]. This showsthat i ∗ is order preserving and maps non-trivial conditions to non-trivial condi-tions. In particular we can also conclude that i ∗ maps compatible conditions tocompatible conditions. i ∗ preserves the incompatibility relation: We prove this by contraposition. As-sume j h : Q h / i Q h [ G ] ∼ = R h → Q for h = 0 , Q / i Q [ G ] and Q / i Q [ G ] are compatible in (Γ) V [ G ] . We can assume that Q ∼ = C / i C [ G ] .By Proposition 6.6 applied for both h = 0 , B , i Q h , j h we have that j h = l h / G for some Γ-correct homomorphism l h : Q h → C h in V such that: • l h ◦ i Q h = i C h for both h = 0 , • C / i C G ] ∼ = Q ∼ = C / i C G ] in V [ G ], • C h i C h [ G ] for both h = 0 , s j / ∈ i C j [ J ] such that C ↾ s and C ↾ s areisomorphic. Without loss of generality we can suppose that C h ↾ s h = C ∈ Γ. Thisgives that (modulo the refinement via s h ) l h ◦ i Q h = i C for both h = 0 ,
1, sinceboth l h ◦ i Q h factor through k which is Γ-freezing B .In particular each l h witnesses in V that Q h ≥ Γ C and are both such that1 C i C [ J ]. Find in V R ≤ ∗ Γ C with R ∈ Rig Γ . Then i R [ J ] = u ◦ i C [ J ] for some (any)Γ-correct injective u : C → R . Hence 1 R i R [ J ], else 1 C ∈ u − [ i R [ J ]] = i C [ J ].This grants that R is a non-trivial condition in ( Rig Γ ↾ B ) V / k [ G ] refining Q h forboth h = 0 , (cid:3) We also obtain the following completeness result as a corollary of Theorems 3.14 and4.9.
Corollary 4.10.
Assume h V, Vi satisfies • MK , • λ Γ is a regular uncountable cardinal, and ncompatible bounded category forcing axioms 33 • Γ is absolutely well-behaved.Suppose B , C ∈ Γ are such that V B | = BCFA (Γ) and V C | = BCFA (Γ) . If G and H aregeneric filters over V for, respectively, B and C , then H V [ G ] λ +Γ and H V [ H ] λ +Γ have the sametheory. Absolutely well-behaved classes
We organize this part of the paper as follows: • We start giving the necessary definitions in 5.1. • We state our main results in 5.2. Specifically we assert that there are uncountablymany absolutely well-behaved definable classes of forcing notions with λ Γ = ω ,whose bounded category forcing axioms yield pairwise incompatible theories for H ω (this is incompatibility in first order logic). • In 5.3 we give the proofs, specifically: – In 5.3.1 we isolate four types of freezing posets which will be used to establishthe freezeability property. – In 5.3.2 we present the iteration lemmas that will be used to establish theiterability property (all of which were already known). – In 5.3.3 we give the proof that there are ℵ -many definable classes Γ of forcingnotions which are absolutely well-behaved with λ Γ = ω . – In 5.3.4 we prove that bounded category forcing axioms for the uncountablymany absolutely well-behaved classes we produced in 5.3.3 yield pairwiseincompatible theories for H ω .5.1. Forcing classes.
We will now define the main classes of forcing notions consideredin this paper. Most of these classes are completely standard, but we nevertheless includetheir definition here for the benefit of some readers.
Definition 5.1.
A poset has the countable chain condition (is c.c.c. , for short) if and onlyif it has no uncountable antichains.Given an ordinal ρ , we will call a sequence ( X i ) i ≤ ρ a continuous chain (or a continuous ρ -chain , if we want to bring in the length) if • ( X k ) k ≤ i ∈ X i +1 whenever i + 1 ≤ ρ and • X i = S k
Given a countable indecomposable ordinal ρ , a poset P is ρ -proper ifand only if there is a cardinal θ such that P ∈ H θ and there is a club D ⊆ [ H θ ] ℵ withthe property that for every continuous chain ( N i ) i ≤ ρ of countable elementary submodelsof H θ containing P and every p ∈ N ∩ P , there is an extension q of p such that q is( N i , P ) -generic for all i ≤ ρ , i.e., for every i ≤ ρ and every dense subset D of P , D ∈ N i , q (cid:13) P D ∩ ˙ G ∩ N i = ∅ . Remark . P is ρ -proper if and only if for every cardinal θ such that P ∈ H θ there issuch a club D ⊆ [ H θ ] ℵ as in the above definition. ρ - PR denotes the class of ρ -proper posets. We write <ω - PR to denote the class of thoseposets that are in ρ - PR for every indecomposable ρ < ω . We say that P is proper if it is1-proper, and denote 1- PR also by PR .The following is a simple but crucial observation: Here, and elsewhere in the remainder of the paper, ω τ denotes ordinal exponentiation. Fact 5.4.
For any countable indecomposable ordinal ρ , ‘ RO ( P ) is ρ -proper ’ is both a Σ property in parameters ρ and ω and a Π property in the same parameters, and the sameholds for ‘ RO ( P ) is <ω -proper ’.Proof. Let θ be large enough such that RO ( P ) ∈ H λ for some λ < θ . Then ‘ RO ( P ) is ρ -proper ’ holds in V if and only if it holds in any (some) transitive set X ⊇ H θ . Hence, RO ( P ) is ρ -proper if and only if there is some regular cardinal θ and some transitive X ⊇ H θ such that ( X, ∈ ) | = ‘ RO ( P ) is ρ -proper’.Now: • The formulae ( X, ∈ ) | = RO ( P ) is ρ -proper and X is transitive are ∆ in theparameters X , P , ρ , ω . • The formula θ is a regular cardinal is Π in parameter θ . • The formula X ⊇ H θ is Π in parameters X, θ since it can be stated as ∀ w ( | trcl( w ) | < θ → w ∈ X ) , where trcl( w ) is the ∆ -definable operation assigining to the set w its transitiveclosure.It is now easy to check that RO ( P ) is ρ -proper is a Σ property in parameters ρ and ω .We leave it to the reader to check that it is also Π in the same parameters. (cid:3) Being ρ -proper, for a forcing P , is equivalent to P preserving a certain combinatorialproperty: Given a set X , we say that S ⊆ ρ ([ X ] ℵ ) is ρ -stationary if for every club D ⊆ [ X ] ℵ there is a continuous ρ -chain σ of members of D such that σ ∈ S .Recalling the standard characterization of properness, the following is not difficult tosee. Fact 5.5.
Given an indecomposable ordinal ρ < ω , the following are equivalent for everyposet P .(1) P is ρ -proper.(2) For every set X , P preserves ρ -stationary subsets of ρ ([ X ] ℵ ) ; i.e., if S ⊆ ρ ([ X ] ℵ ) is ρ -stationary, then (cid:13) P S is ρ -stationary. Using the above fact we can prove:
Fact 5.6.
For every countable indecomposable ordinals ρ , ρ - PR and <ω - PR are closed un-der isomorphisms, two-step iterations, lottery sums, restrictions and complete subalgebras,and contain all countably closed forcings. Definition 5.7.
A forcing notion P is ρ -semiproper iff there is a cardinal θ such that P ∈ H θ for which there is a club D ⊆ [ H θ ] ℵ with the property that for every continuouschain ( N i ) i ≤ ρ of countable elementary submodels of H θ containing P and every p ∈ N ∩P ,there is an extension q of p such that q is ( N i , P ) -semi-generic for all i ≤ ρ . This meansnow that for every i ≤ ρ and every P -name ˙ α ∈ N i for an ordinal in ω V , q (cid:13) P ˙ α ∈ N i . Remark . P is ρ -semiproper if and only if for every cardinal θ such that P ∈ H θ thereis a club D ⊆ [ H θ ] ℵ as in the above definition. ρ - SP denotes the class of ρ -semiproper posets. Also, we write <ω - SP to denote theclass of those posets that are in ρ - SP for every indecomposable ordinal ρ < ω . We saythat P is semiproper if it is 1-semiproper, and denote 1- SP also by SP .As before we have: Fact 5.9. ‘ RO ( P ) is ρ -semiproper ’ is both a Σ property in parameters ρ and ω and a Π property in the same parameters. The same holds for ‘ RO ( P ) is <ω -semiproper ’. ncompatible bounded category forcing axioms 35 Let X be a set such that ω ⊆ X . We say that S ⊆ ρ ([ X ] ℵ ) is ρ -semi-stationary if forevery club D ⊆ [ X ] ℵ there are continuous ρ -chains σ = ( x i : i ≤ ρ ) and σ ′ = ( x ′ i : i ≤ ρ )such that • σ ∈ S , • range( σ ′ ) ⊆ D , and • for each i ≤ ρ , x i ⊆ x ′ i and x i ∩ ω = x ′ i ∩ ω .We have the following characterization of ρ -semiproperness (for any given indecompos-able ρ < ω ). Fact 5.10.
Given an indecomposable ordinal ρ < ω , the following are equivalent for everyposet P .(1) P is ρ -semiproper.(2) P preserves ρ -semi-stationary subsets of ρ ([ X ] ℵ ) for every set X ; i.e., if S ⊆ ρ ([ X ] ℵ ) is ρ -semi-stationary, then S remains ρ -semi-stationary after forcing with P . Again we get that:
Fact 5.11.
For all countable indecomposable ordinals ρ , ρ - SP and <ω - SP are closed underisomorphisms, two-step iterations, lottery sums, restrictions and complete subalgebras, andcontain all countably closed forcings. Definition 5.12.
Given a regular cardinal κ ≥ ω , a poset P preserves stationary subsetsof κ if every stationary subset of κ remains stationary after forcing with P . SSP denotes the class of partial orders preserving stationary subsets of ω . More gen-erally, given a cardinal λ , SSP ( λ ) denotes the class of partial orders preserving stationarysubsets of κ for every uncountable regular cardinal κ ≤ λ .Recall that a Suslin tree is an ω -tree T (i.e., T is a tree of height ω all of whoselevels are countable) without uncountable chains or antichains (a subset of T is calledan antichain iff it consists of pairwise incomparable nodes). We will consider the aboveproperties in conjunction with the preservation of some combination of the two followingproperties. Definition 5.13.
A poset P preserves Suslin trees if (cid:13) P T is Suslin for every Suslin tree T in the ground model. Definition 5.14.
