aa r X i v : . [ m a t h . L O ] F e b The Slicing Axioms
Ziemowit Kostana * , Saharon Shelah †‡ February 24, 2021
Abstract
We introduce the family of axioms
Slice κ , which claim existence of nontrivial de-compositions of the form { <κ ∩ V α | α < κ } , for sets <κ , where { V α | α < κ } is asequence of transitive models of set theory. We study compatibility of these axioms withversions of Martin’s Axiom, and in particular show that Slice ω is compatible only withsome very weak form of MA . We introduce and study the family of axioms
Slice κ for cardinal numbers κ . The axiom Slice κ basically claims that there exists an increasing sequence of transitive models { V α | α < κ } ,which decomposes <κ into an increasing union V α ∩ <κ . Our initial motivation was to finda single model of Martin’s Axiom, which doesn’t satisfy typical consequences of P F A . Thiswas in turn motivated by the following intuition:If the universe is sufficientely complete, in the sense that it has many generic filters, then anytransitive submodel containing enough reals, contain all the reals.This intuition is supported for example by the following result:
Theorem 1 (Thm. 8.6, [8]) . If M M holds, then any inner model with correct ω containsall reals. The conclusion is quite strong, so it makes sense to ask what is left if we weaken
M M to M A ω . This motivated us to formulate the axiom Slice ω , which turned out to be inconsistentwith M A ω . The main results of this paper are the following Theorem (Thm. 2) . Slice ω = ⇒ ¬ MA ω ( σ -centred ) . Theorem (Thm. 5) . If κ is a regular cardinal such that κ ω = κ , then the following theory isconsistent ZF C + M A ( Suslin ) + Slice ω +”2 ω = κ ” . Theorem (Thm. 7) . Assume that ω < κ ≤ θ are regular cardinals, such that θ <κ = θ . Thenthe following theory is consistent ZF C + M A <κ + Slice κ +”2 ω = θ ” . The first of these results provides another argument in favor of the informal claim fromthe beginning. The class of Suslin forcings is a class of c.c.c. forcings, which admit sim-ple (analytic) definitions (see Definition 2). This class is more extensively described in [2].Martin’s Axiom for this class is a considerable weakening of the full
M A . * Research of Z. Kostana was supported by the GAˇCR project EXPRO 20-31529X and RVO: 67985840. † Research of second author partially supported by Israel Science Foundation (ISF) grant no: 1838/19. ‡ Research of both authors partially supported by NSF grant no: DMS 1833363. heorem ([6]) . M A ( Suslin ) implies each of the following:1. Add( N ) = 2 ω ,2. Add( SN ) = 2 ω ,3. ω is regular,4. each MAD family of subsets of ω has size ω . It follows from 1. that all cardinal characteristics in the Cicho´n’s diagram have value ω . SN stands for the class of strong measure zero sets. Theorem ([6]) . M A ( Suslin ) does not imply any of the following:1. t = 2 ω ,2. s = 2 ω ,3. ∀ κ < ω κ = 2 ω ,4. there is no Suslin tree. For an elaborated discussion of cardinal invariants of the continuum we refer the readerto [4].
All non-standard notions are introduced in the subsequent sections. By reals we denote el-ements of the sets ω ω , ω , or seldom R . When we write P = { P α ∗ ˙ Q α | α < θ } for afinite-support iteration of forcings, we sometimes denote by P the final step of the iteration,that is P = P θ . When dealing with infinite iterations, we assume that P the trivial forcing.A function i : P ֒ → P is a complete embedding if the following assertions hold:1. ∀ p, q ∈ P p ≤ p = ⇒ i ( p ) ≤ i ( p ) ,2. ∀ p, q ∈ P p ⊥ p = ⇒ i ( p ) ⊥ i ( p ) ,3. If A ⊆ P is a maximal antichain, then i [ A ] ⊆ P is a maximal antichain.We write P ⋖ P if P ⊆ P and the inclusion is a complete embedding. We will befrequently using the following observation Proposition 1. If V is a countable transitive model of ZF C , P , P ∈ V , and P ⊆ P is aninclusion of partial orders, then the following conditions are equivalent:1. P ⋖ P ,2. If a filter G ⊆ P is P -generic over V then G ∩ P is P -generic over V . Definition 1.