A poset P is ω ω -bounding iff every function f : ω −→ ω added by P is bounded by a function g : ω −→ ω in the ground model; i.e., iff for every P -genericfilter G and every f : ω −→ ω , f ∈ V [ G ], there is some g : ω −→ g , g ∈ V , such that f ( n ) < g ( n ) for all n . STP denotes the class of all posets preserving Suslin trees and ω ω -bounding the classof ω ω -bounding posets.By the same arguments we gave for ρ -properness one gets: Fact 5.15. ‘ RO ( P ) preserves Suslin trees ’ and ‘ P is ω ω -bounding ’ are Σ properties inparameters ω and ω ω , and also Π properties in the same parameters. Moreover, STP and ω ω -bounding are both closed under isomorphisms, two-step iterations, lottery sums,restrictions and complete subalgebras, and contain all countably closed forcings. In [21, XI] Shelah isolates a property he calls S -condition, and for which he proves thefollowing. Lemma 5.16.
Assume P is a forcing notion satisfying the S -condition. Then: (1) P preserves stationary subsets of ω ; (2) if CH holds, then P adds no new reals. As shown in [21, XI–4], among the forcing notions satisfying the S -condition are Nambaforcing (and natural variations thereof), all countably closed forcing notions, and thenatural poset which, for a fixed stationary S ⊆ { α < ω : cf( α ) = ω } , adds an ω -clubthrough S with countable conditions.Given a tree T and a node η of T , let succ T ( η ) denote the set of immediate successorsof η in T . It will be convenient to define the following game P P p (for a partial order P anda P -condition p ). Definition 5.17.
Given a partial order P such that |P| ≥ ℵ , G P is the following gameof length ω between players I and II, with player I playing at even stages and player IIplaying at odd stages.(1) At any given stage n of the game, the corresponding player picks a pair T n ,( p nη ) η ∈ T n , where T n is a tree consisting of finite sequences of ordinals in |P| withoutinfinite branches and where ( p nη ) η ∈ T n is a sequence of conditions in P extending p such that p nν extends p nη in P whenever ν extends η in T n .(2) If n >
0, then(a) T n and ( p nη ) η ∈ T n end-extend T n − and ( p n − η ) η ∈ T n − , respectively,(b) every terminal node in T n − has a proper extension in T n , and(c) every node in T n \ T n − extends a unique terminal node in T n − .(3) Player I starts by playing T = {∅} and p ∅ ∈ P .(4) At any given even stage n > η of T n − , a finite sequence ν η of ordinals in |P| such that ν η extends η properly.He then builds T n as T n − ∪ { ν η ↾ k : k ≤ | ν η | , η a terminal node of T n − } . Player I also has to choose of course ( p nη ) η ∈ T n in such a way that (1) and (2) aresatisfied.(5) At any given odd stage n of the game, player II chooses, for every terminal node η of T n − , a regular cardinal κ nη ∈ [ ℵ , |P| ], and builds T n from T n − by adding to T n − a next level where, for each terminal node η of T n − , the set of immediatesuccessors of η in T n is { η a h α i : α < κ nη } . Player II also has to choose of course( p nη ) η ∈ T n in such a way that (1) and (2) are satisfied.After ω moves, the players have naturally built a tree T = S n T n of height ω whosenodes are finite sequences of ordinals in |P| , together with a sequence ( p η ) η ∈ T = S ( p nη ) η ∈ T n of P -conditions such that for all nodes η , ν in T , if ν extends η in T , then p ν extends p η in P .Finally, player II wins the game iff for every subtree T ′ of T in V , if | succ T ′ ( η ) | = | succ T ( η ) | for every η ∈ T ′ , then there is a condition in P forcing that there is an ω -branch b through T ′ such that p b ↾ n ∈ ˙ G for all n < ω .The definition of the S -condition is the following. Definition 5.18.
A partial order P satisfies the S -condition if and only if |P| ≥ ℵ andplayer II has a winning strategy σ in the game G P such that for every partial run of thegame, the output of σ at any given sequence η ∈ <ω |P| depends only on η , ( p η ↾ k ) k ≤| η | and { k < | η | : | succ T ( η ↾ k ) | > } , where T denotes the tree built by the players up to thatpoint. See [21, XI–Thm. 3.6]. This theorem says that forcing notions with the S -condition do notcollapse ω . However, its proof actually establishes that such forcings in fact preserve stationarysubsets of ω . Shelah’s definition is more general, but the present form suffices for our purposes. ncompatible bounded category forcing axioms 37 S -cond is the class of complete boolean algebras B satisfying the S -condition.One has: Fact 5.19. ‘ RO ( P ) satisfies the S -condition ’ is a Σ property in parameter ω and alsoa Π property in the same parameter. S -cond is closed under isomorphisms, two-step iterations, lottery sums, restrictions andcomplete subalgebras, and contains all countably closed forcings.Proof. As in the case of all other classes dealt with in this section, if RO ( P ) ∈ H θ , then H θ | = ‘ RO ( P ) satisfies the S -condition’ if and only if RO ( P ) satisfies the S -condition.The remaining properties of S -cond other than the closure under complete subalgebrasare left to the reader.We prove now that S -cond is a class closed under complete subalgebras; it will be con-venient for this to resort to the algebraic properties of complete injective homomorphismswith adjoints outlined in Theorem 6.7 of the appendix.Assume B is a complete subalgebra of some C satisfying the S -condition. Let π : C → B be the adjoint map of the inclusion map of B into C . Let σ be the winning strategy forplayer II in G C + . Define σ ′ to be the strategy for player II in G B + obtained by the followingprocedure: • Player I and II build partial plays h T n , { b η : η ∈ T n }i in G B + and partial plays h T n , { c η : η ∈ T n }i in G C + according to these prescriptions: – for all η ∈ T n and all n we have that π ( c η ) = b η ; – for all terminal nodes η ∈ T n with η = h γ , . . . , γ m i , we have that c η = b η ∧ c h γ ,...,γ m − i ; – h T n +1 , { c η : η ∈ T n +1 }i = σ ( h T n , { c η : η ∈ T n }i ); • Player II defines σ ′ ( h T n , { b η : η ∈ T n }i ) = h T n +1 , { b η : η ∈ T n +1 }i .Now assume h T, { b η : η ∈ T }i is built according to a play of G B + in which II follows σ ′ .Fix a subtree T ′ ⊆ T as given by the winning condition for II in G B + . Given a V -genericfilter H for B , we must find some infinite branch η of T ′ in V [ H ] such that b η ↾ n ∈ H for all n .To find this branch let h T, { c η : η ∈ T }i be the tree built in tandem with h T, { b η : η ∈ T }i according to the rules we used to define σ ′ . Fix G ⊇ H V -generic for C . Since C satisfiesthe S -condition, we can find some infinite branch η of T ′ in V [ G ] such that c η ↾ n ∈ G for all n . This gives that b η ↾ n = π ( c η ↾ n ) ∈ H for all n . Hence, in V [ G ] there is an infinite branch η of T ′ such that b η ↾ n ∈ H for all n . Therefore the tree T ∗ = { η ∈ T ′ : b η ∈ H } ∈ V [ H ] isill-founded in V [ G ]. But then the same is true in V [ H ] by absoluteness of ill-foundedness.Finally, any infinite branch of T ∗ witnesses the winning condition for II using σ ′ relativeto T, T ′ , H .Since this can be done for all possible choices of T ⊇ T ′ in V with T constructed using σ ′ , and for all V -generic filters H for B , we have that B satisfies the S -condition. (cid:3) Incompatible bounded category forcing axioms.
In this section we isolate ℵ -many classes Γ of forcing notions with λ Γ = ω , all of which are absolutely well-behaved(in some cases modulo the existence of unboundedly many measurable cardinals), andwhich are pairwise incompatible. Our main results are the following. Theorem 5.20. (1) Each of the following classes Γ is absolutely well-behaved andsuch that λ Γ = ω .(a) PR (b) PR ∩ STP (c) PR ∩ ω ω -bounding (d) PR ∩ STP ∩ ω ω -bounding (e) ρ - PR for every countable indecomposable ordinal ρ such that < ρ < ω . (f ) <ω - PR (g) S -cond (2) Suppose there is a proper class of measurable cardinals. Then each of the followingclasses Γ is absolutely well-behaved and such that λ Γ = ω .(a) ρ - SP for every countable indecomposable ordinal ρ < ω .(b) ( ρ - SP ) ∩ STP for every countable indecomposable ordinal ρ < ω .(c) ( ρ - SP ) ∩ ω ω -bounding for every countable indecomposable ordinal ρ < ω .(d) ( ρ - SP ) ∩ STP ∩ ω ω -bounding for every countable indecomposable ordinal ρ <ω .(e) <ω - SP (f ) ( <ω - SP ) ∩ STP (g) ( <ω - SP ) ∩ ω ω -bounding (h) ( <ω - SP ) ∩ STP ∩ ω ω -bounding Theorem 5.21.
Suppose there is a supercompact cardinal δ such that V δ ≺ V . Suppose Γ and Γ ′ are any two different classes of forcing notions mentioned in Theorem 5.20. Then BCFA (Γ) implies ¬ BCFA (Γ ′ ) .Remark . For some choices of Γ and Γ ′ , the incompatibility of BCFA (Γ) and
BCFA (Γ ′ )can be proved just assuming the existence of an inaccessible cardinal δ such that V δ ≺ V ,or just the existence of an inaccessible δ such that V δ ≺ V together with the existence ofa proper class of Woodin cardinals.As will be clear from the proofs, Theorems 5.20 and 5.21 are just selected samples ofa zoo of possibly incompatible instances of BCFA (Γ). In particular, it should be possibleto combine (some of) the classes mentioned in Theorem 5.20 with other classes of forcingnotions, besides
STP and ω ω -bounding, so long as these classes have a suitable iterationtheory and reasonable closure properties, are both Σ definable and Π definable, and theresulting classes contain SSP -freezing posets.We should point out that the following natural question—in the present context—remains open.
Question 5.23.
Is there, under any reasonable large cardinal, any indecomposable ρ < ω , ρ >
1, for which any of the following classes is absolutely well-behaved?(1) ( ρ - PR ) ∩ STP (2) ( ρ - PR ) ∩ ω ω -bounding (3) ( ρ - PR ) ∩ STP ∩ ω ω -bounding (4) ( <ω - PR ) ∩ STP (5) ( <ω - PR ) ∩ ω ω -bounding (6) ( <ω - PR ) ∩ STP ∩ ω ω -bounding The following question, of a more foundational import, addresses the possibility of therebeing absolutely well-behaved classes Γ such that λ Γ > ω . Question 5.24.