Let κ be any uncountable cardinal. We will say that Slice κ holds if there existsa sequence of transitive classes (not necessarily proper) { V α | α < κ } , such that the followingconditions are satisfied• ∀ α<κ V α | = ZF C ,• ∀ α<ω ω V α = ω ,• <κ = [ α<κ <κ ∩ V α ,• ∀ α<β<κ <κ ∩ V α ( <κ ∩ V β .We will say that the sequence { V α | α < κ } preserves cardinals if κ ∈ V and for eachcardinal λ ∈ V and each α < κ , λ V α is a cardinal.The most important of the slicing axioms is perhaps Slice ω , since it claims that the realline can be decomposed into an increasing union of ω many sets, which belong to biggerand bigger models. The fact that M A ω is inconsistent with Slice ω shows, that the Martin’sAxiom on ω imposes certain compactness on the real line.2 Slicing the real line
We begin with showing that Martin’s Axiom on ω is not compatible with Slice ω . Theorem 2.
Slice ω = ⇒ ¬ MA ω ( σ -centred ) . In the proof we will utilize the known result from [5]. Recall that a set A ⊆ R is a Q -set ,if each subset of A is a relative F σ . Theorem 3 ([5]) . MA( σ -centred ) implies that each set of cardinality less than c is a Q -set.Proof of Theorem 2. Assume that MA ω holds, and ( V α ) α<ω is a sequence of models wit-nessing Slice ω . V | = " ω is uncountable" , so there exists a sequence of pairwise distinct re-als X = { x α | α < ω } ∈ V (note that this sequence is really of legnth ω ). Let f : ω ֒ → ω be a function such that ∀ α < ω f ( α ) / ∈ V α . We will obtain a contradiction, by showing thatthere exists some η < ω , for which rg( f ) ⊆ V η .For every natural number m , let A m = { x α | f ( α )( m ) = 1 } = X ∩ F m , where F m is an F σ subset of reals. Since the sequence ( F m ) m<ω can be coded by a real, clearly it belongsto some model V η . It is enough to show that using this sequence we can give a definition of rg( f ) . But rg( f ) = { x ∈ X | ∃ α < ω ∀ m < ω x α ∈ F m ⇐⇒ x ( m ) = 1 } . It is compatible with any value of ω , that Slice ω holds and is witnessed by a cardinalpreserving sequence. Proposition 2.
Let P be any finite-support product of c.c.c. forcings adding reals, of length atleast ω . Then P (cid:13) Slice ω , and the corresponding sequence of models is cardinal preserving.Proof. Let us consider a finite-support product of c.c.c. forcings P = Y i ∈ I P i , where each P i adds some real number, and | I | ≥ ω . We can decompose I into a strictlyincreasing union I = [ γ<ω I γ . For each α < ω the product Y i ∈ I α P i can be identified with acomplete suborder of P .If G ⊆ P is generic over some model V , then Slice ω is witnessed by the sequence V α = V [ G ∩ Y i ∈ I α P i ] . The following was proved by Baumgartner in [3].
Theorem ([3]) . It is consistent with
M A ω , that all ω -dense subsets of reals are order-isomorphic. In particular, each ω -dense set of reals has a non-trivial order-automorphism. The natural question whether this assertion follows from
M A ω was resolved by Avrahamand the second author [1]. Theorem ([1]) . It is consistent with
M A ω , that there exists a rigid ω -dense real order type. This is also an easy consequence of
Slice ω . Theorem 4.