Are there, under some reasonable large cardinal assumption, any cardinal λ ≥ ω and any class Γ of forcing notions such that Γ is absolutely well-behaved and suchthat λ Γ = λ ? Are there, again under some reasonable large cardinal assumption, anycardinal λ ≥ ω and any class Γ of forcing notions with λ Γ = λ and such that Γ isabsolutely well-behaved and such that BCFA (Γ) is compatible with—or, even, implies—
BCFA (Γ ′ ) for any absolutely well-behaved class Γ ′ with λ Γ ′ = ω ?5.3. Proof of theorems 5.20 and 5.21.
Four freezing posets.
In this section we introduce four instances of
SSP -freezingposets. We feel free to confuse posets with complete boolean algebras, as the context ncompatible bounded category forcing axioms 39 will dictate which is the correct intended meaning of the concept. When proving
SSP -freezability, we will actually be showing the following sufficient condition (for λ = ω ). Lemma 5.25.
Let λ ≥ ω be a cardinal, B a forcing notion, and ˙ C a B -name for a forcingnotion. Suppose that p is a set, and that if G is a B -generic filter, then C = ˙ C G forcesthat there is some A G ⊆ λ coding G in an absolute way mod. SSP ( λ ) , in the sense thatthere is some Σ formula ϕ ( x, y, z ) such that, if H is a B ∗ ˙ C -generic filter over V suchthat H ∩ B = G , then(1) ( H λ + ; ∈ , NS λ ) V [ H ] | = ϕ ( G, A G , p ) , and(2) in every outer model M of V [ H ] such that P ( λ ) V [ H ] ∩ ( NS λ ) M = ( NS λ ) V [ H ] , if ( H λ + ; ∈ , NS λ ) M | = ϕ ( G , A G , p ) and ( H λ + ; ∈ , NS λ ) M | = ϕ ( G , A G , p ) , then G = G .Then the natural inclusion i : B −→ B ∗ ˙ CSSP ( λ ) -freezes B .Proof. Suppose, towards a contradiction, that b , b ∈ B are incompatible, D is a completeboolean algebra, k : ( B ∗ ˙ C ) ↾ b −→ D , k : ( B ∗ ˙ C ) ↾ b −→ D are complete homomor-phisms, K is D -generic and, for each ǫ ∈ { , } , H ǫ = k − ǫ ( K ) and every stationary subsetof λ in V [ H ǫ ] remains stationary in V [ K ]. For each ǫ ∈ { , } , let G ǫ be the filter on B generated by H ǫ ∩ ( B ↾ b ǫ ), and let A ǫ ⊆ λ be such that( H λ + ; ∈ , NS λ ) V [ H ǫ ] | = ϕ ( G ǫ , A ǫ , p )Since ( H λ + ; ∈ , NS κ ) V [ K ] | = ϕ ( G , A , p ) ∧ ϕ ( G , A , p ) , we have that G = G by (2). But this is impossible since b ∈ G , b ∈ G , and since b and b are incompatible conditions in B . (cid:3) Our first freezing poset comes essentially from [18].Given a set X , the Ellentuck topology on [ X ] ℵ is the topology on [ X ] ℵ generated by thesets of the form [ s, Y ], for Y ∈ [ X ] ℵ and s ∈ [ Y ] <ω , where [ s, Y ] = { Z ∈ [ Y ] ℵ : s ⊆ Z } .The following lemma, except for the conclusion that P preserves Suslin trees, is due toMoore [18]. The conclusion that P preserves Suslin trees is due to Miyamoto [15]. Lemma 5.26.
Let X be a set, θ a cardinal such that X ∈ H θ , and Σ a function withdomain [ H θ ] ℵ such that for every countable M [ H θ ] ℵ , • Σ( M ) ⊆ [ X ] ℵ is open in the Ellentuck topology, and • Σ( M ) is M -stationary (meaning that for every function F : [ X ] <ω −→ X , if F ∈ M , then there is some Z ∈ Σ( M ) ∩ M such that F “[ Z ] <ω ⊆ Z ).Let P = P X,θ, Σ be the set, ordered by reverse inclusion, of all countable ⊆ -continuous ∈ -chains p = ( M pi ) i ≤ ν of countable elementary substructures of H θ such that for everylimit ordinal i ≤ ν there is some i < i with the property that M pk ∩ X ∈ Σ( M pi ) for all k such that i < k < i .Then(1) P is proper, preserves Suslin trees, and does not add new reals.(2) Whenever G is P -generic over V and M Gi = M pi for p ∈ G and i ∈ dom( p ) , ( M Gi ) i<ω is in V [ G ] the ⊆ -increasing enumeration of a club of [ H Vθ ] ℵ and issuch that: For every limit ordinal i < ω there is some i < i with the property that M Gk ∩ X ∈ Σ( M Gi ) for all k such that i < k < i .Remark . In most interesting cases, the forcing P X,θ, Σ in the above lemma is not ω -proper.In [18], Moore defines the Mapping Reflection Principle ( MRP ) as the following state-ment: Given X , θ , and Σ as in the hypothesis of Lemma 5.26, there is a ⊆ -continuous ∈ -chain ( M i ) i<ω of countable elementary substructures of H θ such that for every limitordinal i < ω there is some i < i with the property that M k ∩ X ∈ Σ( M i ) for every k such that i < k < i .It follows from Lemma 5.26 that MRP is a consequence of
PFA , and of the forcingaxiom for the class of forcing notions in PR ∩ STP not adding new reals.We will call a partial order R an MRP -poset if there are X , θ and Σ as in the hypothesisof Lemma 5.26 such that R = P X,θ, Σ . Proposition 5.28.
Given a forcing notion P , there is P -name ˙ Q for a forcing notionsuch that(1) ˙ Q is forced to be of the form Coll( ω , P ) ∗ ˙ R , where ˙ R is a Coll( ω , P ) -name foran MRP -poset, and(2)
P ∗ ˙ Q SSP -freezes P , as witnessed by the inclusion map.Proof. By Lemma 5.25, it suffices to prove that P forces that in V Coll( ω , P ) there is an MRP -poset ˙ R such that Coll( ω , P ) ∗ ˙ R codes the generic filter for P in an absolute way mod. SSP in the sense of that lemma. For this, let us work in V P∗ Coll( ω , P ) . Let ˙ B G be a subsetof ω coding the generic filter for P in some canonical way, let ~C = ( C δ : δ ∈ Lim( ω )) ∈ V be a ladder system on ω (i.e., every C δ is a cofinal subset of δ of order type ω ), and let( S α ) α<ω ∈ V be a partition of ω into stationary sets. Given X ⊆ Y , countable sets ofordinals, such that Y ∩ ω and ot( Y ) are both limit ordinals and such that X is boundedin sup( Y ), let c ( X, Y ) mean | C ot( Y ) ∩ sup( π Y “ X ) | < | C Y ∩ ω ∩ X ∩ ω | , where π Y is the collapsing function of Y .Let θ be a large enough cardinal and let Σ be the function sending a countable N H θ to the set of Z ∈ [ ω ∩ N ] ℵ such that c ( X, ω ∩ N ) iff the unique α < ω such that N ∩ ω ∈ S α is in B ˙ G . Now, X = ω , θ and Σ satisfy the hypothesis of Lemma 5.26(s. [18]). Let R = P ω ,θ, Σ . By Lemma 5.26, R adds ( Z ˙ Gi ) i<ω , a strictly ⊆ -increasingenumeration of a club of [ ω V ] ℵ , such that for every limit ordinal i < ω , if Z ˙ Gi ∩ ω ∈ S α ,then there is a tail of k < i such that c ( Z ˙ Gk , Z ˙ Gi ) if and only if α ∈ B ˙ G .Let κ = ω , let H be ( P ∗
Coll( ω , P )) ∗ ˙ R -generic, let G = H ∩ P , and let A G be asubset of ω which canonically codes B G and ( Z Gi ) i<ω . If M is any outer model such thatevery stationary subset of ω in V [ H ] remains stationary in M , then B G is the uniquesubset B of ω for which there is, in M , a set A ⊆ ω coding B together with an ⊆ -increasing enumeration ( Z i ) i<ω of a club of [ κ ] ℵ with the property that for every limitordinal i < ω , if Z i ∩ ω ∈ S α , then there is a tail of k < i such that c ( Z k , Z i ) if and onlyif α ∈ B . Indeed, If B ′ ∈ M were another such set, as witnessed by A ′ ⊆ M , α ∈ B ∆ B ′ ,and ( Z ′ i ) i<ω ∈ M were an ⊆ -increasing enumeration of a club of [ κ ] ℵ with the propertythat for every limit ordinal i < ω , if Z ′ i ∩ ω ∈ S α , then there is a tail of k < i such that c ( Z k , Z i ) if and only if α ∈ B ′ , then we would be able to find some i such that Z i = Z ′ i , Z i ∩ ω ∈ S α , and such that Z k = Z ′ k for all k in some cofinal subset J of i . But then wewould have that c ( Z k , Z i ), for all k in some final segment of J , both holds and fails. Σ is in essence the mapping used by Moore in [18] to prove that
BPFA implies 2 ℵ = ℵ . ncompatible bounded category forcing axioms 41 Finally, it is immediate to see that there is a Σ formula ϕ ( x, y, z ) such that ϕ ( G, A G , p )expresses the above property of G and A G over ( H ω ; ∈ ) M for any M as above, for p =( κ, ~C, ( S α ) α<ω ). (cid:3) Using coding techniques from [7], one can prove the following stronger version of Lemma5.28. However, we do not have any use for this stronger form, so we will not give the proofhere.
Lemma 5.29.
Given a partial order P there is P -name ˙ Q for a partial order with thefollowing properties.(1) ˙ Q is forced to be of the form Coll( ω , P ) ∗ ˙ R where ˙ R is a Coll( ω , P ) -name for aforcing of the form ˙ R ∗ ˙ R , where ˙ R has the countable chain condition and ˙ R is forced to be an MRP -poset.(2) Suppose b , b ∈ RO ( P ) are incompatible, B is a complete boolean algebra, and k ǫ : RO ( P ∗ ˙ Q ) ↾ b ǫ −→ B are complete homomorphisms for ǫ ∈ { , } . Then B collapses ω . Our second freezing poset comes from [23, Section 1], where the following is proved,using a result of Todorˇcevi´c from [22].