Slice ω implies that there is an ω -dense rigid subset of the real line.Proof. Let ( V α ) α<ω be a sequence witnessing Slice ω . For each α , we choose x α ∈ R ∩ ( V α \ [ β<α V β ) . We can easily arrange the construction, so that we hit each open interval ω -many times. Theset X = { x α | α < ω } is ω -dense, and it remains to prove, that it is also rigid. Supposethat f : X → X is an order isomorphism. f extends uniquely to a continuous function f ′ : R → R , and each such function can be coded by a real number. Therefore there is some η < ω , such that f ′ ∈ V η . Now, for any ξ > η , it is not possible that f ( x η ) = x ξ , becauseit would mean x ξ ∈ V η , contrary to the choice of x ξ . But, likewise, it is not possible that f − ( x η ) = x ξ . The conclusion is that for all ξ > η , f ( x ξ ) = x ξ . But this means that f isidentity on a dense set, and therefore everywhere.3 Slicing the real line while preserving MA(Suslin)
We are going to show that
Slice ω is consistent with a version of Martin’s Axiom which takesinto account only partial orders representable as analytic sets (see [2], Ch. 3.6, or [6]). Definition 2.
A partial order ( P , ≤ ) has a Suslin definition if P ∈ Σ ( ω ω ) , and both orderingand incompatibility relations in P are analytic relations on ω ω . P is Suslin if it has a Suslindefinition and is c.c.c.The following is the main result of this Section.
Theorem 5. If κ is a regular cardinal such that κ ω = κ , then the following theory is consis-tent ZF C + M A ( Suslin ) + Slice ω +”2 ω = κ ” . Let ψ ( − , − , − , − ) be a universal analytic formula, i.e. a Σ formula with the propertythat for each analytic set P ⊆ ω ω × ω ω × ω ω there exists r ∈ ω ω such that P = { x ∈ ω ω × ω ω × ω ω | ψ ( x, r ) } . We want to use ψ to add generic filters to all possible Suslin forcings. We will say that ψ ( − , − , − , ˙ r α ) defines ˙ Q α if ˙ r α is a P α -name for a real and P α forces each of the following ˙ Q α is a separative partial order with the greatest element ,ψ ( x, , , ˙ r α ) ⇐⇒ x ∈ ˙ Q α ,ψ ( x, y, , ˙ r α ) ⇐⇒ x ≤ ˙ Q α y,ψ ( x, y, , ˙ r α ) ⇐⇒ x ⊥ ˙ Q α y. We will write ψ ∈ ( x, z ) for ψ ( x, , , z ) , ψ ⊥ ( x, y, z ) for ψ ( x, y, , z ) , and ψ ≤ ( x, y, z ) for ψ ( x, y, , z ) .We are going to iterate all Suslin forcings, each of them cofinally many times. Moreprecisely, we define by induction a finite-support iteration { P α ∗ ˙ Q α | α < κ } :• P = { } ,• P α (cid:13) ” ˙ Q α = { x ∈ ω ω | ψ ( x, ˙ r α ) } if this formula defines a Suslin forcing; else ˙ Q = { } ” ,The variable ˙ r α ranges over all reals, and all possible names for reals, each of themcofinally many times. In order to iterate through all possible parameters using a suitablebookkeeping, we introduce the class of simple conditions, following [2]. Definition 3.