Lemma 5.30.
There is a sequence (( K ξ , K ξ ) : ξ < ω ) of colourings of [ κ ] , for κ =cf(2 ℵ ) , with the property that in any ω -preserving outer model in which | κ | = ℵ , if B ⊆ ω , then there is a c.c.c. partial order R forcing the existence of ℵ -many decompositions κ = S n<ω X ξn , for ξ < ω , such that for all ξ < ω : • for some fixed i ξ = 0 , , X ξn is K ξi ξ -homogeneous for all n < ω ; • ξ ∈ B if and only if i ξ = 0 . Proposition 5.31.
Given a forcing notion P , there is P -name ˙ Q for a forcing notionwith the following properties.(1) Letting µ = |P| + cf(2 ℵ ) , ˙ Q is forced to be a forcing of the form Coll( ω , µ ) ∗ ˙ R ,where ˙ R is a Coll( ω , µ ) -name for a c.c.c. forcing.(2) P ∗ ˙ Q SSP -freezes P , as witnessed by the inclusion map. In fact, if b , b ∈ RO ( P ) are incompatible, B is a complete boolean algebra, and k ǫ : RO ( P ∗ ˙ Q ) ↾ b ǫ −→ B is a complete homomorphism for ǫ ∈ { , } , then B collapses ω .Proof. Let us work in V P∗ Coll( ω , µ ) . Let ˙ B G be a subset of ω coding the generic filter for P in some canonical way, let ~K = (( K ξ , K ξ ) : ξ < ω ) be a sequence of colourings of [ κ ] ,for κ = cf V (2 ℵ ), as given by Lemma 5.30, and let R be a c.c.c. partial order forcing theexistence of ℵ -many decompositions κ = S n<ω X ξn such that for all ξ < ω , • there is i ξ such that [ X ξn ] ⊆ K ξi ξ for all n < ω and • ξ ∈ B if and only if for all n < ω , [ X ξn ] ⊆ K ξ .Let A G be a subset of ω which canonically codes B G and n ( X ξn ) n<ω : ξ < ω o . If M is any outer model in which ω V has not been collapsed, then B G is the unique B ⊆ ω for which there is, in M , a set A ⊆ ω coding B together with ℵ -many decompositions n ( X ξn ) n<ω : ξ < ω o of κ such that for all n < ω and ξ < ω , ξ ∈ B if and only if[ X ξn ] ⊆ K ξ . Indeed, if B ′ ∈ M were another such set, as witnessed by A ′ ⊆ M , ξ ∈ B ∆ B ′ ,and ℵ -many decompositions n ( Y ξn ) n<ω : ξ < ω o ∈ M of κ , then there would be some n and m such that X ξn ∩ Y ξm has more than one element, and is in fact uncountable. Butthen, for every s ∈ [ X ξn ∩ Y ξm ] , we would have that s is both in K ξ and K ξ , which isimpossible. Finally, it is immediate to see that there is a Σ formula ϕ ( x, y, z ) such that ϕ ( G, A G , p )expresses the above property of G and A G over ( H ω ; ∈ ) M for any M as above, for p = ~K . (cid:3) The following principle, as well as Lemma 5.33, are due to Woodin ([30]).
Definition 5.32. ψ AC is the following statement: Suppose S and T are stationary andco-stationary subsets of ω . Then there are α < ω and a club C of [ α ] ℵ such that forevery X ∈ C , X ∩ ω ∈ S if and only if ot( X ) ∈ T .The AC -subscript in the above definition hints at the fact that ψ AC implies L ( P ( ω )) | = AC (which comes from an argument similar to the one in the proof of Lemma 5.34).Our third freezing poset is essentially the following forcing for adding a suitable instanceof ψ AC by initial segments, using a measurable cardinal κ (i.e., turning κ into an ordinal α as required by the conclusion of ψ AC ). Lemma 5.33.
Let κ be a measurable cardinal and let S and T be stationary and co-stationary subsets of ω . Let Q = Q κ,S,T be the set, ordered by reverse inclusion, of allcountable ⊆ -continuous ∈ -chains p = ( M pi ) i ≤ ν of countable elementary substructures of H κ such that for every i ≤ ν , M pi ∩ ω ∈ S if and only if ot( M pi ∩ κ ) ∈ T .(1) Q is <ω -semiproper, preserves Suslin trees, and does not add new reals.(2) if G is Q -generic over V and M Gi = M pi whenever p ∈ G and i ∈ dom( p ) , then ( M Gi ) i<ω is the ⊆ -increasing enumeration of a club of [ H Vκ ] ℵ such that for everylimit ordinal i < ω , M Gi ∩ ω ∈ S if and only if ot( M Gi ∩ κ ) ∈ T .Proof. The proofs of all assertions, except the fact that Q preserves Suslin trees, arestandard. For the reader’s convenience, we sketch the proof that Q is <ω -semiproper,though. We also prove that Q preserves Suslin trees.We get the <ω -semiproperness of Q as follows: the main point is that if U is a normalmeasure on κ , N is an elementary submodel of some H θ such that U ∈ N and | N | < κ ,and η ∈ T ( U ∩ N ), then N [ η ] := { f ( η ) : f ∈ N, f a function with domain κ } is an elementary submodel of H θ such that N ∩ κ is a proper initial segment of N [ η ] ∩ κ ( η ∈ N [ η ] is above every ordinal in N ∩ κ , and any γ ∈ N [ η ] ∩ η is of the form f ( η ), for someregressive function f : κ −→ κ in N which, by normality of U , is constant on some set in U ∩ N ). If N is countable, then by iterated applications of this construction, taking unionsat nonzero limit stages, one obtains a ⊆ -continuous and ⊆ -increasing sequence ( N ν ) ν<ω of elementary submodels of H θ such that N = N and such that N ν ′ ∩ κ is a properend-extension of N ν ∩ κ for all ν < ν ′ < ω . Since (ot( N ν ∩ κ ) : ν < ω ) is then a strictlyincreasing and continuous sequence of countable ordinals, we may find, by stationarity of S and ω \ S , some ν < ω such that N ∩ ω ∈ S if and only if ot( N ν ) ∈ T .This observation yields the <ω -semiproperness of Q since, given α < ω and an ∈ -chain( N ξ ) ξ<α of countable elementary submodels of some H θ such that U , S , T ∈ N , one canrun the above construction for each N ξ by working inside N ξ +1 .The preservation of Suslin trees can be proved by the following version of the argumentin [15] for showing that MRP -forcings preserve them. Suppose U is a Suslin tree, ˙ A isa Q -name for a maximal antichain of U , and N is a countable elementary submodel ofsome large enough H θ containing U , ˙ A , and all other relevant objects. By moving to an ω -end-extension of N if necessary as in the proof of <ω -semiproperness, we may assumethat N ∩ ω ∈ S if and only if ot( N ∩ κ ) ∈ T . Let ( u n ) n<ω enumerate all nodes in U ofheight N ∩ ω . Given a condition p ∈ Q in N , we may build an ( N, Q )-generic sequence( p n ) n<ω of conditions in N extending p and such that for every n there is some v ∈ U below u n such that p n +1 forces v ∈ ˙ A . By the choice of N , we have in the end that ncompatible bounded category forcing axioms 43 p ∗ = S n p n ∪ { ( N ∩ ω , N ∩ κ ) } is a condition in Q extending p . But, by construction of( p n ) n<ω , p ∗ forces ˙ A to be contained in the countable set U ∩ N : If u ∈ ˙ A \ N , and u n isthe unique node of height N ∩ ω such that u n ≤ U u , then u n is forced to extend somenode in ˙ A of height less than N ∩ ω , which is a contradiction since ˙ A was supposed to bea name for an antichain.It remains to show how to find p n +1 given p n . Working in N , we first extend p n to some p ′ n in some suitable dense subset D ∈ N of Q . Since U is a Suslin tree, we have that u n is totally ( U, N )-generic, in the sense that for every antichain B of U in N , u n extends aunique node in B . Also, the set E ∈ N of u ∈ U for which there is some v ∈ U below u and some q ∈ Q extending p ′ n and forcing that v ∈ ˙ A is dense in U . It follows that wemay find some u ∈ E ∩ N below u n , as witnessed by some q ∈ Q ∩ N and some v ∈ U ∩ N .But then we may let p n +1 = q . (cid:3) Given a measurable cardinal κ , we will call a partial order R a ψ κ AC -poset if R is of theform Q κ,S,T for stationary and co-stationary subsets S , T of ω . Proposition 5.34.
Given a forcing notion P and a measurable cardinal κ , there is P -name ˙ Q for a forcing notion such that(1) ˙ Q is forced to be of the form Coll( ω , P ) ∗ ˙ R , where ˙ R is a Coll( ω , P ) -name fora ψ κAC -poset, and(2) P ∗ ˙ Q SSP -freezes P , as witnessed by the inclusion map.Proof. Let ( S α : α < ω ) be a partition of ω into stationary sets and let T be a stationaryand co-stationary subset of ω . Working in V P∗ Coll( ω , P ) , let B ˙ G = ∅ be a subset of ω coding ˙ G in a canonical way. We may assume that B ˙ G = ω . Let ˙ R be Q κ,S,T for S = S α ∈ B ˙ G S α . By Lemma 5.33, ˙ R adds a club C ˙ G of [ κ ] ℵ with the property that foreach X ∈ C ˙ G , X ∩ ω ∈ S α ∈ B ˙ G S α if and only if ot( X ) ∈ T . Now it is easy to see that P ∗ ˙ R codes ˙ G in an absolute way in the sense of Lemma 5.25. The main point is thatif H is a ( P ∗
Coll( ω , P )) ∗ ˙ R -generic filter, G = H ∩ P , and M is an outer model suchthat every stationary subset of ω in V [ H ] remains stationary in M , then in M there isno B ′ ⊆ ω such that B ′ = B G and such that there is a club C of [ κ ] ℵ with the propertythat for all X ∈ C , X ∩ ω ∈ S α ∈ B ′ S α if and only if ot( X ) ∈ T . Otherwise, if α ∈ B ′ ∆ B G ,then there would be some X ∈ C ∩ C G such that X ∩ ω ∈ S α . But then we would havethat ot( X ) is both in T and in ω \ T . (cid:3) Let us move on now to our fourth freezing poset.Given cardinals µ < λ with µ regular, let S λµ = { ξ < λ : cf( ξ ) = µ } Let ~S = ( S α ) α<ω be a sequence of pairwise disjoint stationary subsets of S ω ω , let U ⊆ S ω ω be such that both U and S ω ω \ U are stationary, and let B ⊆ ω . Then S ~S,U,B is the partialorder, ordered by end-extension, consisting of all strictly ⊆ -increasing and ⊆ -continuoussequences ( Z ν ) ν ≤ ν , for some ν < ω , such that for all ν ≤ ν and all α < ω , • Z ν ∈ [ ω ] ℵ , and • if sup( Z ν ∩ ω ) ∈ S α , then sup( Z ν ) ∈ U if and only if α ∈ B .The proof of the following lemma appears in [9] essentially. Lemma 5.35.