By induction on α we define simple conditions in P α .• α = 0 . P = { } , and we declare to be simple.• α + 1 . ( p, ˙ q ) ∈ P α +1 is simple if p ∈ P α is simple and ˙ q = { ( m, n, p mn ) | m, n < ω, p mn ∈ P α } , where each p mn is a simple condition in P α . (for each m ∈ ω , the set { p mn | n < ω } is amaximal antichain deciding ˙ q ( m ) , i.e. p mn (cid:13) ˙ q ( m ) = n )• lim α. p ∈ P α is simple if for each β < α , p ↾ β ∈ P β is simple.It is straightforward to check by induction, that the set of simple P α -conditions is densein P α , and that each P α at most κ many names for reals (if we restrict to names with simpleconditions). Proposition 3. If ω ≤ κ is an uncountable regular cardinal such that κ ω = κ , then P κ (cid:13) M A ( Suslin ) + ”2 ω = κ ” . roof. Let us denote by W α the corresponding extensions of V by P α . Let ( S, ≤ ) be a Suslinforcing in W κ . Assume S is defined by the formula ψ ( − , r ) . We fix a family { A γ | γ < λ } of maximal antichains in S , where λ < κ . By the Löwenheim-Skolem theorem, we can findan elementary substructure of ( S, ≤ , A γ ) γ<λ of size λ . For simplicity of notation we canassume that S is this substructure, and so | S | ≤ λ . Therefore ( S, A γ , ≤ ) γ<λ ∈ W δ , for some δ < κ , and we can enlarge δ so that P δ (cid:13) ˙ r δ = r. By absoluteness of the formulae ψ ∈ ( − , r ) , ψ ⊥ ( − , r ) , and ψ ≤ ( − , r ) , the partial order definedby ψ ( − , r ) in W δ is a suborder of the one defined by this formula in W κ (even a completesuborder, which is not relevant here). Therefore the generic filter added for ˙ Q δ in W δ will bea filter intersecting the sets A γ in S .If N is a transitive class containing κ , we can define by induction the relativized iteration P Nκ ⊆ P κ , taking into account only names from N .• P N = { } ,• P Nα (cid:13) ” ˙ Q Nα = { x ∈ ω ω | ψ ∈ ( x, ˙ r α ) } if this formula defines a Suslin forcing, ˙ r α ∈ N , and ˙ r α is a P Nα -name; else ˙ Q Nα = { } ” ,• P Nα +1 = P Nα ∗ ˙ Q Nα .If we take direct limits in the limit step, it is clear that P Nα is really a subset of P α . Note,that we do not define names ˙ r α inductively along the way, since they have already beendefined in the construction of P κ , which we take as granted. This construction is inspired bythe lemmas 1.4 and 1.5 from [6], and conceptually is very similar. In order for it to work asdesired, we prove by induction the some properties of P Nα . Theorem 6. If N is a transitive class containing κ , then for all α ≤ κ P Nα ⋖ P α . Specifically:1. If p ⊥ p in P Nα , then p ⊥ p in P α .2. If p ≤ p in P Nα , then p ≤ p in P α .3. If G ⊆ P α is a filter generic over V , then G ∩ P Nα ⊆ P Nα is also generic over V .Proof. α = 0 . Clear.