Let ~S = ( S α ) α<ω be a sequence of pairwise disjoint stationary subsets of S ω ω , let U ⊆ S ω ω be such that both U and S ω ω \ U are stationary, and let B ⊆ ω . Then S ~S,U,B preserves stationary subsets of ω , as well as the stationarity of all S α , and forcesthe existence of strictly ⊆ -increasing and ⊆ -continuous enumeration ( Z ν : ν < ω ) of a See also the argument in the proof of Proposition 5.36 that ˙ Q satisfies the S -condition. club of [ ω V ] ℵ such that for all ν , α < ω , if sup ( Z ν ∩ ω V ) ∈ S α , then sup ( Z ν ) ∈ U if andonly if α ∈ B . Proposition 5.36.
For every partial order P and for all cardinals κ > κ ≥ δ ≥ |P| ,if (cid:13) P∗ Coll( ω , δ ) κ = ω , (cid:13) P∗ Coll( ω , δ ) κ = ω , ( S α ) α<ω ∈ V is a sequence of pairwisedisjoint stationary subsets of S κ ω , and U ⊆ S κ ω is a a stationary set in V such that S κ ω \ U is also stationary in V , then there is a P ∗
Coll( ω , δ ) -name ˙ B for a subset of ω such that(1) P forces Coll( ω , δ ) ∗ ˙ S ~S,U, ˙ B to have the S -condition, and such that(2) P ∗ (Coll( ω , δ ) ∗ ˙ S ~S,U, ˙ B ) SSP -freezes P , as witnessed by the inclusion map i : P −→ P ∗ (Coll( ω , δ ) ∗ ˙ S ~S,U, ˙ B ) Proof.
Working in V P∗ Coll( ω , δ ) , let ( S α ) α<ω ∈ V and U ∈ V be as stated, and let B ˙ G bea subset of ω coding the generic filter G for P in a canonical way. Let ˙ R be a P -namefor Coll( ω , δ ) ∗ ˙ S ~S,U,B ˙ G . Claim 5. P forces that ˙ R has the S -condition.Proof. Since Coll( ω , δ ) has the S -condition, it suffices to prove that P ∗
Coll( ω , δ ) forces˙ S ~S,U, ˙ B to have the S -condition. Let us work in V P∗ Coll( ω , δ ) . Let σ be the followingstrategy for player II in G ˙ R : Whenever it is her turn to play, player II will alternatebetween the following courses of action (a), (b) (i.e., she will opt for (a) or (b) dependingon the parity of the finite set { k < | η | : | succ T ( η ↾ k ) | > } , with the notation used inDefinition 5.18).(a) Player II chooses κ η = κ , succ T ( η ) = { η a h α i : α < κ } , and ( p η a h α i ) α<κ where, for each α < κ , p η a h α i is a condition extending p η and such that α ∈ S range( p η a h α i ).(b) Player II chooses κ η = κ , succ T ( η ) = { η a h α i : α < κ } , and ( p η a h α i ) α<κ where, for each α < κ , p η a h α i is a condition extending p η and such that α ∈ S range( p η a h α i ).Let now T be the tree built along a run of G ˙ R in which player II has played accordingto σ , let T ′ be a subtree of T such that | succ T ′ ( η ) | = | succ T ( η ) | for every η ∈ T ′ , and let N be a countable elementary substructure of some large enough H θ containing all relevantobjects (which includes our run of G ˙ R and T ′ ), such that sup( N ∩ κ ) ∈ S , and such thatsup( N ∩ κ ) ∈ U if 0 ∈ B ˙ G and sup( N ∩ κ ) / ∈ U if 0 / ∈ B ˙ G .Such an N can be easily found (s. [9]): Indeed, suppose, for concreteness, that 0 ∈ B ˙ G .Then, letting F : [ H θ ] <ω −→ H θ be a function generating the club of countable elementarysubmodels of H θ containing all relevant objects, we may find, using the stationarity of U ,an ordinal α ∈ U such that the closure X of [ α ] <ω under F is such that X ∩ κ = α .We may of course assume that α > κ . Since cf( α ) = ω , we may pick a countable cofinalsubset Y of α . Using now the stationarity of S , we may find β ∈ S with the propertythat the closure X of [ β ∪ Y ] <ω under F is such that X ∩ κ = β . Since cf( β ) = ω , wemay now pick a countable subset Z of β . But then, letting N be the closure of Y ∪ Z under F , we have that sup( N ∩ κ ) = β and sup( N ∩ κ ) = α , and so N is as desired.Letting ( p η ) η ∈ T ′ be the tree of ˙ R -conditions corresponding to T ′ , it is now easy to finda cofinal branch b through T ′ such that for all n < ω , b ↾ n ∈ N , and such thatsup( [ n<ω ( ∪ range( p b ↾ n ) ∩ κ )) = sup( N ∩ κ )and sup( [ n<ω ( ∪ range( p b ↾ n ))) = sup( N ∩ κ ) . ncompatible bounded category forcing axioms 45 Let ν = sup { dom( p b ↾ n ) : n < ω } and X = [ n<ω ( ∪ range( p b ↾ n ) ∩ κ ) . It follows now that, letting p b = S n p b ↾ n ∪ {h ν, X i} , p b is a condition in ˙ R forcing that p b ↾ n is in the generic filter for all n . (cid:3) Going back to V , the proof that P ∗ (Coll( ω , δ ) ∗ ˙ S ~S,U, ˙ B ) SSP -freezes P (as witnessed bythe inclusion map) is very much like the proofs of Propositions 5.28 and 5.34. Suppose H is a generic filter for P ∗ (Coll( ω , δ ) ∗ ˙ S ~S,U, ˙ B ), G = H ∩ P , and M is any outer model suchthat every stationary subset of ω in V [ H ] remains stationary in M . Suppose, towardsa contradiction, that in M there is some subset B ′ = B G of ω for which there is an ⊆ -increasing and ⊆ -continuous enumeration ( Z ′ ν : ν < ω ) of a club of [ κ ] ℵ such thatfor all ν , α < ω , if sup( Z ′ ν ∩ κ ) ∈ S α , then sup( Z ′ ν ) ∈ U if and only if α ∈ B ′ . If α ∈ B ′ ∆ B G , there is some ν such that Z ν = Z ′ ν and sup( Z ν ∩ κ ) ∈ S α . But then we haveboth sup( Z ν ) ∈ U and sup( Z ν ) / ∈ U .Finally, the existence of a Σ definition—with p = ( λ, ~S, U ) as parameter—as requiredby Lemma 5.25 is easy. (cid:3) Iteration Lemmas.
We need the following preservation lemmas, due to Shelah ([21,III, resp. VI], see also [10]).
Lemma 5.37.
Suppose ρ < ω is an indecomposable ordinal and ( P α , ˙ Q β : α ≤ γ, β < γ ) is a countable support iteration such that for all β < γ , (cid:13) P β ˙ Q β ∈ ρ - PR Then P γ ∈ ρ - PR . Lemma 5.38.
Suppose ρ < ω is an indecomposable ordinal and ( P α , ˙ Q β : α ≤ γ, β < γ ) is a countable support iteration such that for all β < γ , (cid:13) P β ˙ Q β ∈ ( ρ - PR ) ∩ ω ω -bounding Then P γ ∈ ( ρ - PR ) ∩ ω ω -bounding . We will also use the following preservation result due to Miyamoto.
Lemma 5.39.
Suppose ρ < ω is an indecomposable ordinal and ( P α , ˙ Q β : α ≤ γ, β < γ ) is a countable support iteration such that for all β < γ , (cid:13) P β ˙ Q β ∈ ( ρ - PR ) ∩ STP
Then P λ ∈ ( ρ - PR ) ∩ STP . Incidentally, note that we cannot guarantee that ν = N ∩ ω . Shelah defines a certain variant of the notion of countable support iteration, which hecalls revised countable support (RCS) iteration . Variants of the notion of RCS iterationhave been proposed by Miyamoto and others (for example a detailed account of RCS-iterations in line with Donder and Fuchs’ approach is given in [4]). In the following, anymention of revised countable support iteration will refer to either Shelah’s or Miyamoto’sversion.The first preservation result involving RCS iterations we will need is the following lemma,proved in [21, XI].
Lemma 5.40.
Suppose hP α , ˙ Q β : α ≤ γ, β < γ i is an RCS iteration such that thefollowing holds for all β < γ .(1) If β is even, (cid:13) P β ˙ Q β = Coll (2 |P β | , ω ) .(2) If β is odd, (cid:13) P β ˙ Q β has the S -condition.Then P γ has the S -condition. The following is a well-known result of Shelah.
Lemma 5.41.
Suppose ρ < ω is an indecomposable ordinal and ( P α , ˙ Q β : α ≤ γ, β < γ ) is a revised countable support iteration such that for all β < γ , (cid:13) P β ˙ Q β ∈ ρ - SP Then P γ ∈ ρ - SP . We will also need the following lemmas due to Miyamoto [16, 17].
Lemma 5.42.
Suppose CH holds, ρ < ω is an indecomposable ordinal, and ( P α , ˙ Q β : α ≤ γ, β < γ ) is a revised countable support iteration such that for all β < γ , (cid:13) P β ˙ Q β ∈ ( ρ - SP ) ∩ ω ω -bounding Then P γ ∈ ( ρ - SP ) ∩ ω ω -bounding . Lemma 5.43.