• α + 1 . We can assume that ˙ Q Nα is defined by the formula ψ ( − , ˙ r α ) , for otherwise P Nα +1 = P Nα , and we are done by the induction hypothesis. Fix two incomparableconditions p , p ∈ P Nα +1 . Then p = ( p ′ , ˙ q ) , p = ( p ′ , ˙ q ) , where p ′ , p ′ ∈ P Nα , and p ′ (cid:13) ψ ∈ ( ˙ q , ˙ r α ) ,p ′ (cid:13) ψ ∈ ( ˙ q , ˙ r α ) . The forcing relation used above is a relation from P Nα , however since ˙ r α , ˙ q and ˙ q are P Nα -names, this is the same relation as coming from P α (remember that P Nα ⋖ P α ). Weaim to show that p ⊥ p in P α +1 .If p ′ ⊥ p ′ in P α , then clearly p ⊥ p in P α +1 , so assume otherwise, and fix p ≤ p ′ , p ′ (in P α ). Let p ∈ G ⊆ P α be a filter generic over V . Conditions p and p wereincomparable in P Nα +1 and, by the induction hypothesis, G ∩ P Nα ⊆ P Nα is generic over V , therefore V [ G ∩ P Nα ] | = ψ ⊥ ( ˙ q [ G ] , ˙ q [ G ] , ˙ r α [ G ]) . By absoluteness V [ G ] | = ψ ⊥ ( ˙ q [ G ] , ˙ q [ G ] , ˙ r α [ G ]) . Since p was arbitrary, it follows that p ⊥ p in P α +1 .5 lim α . Follows from the induction hypothesis, since conditions have finite supports.2.• α = 0 . Clear.• α + 1 . Again, we can assume that ˙ Q Nα is defined by the formula ψ ( − , ˙ r α ) . Fix twoconditions p ≤ p ∈ P Nα +1 . Then p = ( p ′ , ˙ q ) , p = ( p ′ , ˙ q ) , where p ′ , p ′ ∈ P Nα ,and p ′ (cid:13) ψ ∈ ( ˙ q , ˙ r α ) ,p ′ (cid:13) ψ ∈ ( ˙ q , ˙ r α ) . By the induction hypothesis p ′ ≤ p ′ in P α . Moreover ˙ r α , ˙ q and ˙ q are P Nα -names, sothe forcing relation p ′ (cid:13) ˙ q ≤ ˙ q holds in P Nα as well as in P α .• lim α . Follows from the induction hypothesis, since conditions have finite supports.3.• α = 0 . Clear.• lim α . Let { p n | n < ω } be a maximal antichain in P Nα , and p ∈ P α . There is some γ < α such that p ∈ P γ . { p n ↾ γ | n < ω } might not be an antichain in P Nγ , howevereach condition in P Nγ is compatible with some p n ↾ γ . We can refine { p n ↾ γ | n < ω } to an antichain in P Nγ , and this antichain will remain maximal in P γ by the inductionhypothesis. Therefore { p n ↾ γ | n < ω } intersects every condition in P γ , and inparticular some p n ↾ γ is compatible with p in P γ . But then p n is compatible with p in P α .• α +1 . We aim to show that for any G ⊆ P α +1 generic over V , G ∩ P Nα +1 is also genericover V . Lemma 1. If G ⊆ P α is generic over V , and H ⊆ ˙ Q α [ G ] is generic over V [ G ] , then H ∩ ˙ Q Nα [ G ] ⊆ ˙ Q Nα [ G ] is generic over V [ G ∩ P Nα ] . Why is this sufficient? Let G ⊆ P α ∗ ˙ Q α be a filter generic over V . Recalling thenotation from [7], G = G ∗ H = { ( p, ˙ q ) | p ∈ G, ˙ q [ G ] ∈ H } , where G = { p ∈ P α | ∃ ˙ q ∈ ˙ Q ( p, ˙ q ) ∈ G } , and H = { ˙ q [ G ] | ∃ p ∈ G ( p, ˙ q ) ∈ G } . It is known that for any iteration P ∗ ˙ Q , if G ⊆ P is generic over V and H ⊆ ˙ Q [ G ] is generic over V [ G ] , then G ∗ H is generic for P ∗ ˙ Q over V (for details consult forexample [7], Section 5, Chapter VIII). Let G ′ = G ∩ P Nα . It is generic for P Nα over V by the induction hypothesis. Now for filters G and H defined above ( G ∗ H ) ∩ ( P Nα ∗ ˙ Q Nα ) = { ( p, ˙ q ) | p ∈ G ′ , ˙ q [ G ] ∈ H, ˙ q ∈ ˙ Q Nα } = { ( p, ˙ q ) ∈ P Nα ∗ ˙ Q Nα | p ∈ G ′ , ˙ q [ G ′ ] ∈ H } = G ′ ∗ ( H ∩ ˙ Q Nα [ G ′ ]) . But if the conclusion of Lemma 1 holds, this is a P Nα ∗ ˙ Q Nα -generic filter over V .We turn to the proof of Lemma 1. Proof.