Suppose ρ < ω is an indecomposable ordinal and ( P α , ˙ Q β : α ≤ γ, β < γ ) is a revised countable support iteration such that for all β < γ , (cid:13) P β ˙ Q β ∈ ( ρ - SP ) ∩ STP
Then P γ ∈ ( ρ - SP ) ∩ STP . Absolutely well-behaved classes.
Lemma 5.44.
The following classes Γ are absolutely well-behaved and such that λ Γ = ω .(1) ρ - PR for every countable indecomposable ordinal ρ .(2) <ω - PR (3) PR ∩ STP (4) PR ∩ ω ω -bounding (5) PR ∩ STP ∩ ω ω -bounding It is not clear whether these notions are equivalent in any reasonable sense. ncompatible bounded category forcing axioms 47
Proof.
Given any of these classes Γ, all conditions in the definition of absolutely well-behaved class—except for the fact that Γ has the Γ-freezability property—are clearlysatisfied for Γ. In particular, Γ is defined by both a Σ property and a Π property byFact 5.4, it is closed under isomorphisms, two-step iterations, lottery sums, restrictionsand complete subalgebras, and it contains all countably closed forcings by Fact 5.6. Theiterability property follows from Lemmas 5.37, 5.38 and 5.39: the winning strategy forplayer II is to play the countable support limit at all limit stages (notice that at stages ofcofinality ω this limit is the direct limit). As to the freezability property, it turns out thatΓ has in fact the SSP -freezability property. This follows immediately from Proposition5.28 together with Lemma 5.26 for PR , as well as for the classes in (3), (4) and (5), andfrom Proposition 5.31 for ρ - PR , for any given indecomposable ordinal ρ < ω such that ρ >
1, and for <ω - PR . Finally, it is clear that λ Γ = ω holds for each of these classesΓ. (cid:3) We move on now to our first class not contained in PR . Lemma 5.45. S -cond is an absolutely well-behaved class Γ with λ Γ = ω .Proof. Let Γ = S -cond . Except for the freezability condition, all conditions in the defini-tion of absolutely well-behaved class are clearly satisfied by Γ: Γ is defined by both a Σ property and a Π property, it is closed under isomorphisms, two-step iterations, lotterysums, restrictions and complete subalgebras, and it contains all countably closed forcingsby Fact 5.19. The iterability condition follows immediately from Lemma 5.40: the win-ning strategy for player II is to play the revised countable support limit at all limit stages(notice that at stages of cofinality ω this limit is the direct limit), and to play at all non-limit stages α + 2 n the algebra B α +2 n = B α +2 n − ∗ ˙ C , where ˙ C is a B α +2 n − -name for theboolean completion of Coll( ω , | B α +2 n − | ). As to the freezability condition, we have that S -cond has in fact, by Proposition 5.36, the SSP -freezability condition—which implies theΓ-freezability condition by Lemma 5.16 (1). Finally, it is clear that λ Γ = ω . (cid:3) Lemma 5.46.
Suppose there is a proper class of measurable cardinals. Given any inde-composable ordinal ρ < ω , each of the following classes Γ is absolutely well-behaved andsuch that λ Γ = ω .(1) ρ - SP (2) ( ρ - SP ) ∩ STP (3) ( ρ - SP ) ∩ ω ω -bounding (4) ( ρ - SP ) ∩ STP ∩ ω ω -bounding Also, each of the following classes is ω -suitable with respect to the same theory.(1) <ω - SP (2) ( <ω - SP ) ∩ STP (3) ( <ω - SP ) ∩ ω ω -bounding (4) ( <ω - SP ) ∩ STP ∩ ω ω -bounding Proof.
Each of these Γ is defined both by a Σ property and by a Π property by Fact 5.9,and is closed under preimages by complete injective homomorphisms, two-step iterationsand products, and contains all countably closed forcings by Fact 5.11. The iterabilitycondition for each of these classes follows from (some combination of) Lemmas 5.41, 5.42,and 5.43: the winning strategey for player II is to play the revised countable support limitat all limit stages (notice that at stages of cofinality ω this limit is the direct limit). Thefreezability condition follows from Lemma 5.33, together with Proposition 5.34 (for thecase ρ = 1, one could as well invoke Lemma 5.26 together with Proposition 5.28 instead).Finally, the fact that λ Γ = ω is again immediate. (cid:3) By Remark 3.12 this is enough to guarantee clause (5) in the definition of well-behaved class.And of course the same applies in the proofs of Lemmas 5.45 and 5.46.
The standard proof, due to Shelah ([20]), that
SPFA implies
SSP = SP actually showsthe following. Proposition 5.47. FA (( <ω - SP ) ∩ STP ∩ ω ω -bounding ) implies SSP = SP .Proof. This follows from the fact that the natural semiproper forcing Q P (s. [20]) suchthat an application of FA ℵ ( {Q P } ) yields the semiproperness of a given SSP P is in fact <ω -semiproper, does not add new reals, and preserves Suslin trees. The proof of thefirst two assertions is straightforward, and the preservation of Suslin trees follows by anargument as in the final part of the proof of Lemma 5.33. (cid:3) Corollary 5.48.
Suppose FA (( <ω - SP ) ∩ STP ∩ ω ω -bounding ) holds. Then:(1) BCFA ( SSP ) holds iff BCFA ( SP ) does.(2) BCFA ( SSP ∩ STP ) holds iff BCFA ( SP ∩ STP ) does.(3) BCFA ( SSP ∩ ω ω -bounding ) holds iff BCFA ( SP ∩ ω ω -bounding ) does.(4) BCFA ( SSP ∩ STP ∩ ω ω -bounding ) holds iff BCFA ( SP ∩ STP ∩ ω ω -bounding ) does. Pairwise incompatibility of
BCFA (Γ) for absolutely well-behaved Γ . Each one of theincompatibilities contained in Theorem 5.21 follows from two or more of the lemmas inthis subsection put together.Given an ordinal α < ω , a function g : ω −→ ω is a canonical function for α if thereis a surjection π : ω −→ α and a club C ⊆ ω such that for all ν ∈ C , g ( ν ) = ot( π “ ν ). Let Club Bounding denote the following statement: For every function f : ω −→ ω there issome α < ω such that { ν < ω : f ( ν ) < g ( ν ) } contains a club whenever g is a canonicalfunction for α . Lemma 5.49.
Suppose δ is an inaccessible cardinal such that V δ ≺ V . Let Γ be anyabsolutely well-behaved class, defined with a parameter in V δ , and such that λ Γ = ω .Suppose Γ ⊆ PR holds after forcing with Add( ω , . If BCFA (Γ) holds, then Club Boundingfails.Proof.
To see that Club Bounding fails in V , we first force with Add( ω , G bethe corresponding generic filter. We then force with Γ V [ G ] δ . By Theorem 3.14, it holds in V [ G ] that Γ V [ G ] δ forces BCFA (Γ). Also, it is immediate to check that G adds a function f : ω −→ ω such that { X ∈ [ α ] ℵ : X ∩ ω ∈ ω , ot( X ) < f ( X ∩ ω ) } is a stationary subset of [ α ] ℵ for every ordinal α . Since every proper forcing will preservethe stationarity of these sets and since Γ V [ G ] δ is proper in V [ G ] by our assumption, it followsthat Club Bounding fails in ( V [ G ]) Γ δ . Since the failure of Club Bounding is expressibleover H ω , and since we have that H V Γ δ ω = H V Add( ω , ∗ ˙Γ δ ω ≺ H V Γ ω and H Vω ≺ H V Γ ω , it followsthat Club Bounding fails in V . (cid:3) Lemma 5.50.
BFA ℵ (( <ω - PR ) ∩ ω ω -bounding ) implies that there are no Suslin trees.Proof. Suppose, towards a contradiction, that T is a Suslin tree and the forcing axiom BFA ℵ (( <ω - PR ) ∩ ω ω -bounding ) holds. Without loss of generality we may assume that T is a normal Suslin tree. We have that T is a c.c.c. forcing which is ω ω -bounding as in factit does not add new reals. But forcing with T adds an ω -branch through T . Hence, by BFA ℵ ( T ), T has an ω -branch and so it is not Suslin, which is a contradiction. (cid:3) Recall that d is the minimal cardinality of a family F ⊆ ω ω with the property that forevery f : ω −→ ω there is some f ∈ F such that g ( n ) < f ( n ) for a tail of n < ω . Lemma 5.51.
Suppose δ is an inaccessible cardinal such that V δ ≺ V . Let Γ be anyabsolutely well-behaved class, defined with a parameter in V δ , and such that λ Γ = ω .Suppose Γ ⊆ ω ω -bounding. If BCFA (Γ) holds, then d = ω . ncompatible bounded category forcing axioms 49 Proof.
We know that Γ δ forces BCFA (Γ), and collapses (2 ℵ ) V to ℵ . Since Γ ⊆ ω ω -bounding,we thus have that V Γ δ | = d = ω . But ‘ d = ω ’ is expressible over H ω , and therefore V | = d = ω by the same absoluteness argument as in the proof of Lemma 5.49. (cid:3) Lemma 5.52.
BFA ℵ (( <ω - PR ) ∩ STP ) implies d > ω .Proof. This is immediate since Cohen forcing, being countable, preserves Suslin trees. (cid:3)
Lemma 5.53.
Suppose
BFA ℵ (( <ω - SP ) ∩ STP ∩ ω ω -bounding ) holds and there is a mea-surable cardinal. Then ψ AC holds, and therefore Club Bounding holds as well.Proof. BFA ℵ (( <ω - SP ) ∩ STP ∩ ω ω -bounding) implies ψ AC by Lemma 5.33. To see that ψ AC implies Club Bounding, see for example Fact 3.1 in [3]. (cid:3) The following lemma is an immediate consequence of Lemma 5.53.
Lemma 5.54.