Fix a maximal antichain
A ⊆ ˙ Q Nα [ G ] = ˙ Q Nα [ G ′ ] , belonging to V [ G ′ ] . As A isa countable set of reals, it can be coded using a single real z ∈ ω ω . Recall that ˙ Q Nα [ G ′ ] is defined in V [ G ′ ] by the formula ψ with the parameter ˙ r α [ G ′ ] = ˙ r α [ G ] . It is standardto check, that the following claim can be written as a Π formula.6 ( x, y ) = ” x is a real coding a maximal antichain in the partial ordering defined bythe formula ψ ( − , − , − , y )” .Now V [ G ′ ] | = φ ( z, ˙ r α [ G ′ ]) , and so by absoluteness V [ G ] | = φ ( z, ˙ r α [ G ]) . But ψ ( − , ˙ r α [ G ]) is the formula defining ˙ Q α [ G ] in V [ G ] . Therefore A remains maximalin ˙ Q α [ G ] , and conclusion of the Lemma easily follows.This concludes the proof.Let us note that even if N is an inner model of ZFC, usually P Nκ / ∈ N . Definition of P Nκ makes use of a list of P Nα -names, for all α < κ , and although some such enumeration belongsto N (as it is a model of choice), this particular might not. In what sense is P Nκ a relativized version of P κ , is explained by the next lemma. Lemma 2.
For any transitive class N containing κ , for each α ≤ κ , N ∩ P α ⊆ P Nα .Proof. We proceed by induction.• α = 0 . Clear.• lim α. If r ∈ N ∩ P α , we choose γ < α containing the support of r . Then r ↾ γ ∈ P γ ∩ N ⊆ P Nγ . It is routine to verify by induction that for all γ ≤ δ ≤ α , r ↾ δ ∈ P Nδ .• α + 1 . If r = ( p, ˙ q ) ∈ N ∩ ( P α ∗ ˙ Q α ) , then p ∈ P Nα , ˙ q ∈ N , and we need onlyto prove that ˙ q is a P Nα -name. But note, that ˙ q = { ( m, n, s mn ) | m, n < ω } , where ∀ m, n < ω s mn ∈ P α ∩ N ⊆ P Nα . Lemma 3.
For each α ≤ κ , if p ∈ P α is simple then p is definable (in the language of settheory) with a parameter from κ ω .Proof. • α = 0 . Clear, since each real is definable with a real parameter.• α + 1 . Let r = ( p, ˙ q ) be simple. We can write ˙ q = { ( m, n, p mn ) | m, n ∈ ω, p mn ∈ P α } , where each p mn is simple. By the induction hypothesis each p mn definable with a param-eter from κ ω , and so is p . Clearly r can be defined from them, and so r is definable withcountably many parameters from κ ω . We can easily code them as a single parameter.• lim α . Fix r ∈ P α . r has finite support, so there exists β < α containing the supportof r . By the induction hypothesis p ↾ β is definable with a parameter from κ ω , and p isdefinable with parameters p ↾ β , β , and α . Proof of Theorem 5.
We start with a model V | = Slice ω +”2 ω = κ ” , and we assume more-over that the sequence { V α | α < ω } witnessing Slice ω satisfies the following strongerproperty: κ ω = [ α<ω κ ω ∩ V α . Such model is easy to get, for example by adding κ many Cohen reals to a model of CH ,and proceeding like in the proof of Proposition 2.Let P = { P α ∗ ˙ Q α | α < κ } be the iteration described above, which forces M A ( Suslin ) + ”2 ω = κ ” . We claim that if G ⊆ P is generic over V , then the sequence V [ G ∩ P V α ] witnesses Slice ω in V [ G ] . For this we need to show two things7. If r ∈ ω ω , then r ∈ V [ G ∩ P V α ] for some α < κ .2. None of the models V [ G ∩ P V α ] contains all reals.Concerning . assume that P κ (cid:13) ˙ r ∈ ω ω . We can assume that ˙ r = { ( m, n, p mn ) | m, n < ω } , and all conditions p mn are simple. By Lemma 3 each condition p mn is definable with a param-eter from κ ω . It follows, that ˙ r is definable with a parameter from κ ω , and by our assumptionthis parameter belongs to some model V α . Therefore ˙ r is a P V α -name, and so ˙ r [ G ] = ˙ r [ G ∩ P V α ] ∈ V [ G ∩ P V α ] . Concerning . fix a real r ∈ ω ω \ V α . There exists a representation of the Cohen forcingas a Borel subset of ω ω , from which the real r is definable. For concreteness, let us put C r = ω <ω ∪ { r } ⊆ ω ω , where ω <ω is ordered by the end-extension and ∀ s ∈ ω <ω s ⊥ r. Since r / ∈ V α , it follows that for some γ < κ P V α γ (cid:13) ˙ Q V α γ = { } , and P γ (cid:13) ˙ Q γ = C r . Therefore we can find a complete embedding of C r into the quotient forcing C r ֒ → P κ / ( P V α κ ∩ G ) , given by the formula x P γ ⌢ (1 P γ , x ) ⌢ P κ \ ( γ +1) . This shows that V [ G ] contains a Cohen real over V [ P V α ∩ G ] . <κ Although
M A ω is inconsistent with Slice ω , it is consistent with Slice κ for any κ > ω .This is the consequence of the following theorem: Theorem 7.