Suppose δ is an inaccessible cardinal such that V δ ≺ V . Suppose there is aproper class of measurable cardinals. Let Γ be an absolutely well-behaved class with λ Γ = ω defined from a parameter in V δ and such that ( <ω - SP ) ∩ STP ∩ ω ω -bounding ⊆ Γ holds inany generic extension by a member of Γ . If BCFA (Γ) holds, then ψ AC , and therefore alsoClub Bounding, hold as well.Proof. Γ δ forces BFA ℵ (Γ) and therefore also BFA ℵ (( <ω - SP ) ∩ STP ∩ ω ω -bounding). Hence, H V Γ δ ω | = ψ AC by Lemma 5.53. But then V | = ψ AC by the usual absoluteness argument. (cid:3) Remark . The conclusion, in Lemma 5.54, that Club Bounding holds is equiconsistentwith the existence of an inaccessible limit of measurable cardinals [8]. In fact, if there isno inner model with an inaccessible limit of measurable cardinals, then Club Boundingfails.Lemma 5.56 follows trivially from the fact that
BFA ℵ ( <ω - PR ) implies MA ω . Lemma 5.56.
BFA ℵ ( <ω - PR ) implies ℵ = 2 ℵ . A partial order P is said to have the σ -bounded chain condition if P = S n<ω P n and foreach n there is some k n < ω such that for every X ∈ [ P n ] k n there are distinct p , p ′ ∈ P n which are compatible in P . Also, a partial order P is Knaster if every uncountable subsetof P contains an uncountable subset consisting of pairwise compatible conditions in P .It is easy to see, and a well-known fact, that random forcing preserves Suslin trees. Thisfollows from the fact that random forcing has the σ -bounded chain condition, that everyforcing with the σ -bounded chain condition is Knaster, and that every Knaster forcingpreserves Suslin trees.Lemma 5.57 follows from the above, together with the fact that random forcing is ω ω -bounding and adds a new real. Lemma 5.57.
BFA ℵ (( <ω - PR ) ∩ STP ∩ ω ω -bounding ) implies ¬ CH . Lemma 5.58.
Suppose δ is an inaccessible cardinal such that V δ ≺ V . Let Γ be anyabsolutely well-behaved class, defined with a parameter in V δ , and such that λ Γ = ω .Suppose Γ ⊆ S -cond holds after forcing with Add( ω , . If BCFA (Γ) holds, then so does CH .Proof. Let G be Add( ω , V [ G ] satisfies CH . Then force BCFA (Γ)over V [ G ] via Γ δ . Let V be this extension. Since, in V [ G ], Γ δ has the S -condition, forcingwith Γ V [ G ] δ over V [ G ] did not add new reals thanks to Lemma 5.16 (2). In particular, V | = CH . But then CH holds in V by the usual absoluteness argument. (cid:3) It tuns out that
BCFA (Γ), where Γ is any absolutely well-behaved class such that λ Γ = ω and Γ ⊆ S -cond holds after adding a Cohen subset of ω actually implies ♦ . The proof is essentially the same as above, using the following recent result due to Magidor, togetherwith the fact that if V ⊆ V ⊆ W are models with the same ω and ~X ∈ V is a ♦ -sequencein W , then ~X is also a ♦ -sequence in V . Theorem 5.59. (Magidor) Suppose ♦ holds. Then there is a ♦ -sequence that remains a ♦ -sequence after any forcing with the S -condition. The following well-known fact can be proved by an argument as in the final part of theproof of Lemma 5.33.
Fact 5.60. If P is countably closed, then P preserves Suslin trees. The proof of the following lemma is like the proofs of Lemmas 5.49, 5.51, and 5.58,using the well-known fact that Add( ω ,
1) adds a Suslin tree T . Lemma 5.61.
Suppose δ is an inaccessible cardinal such that V δ ≺ V . Let Γ be anyabsolutely well-behaved class, defined with a parameter in V δ , and such that λ Γ = ω .Suppose Γ ⊆ STP holds after forcing with
Add( ω , . If BCFA (Γ) holds, then there is aSuslin tree.
It will be convenient to consider the following families of Club-Guessing principles on ω (s. [1]). Definition 5.62.
Let τ < ω be a nonzero ordinal.(1) τ - TWCG denotes the following statement: There is a a sequence ~C = ( C δ : δ = ω τ · η for some nonzero η < ω )such that |{ C δ ∩ γ : γ < ω }| ≤ ℵ for every δ ∈ dom( ~C ), and such that for everyclub C ⊆ ω there is some δ ∈ dom( ~C ) with ot( C δ ∩ C ) = ω τ .(2) τ - TCG denotes the following statement: There is a a sequence ~C = ( C δ : δ = ω τ · η for some nonzero η < ω )such that |{ C δ ∩ γ : γ < ω }| ≤ ℵ for every δ ∈ dom( ~C ), and such that for everyclub C ⊆ ω there is some δ ∈ dom( ~C ) with C δ ⊆ C .In the above definition, TWCG and
TCG stand for thin weak club-guessing and thinclub-guessing , respectively.
Lemma 5.63.
Let τ < ω be a nonzero ordinal. Then BFA ℵ (( ω τ - PR ) ∩ STP ∩ ω ω -bounding ) implies the failure of τ ′ - TWCG for every τ ′ such that τ < τ ′ < ω .Proof. Let us consider the following natural forcing P ~C for killing an instance ~C = ( C δ : δ = ω τ · η for some nonzero η < ω )of τ ′ - TWCG : P ~C is the set, ordered by reverse end-extension, of countable closed subset c of ω such that ot( C δ ∩ c ) < ω τ ′ for every δ ∈ dom( ~C ). It is proved in [1] that P ~C is ω τ -proper, does not add new reals, and adds a club C ⊆ ω such that ot( C δ ∩ C ) < ω τ ′ for every δ ∈ dom( ~C ). Hence, it only remains to prove that P ~C preserves Suslin trees.This can be shown by an argument similar to the main argument in the proof in [15]that MRP -posets preserve Suslin tree. We present the argument here for the reader’sconvenience.Suppose U is a Suslin tree, ˙ A is a Q -name for a maximal antichain of U , and N is acountable elementary submodel of some large enough H θ containing U , ˙ A , and all otherrelevant objects. Let δ = N ∩ ω . As in the last part of the proof of Lemma 5.33, let ncompatible bounded category forcing axioms 51 ( u n ) n<ω enumerate all nodes in U of height δ . Given a condition c ∈ P ~C ∩ N , we aim tobuild an ( N, P ~C )-generic sequence ( c n ) n<ω of conditions in N extending c such that forevery n there is some v ∈ U below u n such that c n +1 forces v ∈ ˙ A . We will make sure that C δ ∩ S n<ω c b ⊆ c , which will guarantee that c ∗ = S N<ω c n ∪ { δ } ∈ P ~C . But this will beenough, as then c ∗ will be an extension of c in P ~C forcing ˙ A ⊆ U ∩ N .It thus remains to show how to find c n +1 given c n . Working in N , we may first fixsome countable M H χ (for some large enough χ ) containing c n and all other relevantobjects (including some relevant dense set D ⊆ P ~C that we need to meet), and such that[ η, δ M ] ∩ C δ = ∅ , where δ M = M ∩ ω . In order to find M , we first consider a strictly ⊆ -increasing and continuous sequence ( M ν ) ν<ω ∈ N of elementary submodels containingall relevant objects. Since ( M ν ∩ ω ) ν<δ is a club of δ of order type δ and ot( C δ ) = ω τ ′ < δ ,we can then find some ν < δ such that M = M ν is as desired. Now, working in M , wemay, first, extend c n to a condition c ′ n such that max( c ′ n ) > η and [max( c n ) , η ] ∩ c ′ n = ∅ ,and then extend c ′ n to a condition c ′′ n in D . Let now ¯ u be the unique node in U below u n of height δ M . Since U is a Suslin tree, we have that u n is totally ( U, M )-generic. Also, theset E ∈ N of u ∈ U for which there is some v ∈ U below u and some ¯ c ∈ P ~C extending c ′′ n and forcing that v ∈ ˙ A is dense in U . It follows that we may find some u ∈ E ∩ M below u n , as witnessed by some ¯ c ∈ P ~C ∩ N and some v ∈ U ∩ M . But then we may let c n +1 = ¯ c . (cid:3) Lemma 5.64.
Suppose δ is an inaccessible cardinal such that V δ ≺ V . Let Γ be anyabsolutely well-behaved class, defined with a parameter in V δ , and such that λ Γ = ω .Let τ < ω be a nonzero ordinal, and suppose Γ ⊆ ω τ - SP holds in any generic extensionby countably closed forcing. If BCFA (Γ) holds, then so does τ - TCG .Proof.
By a result in [31], there is a countably closed forcing notion adding a τ - TCG -sequence. The rest of the argument is as in the proof of Lemma 5.49 (and subsequentlemmas), using the preservation of τ - TCG -sequences by any ω τ -semiproper forcing, whichis a completely standard fact. (cid:3) Bounded category forcing axioms and stronger large cardinal assumptions.
As we know, the theory of H ω given by the bounded category forcing axioms we haveexplored in this section is invariant, in the presence of reasonable large cardinals, relativeto extensions via members in the corresponding class forcing CFA (Γ). In this final subsec-tion we show that the combinatorial theory of H ω given by these axioms is neverthelesssensitive to additional background large cardinal assumptions. Recall that δ is the supremum of the the set of lengths of ∆ -definable pre-well-orderings on R . Proposition 5.65.
Suppose δ is an inaccessible cardinal such that V δ ≺ V . Suppose thereis a proper class of Woodin cardinals. Let Γ be any absolutely well-behaved class, definedwith a parameter in V δ , such that λ Γ = ω and Γ ⊆ PR . Suppose BCFA (Γ) holds. Then δ < ω .Proof. By a result of Neeman and Zapletal [19], the existence of a proper class of Woodincardinals yields that if P is a proper poset and G is P -generic over V , then the identityon L ( R ) V is an elementary embedding between L ( R ) V and L ( R ) V [ G ] . Since Γ δ is proper,it follows that V Γ δ | = δ < ω . Since ‘ δ < ω ’ is expressible over H ω , we then have that V | = δ < ω by the usual absoluteness argument. (cid:3) Recall that the Strong Reflection Principle (
SRP ) is the following assertion: For everyset X such that ω ⊆ X and every S ⊆ [ X ] ℵ there is a strong reflecting sequence ( x i ) i<ω See also Remark 5.55. for S , i.e., x i ∈ [ X ] ℵ , ( x i ) i<ω is strictly ⊆ -increasing and ⊆ -continuous, and for all i , x i / ∈ S if and only if there is no y ∈ S such that x i ⊆ y and y ∩ ω = x i ∩ ω . Lemma 5.66. FA ℵ ( SP ∩ STP ∩ ω ω -bounding ) implies δ = ω .Proof. We have that FA ℵ ( SP ∩ STP ∩ ω ω -bounding) implies SRP since, given S ⊆ [ X ] ℵ ,the standard forcing for adding a strong reflecting sequence for S is semiproper, does notadd reals, and preserves Suslin trees, where the last fact follows from an argument as inthe final part of the proof of Lemma 5.33. Also, SRP implies ¬ (cid:3) κ , for every cardinal κ ≥ ω , and hence implies that the universe is closed under sharps. Since it also impliesthe saturation of NS ω , by a classical result of Woodin ([30]) it implies δ = ω . (cid:3) Corollary 5.67.
Suppose δ is a supercompact cardinal such that V δ ≺ V . Let Γ be anyabsolutely well-behaved class, defined with a parameter in V δ , such that λ Γ = ω and suchthat SP ∩ STP ∩ ω ω -bounding ⊆ Γ forces in every generic extension via any member from Γ . Suppose BCFA (Γ) holds. Then δ = ω .Proof. We know that Γ δ forces FA ℵ (Γ) and therefore, by our assumption, it also forces FA ℵ ( SP ∩ STP ∩ ω ω -bounding). But then V | = δ = ω by Lemma 5.66 together with theusual absoluteness argument. (cid:3) Question 5.68.
Does FA ℵ ( ω - SP ) imply δ = ω ?6. Appendix
We collect here a few results (without proofs) translating the approach to forcing anditerations via posets to that done via complete boolean algebras; for details see the forth-coming [4] or [29] (the latter is available on ArXiv).Given a boolean algebra B and a prefilter G on B (i.e., a family such that V F > B for all finite F ⊆ G ), we denote by B / G the quotient algebra obtained using theideal J = { b : b ∧ c = 0 B for all c ∈ G } . B + denotes the positive elements of B and ˙ G B = (cid:8) h ˇ b, b i : b ∈ B (cid:9) is the canonical B –name for the V –generic filter. Theorem 6.1.
Assume ( P, ≤ ) is a partial order. Let RO ( P ) denote the family of regularopen sets for the topology τ P whose open sets are the downward closed subsets of P . Thefunction i p : P → RO ( P ) given by i P : p Reg ( ↓ p ) (where ↓ p = { q ∈ P : q ≤ p } andReg ( A ) is the interior of the closure of A for the topology generated by the sets ↓ p ) isan order and incompatibility preserving map of ( P, ≤ ) into RO ( P ) + with image dense in RO ( P ) + , and is such that p (cid:13) P φ ( τ , . . . , τ n ) if and only if i P ( p ) ≤ J φ ( τ , . . . , τ n ) K RO ( P ) . Two-step iterations.
Given ˙ C , a B -name for a forcing notion, by B ∗ ˙ C we intendthe two-step iteration as defined in [13, Section VIII.5] (or, equivalently—when ˙ C is a B -name for a cba—according to [12, Lemma 16.3], which is actually more in line with theapproach pursued in [4] or [29]). To simplify notation we also feel free to confuse (in someoccasions) a partial order with its boolean completion. Theorem 6.2. If i : B → C is an injective complete homomorphism of complete booleanalgebras, then B ∗ ( C / i [ ˙ G B ] ) ∼ = C . Conversely, if ˙ Q ∈ V B is a B -name for a complete boolean algebra and G is V -genericfor B , then ( B ∗ ˙ Q ) / i B ∗ ˙ C [ G ] ∼ = ˙ Q G . ncompatible bounded category forcing axioms 53 Proposition 6.3.
Let ˙ C , ˙ C be B -names for complete boolean algebras, and let ˙ k be a B -name for a complete homomorphism from ˙ C to ˙ C . Then there is a complete homo-morphism i : B ∗ ˙ C → B ∗ ˙ C such that r ˙ k = i/ ˙ G B z = B . Moreover if ˙ k is a B name for an injective homomorphism, i is injective.In the following diagrams we assume G is V -generic for B . V [ G ] : ( ˙ C ) G ( ˙ C ) G ( ˙ k ) G V : B B ∗ ˙ C B ∗ ˙ C ii i V [ G ] : ( B ∗ ˙ C ) / i [ G B ] ( B ∗ ˙ C ) / i [ G B ] ( ˙ C ) G ( ˙ C ) G ( ˙ k ) G i/ G B ∼ = ∼ = Proposition 6.4.
Let G be V -generic for B , I be its dual ideal, and i j : B → C j becomplete homomorphisms for j = 0 , .Assume C / i [ G ] and C / i [ G ] are isomorphic complete boolean algebras in V [ G ] . V [ G ] : V : B C C i i C / i [ G ] C / i [ G ] k ∼ = Then C ↾ i ( b ) and C ↾ i ( b ) are isomorphic in V for some b ∈ G and k ∼ = l/ G . V [ G ] : C ↾ i ( b ) / i [ G ] C ↾ i ( b ) / i [ G ] C / i [ G ] C / i [ G ] l/ G ∼ == = k ∼ = V : B C C i i C ↾ i ( b ) C ↾ i ( b ) l ∼ = restrest Lemma 6.5.
Assume Γ is a definable class of forcings. Let B , C , C be complete booleanalgebras, and let G be any V -generic filter for B . Let i , i , j be Γ -correct complete homo-morphisms in V forming a commutative diagram of injective complete homomorphisms ofcomplete boolean algebras as in the following picture: B C C i i j Then in V [ G ] , j/ G : C / G → C / G is still a Γ V [ G ] -correct complete homomorphism. Proposition 6.6.
Assume Γ is a definable class of forcings. Let G be V -generic for somecomplete boolean algebra B . Assume k : B → R is a Γ -correct homomorphism in V and h : R / k [ G ] → Q is a Γ -correct homomorphism in V [ G ] . B R V : k R / k [ G ] Q V [ G ] : h Then there are in V : • C ∈ Γ , • a Γ -correct homomorphism l : B → C , • a Γ -correct homomorphism ¯ h : R → C ,such that: • Q is isomorphic to C / l [ G ] in V [ G ] , • ¯ h/ G ∼ = h (modulo the isomorphism of Q with C / l [ G ] ) holds in V [ G ] , • ¯ h ◦ k = l holds in V , • C l [ G ] . B RC V : k ¯ hl ∼ = B /G R / k [ G ] C /l [ G ] ∼ = Q V [ G ] : ¯ h/ G ∼ = hk/ G ∼ =Id l/ G ∼ =Id Limit length iterations.Theorem 6.7.
Assume i : B → C is a complete injective homomorphism of completeboolean algebra. Then the following holds:(1) i has an adjoint π i : C → B defined by π i ( c ) = V { b ∈ B : i ( b ) ≥ c } (2) For any b ∈ B and c, d ∈ C , we have that:(a) π i is order preserving;(b) ( π i ◦ i )( b ) = b , hence π i is surjective;(c) ( i ◦ π i )( c ) ≥ c ; in particular, π i maps C + to B + ;(d) π i preserves joins, i.e., π i ( W X ) = W π i [ X ] for all X ⊆ C for which the supre-mum W X exists in C ;(e) i ( b ) = W { e : π i ( e ) ≤ b } ;(f ) π i ( c ∧ i ( b )) = π i ( c ) ∧ b = W { π i ( e ) : e ≤ c, π i ( e ) ≤ b } ;(g) π i preserves neither meets nor complements whenever i is not surjective, but π i ( d ∧ c ) ≤ π i ( d ) ∧ π i ( c ) and π i ( ¬ c ) ≥ ¬ π i ( c ) . Definition 6.8. F = { i αβ : B α → B β | α ≤ β < λ } is a complete iteration system ofcomplete boolean algebras iff for all α ≤ β ≤ γ < λ :(1) B α is a complete boolean algebra and i αα is the identity on it;(2) i αβ is a complete injective homomorphism and π αβ : B β → B α , given by c V { b ∈ B : i αβ ( b ) ≥ c } , is its associated adjoint;(3) i βγ ◦ i αβ = i αγ .Let F be a complete iteration system of length λ . Then: ncompatible bounded category forcing axioms 55 • The inverse limit of the iteration islim ←− ( F ) = ( f ∈ Y α<λ B α : ∀ α ∀ β > α ( π αβ ◦ f )( β ) = f ( α ) ) and its elements are called threads . • The direct limit islim −→ ( F ) = (cid:8) f ∈ lim ←− ( F ) : ∃ α ∀ β > α f ( β ) = i αβ ( f ( α )) (cid:9) and its elements are called constant threads . The support of a constant thread,supp( f ), is the least α such that ( i αβ ◦ f )( α ) = f ( β ) for all β ≥ α . • The revised countable support limit is lim rcs ( F ) = (cid:8) f ∈ lim ←− ( F ) : f ∈ lim −→ ( F ) ∨ ∃ α f ( α ) (cid:13) B α cof(ˇ λ ) = ˇ ω (cid:9) . Theorem 6.9.
Assume n P α , ˙ Q β : α ≤ λ, β < λ o is an iteration of posets. Let i αβ : RO ( P α ) → RO ( P β ) be the complete homomorphism induced by the natural inclusion of P α into P β . Then: • F = { i α,β : α ≤ β < λ } is an iteration system of complete injective homomor-phisms of complete boolean algebras. • If λ = ω , lim ←− ( F ) is isomorphic to the boolean completion of the full limit of n P α , ˙ Q β : α ≤ λ, β < λ o . • For any regular λ , lim −→ ( F ) is isomorphic to the boolean completion of the directlimit of n P α , ˙ Q β : α ≤ λ, β < λ o . Theorem 6.10 (Baumgartner) . Let λ be a regular cardinal and F = { i αβ : α ≤ β < λ } bean iteration system such that B α is <λ -cc for all α and S = (cid:8) α : B α ∼ = RO (lim −→ ( F ↾ α )) (cid:9) is stationary. Then lim −→ ( F ) is <λ -cc. These results suffice to prove all the needed equivalences (used in this paper) betweenresults on forcing and iterations proved in the language of partial orders and their corre-sponding formulation in terms of complete boolean algebras.
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David Asper´o, School of Mathematics, University of East Anglia, Norwich NR47TJ, UK
Email address : [email protected] Matteo Viale, Department of Mathematics “Giuseppe Peano”, Univ. Torino, viaCarlo Alberto 10, 10125, Torino, Italy
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