Assume that ω < κ ≤ θ are regular cardinals, such that θ <κ = θ . Then thefollowing theory is consistent ZF C + M A <κ + Slice κ +”2 ω = θ ” . We are going to apply a finite-support iteration of the form P = { P α ∗ ˙ Q α | α < θ } , where for each α < θ P α (cid:13) ˙ Q α = ( λ α , ˙ ≤ α ) , for λ α < κ . We also assume that ∈ λ α is always the largest element in ˙ Q α . We can arrangethe iteration so that each partial order of size < κ will appear cofinally many times (see theproof of Propsition 3). Definition 4.
By induction on α , we define the class of simple P α -conditions.• α = 0 . P = { } , and we declare to be simple.• α + 1 . ( p, ˙ q ) ∈ P α +1 is simple if p ∈ P α is simple and ˙ q = { ( γ n , p n ) | n < ω } , whereconditions p n are simple. 8 lim α. p ∈ P α is simple if for each β < α , p ↾ β ∈ P β is simple.Like in the previous section, is is easy to check that the set of simple conditions is alwaysdense. Lemma 4.
For each α ≤ κ , if p ∈ P α is simple then p is definable (in the language of settheory) with a parameter from κ ω .Proof. • α = 0 . Clear.• α +1 . Let r = ( p, ˙ q ) be simple. We can write ˙ q = { ( γ n , p n ) | n < ω } , where conditions p n are simple. By the induction hypothesis each p n is definable with a parameter from κ ω , and so is p . Clearly r can be defined from them, and so r is definable with countablymany parameters, which we can code as one.• lim α . Fix r ∈ P α . r has finite support, so there exists β < α containing the supportof r . By the induction hypothesis p ↾ β is definable with a parameter from κ ω , and p isdefinable with parameters p ↾ β , β , and α .Like in the previous Section, we can define by induction the relativized forcings P Nκ ⊆ P κ , taking into account only names from N .• P N = { } ,• Assume P Nα is defined. We define a P Nα -name ˙ Q Nα as follows – ˙ Q Nα = ˙ Q α if ˙ Q α ∈ N , and ˙ Q α is a P Nα -name, – ˙ Q Nα = { } otherwise.• P Nα +1 = P Nα ∗ ˙ Q Nα .In limit steps we take direct limits, so P Nκ ⊆ P κ . Repeating the proof of Lemma 2, weobtain Lemma 5. If N is a transitive class containing κ , α ≤ κ , then P α ∩ N ⊆ P Nα . Lemma 6. If N is a transitive class, then for all α ≤ κ P Nα ⋖ P α . Specifically:1. If p ⊥ p in P Nα , then p ⊥ p in P α .2. If p ≤ p in P Nα , then p ≤ p in P α .3. If A ⊆ P Nα is a maximal antichain, then A is maximal in P α .Proof. We proceed by induction on α .1.• α = 0 . Clear.• α + 1 . Assume ( p , ˙ q ) ⊥ ( p , ˙ q ) in P Nα +1 . If p ⊥ p in P Nα , then by the inductionhypothesis p ⊥ p in P α and we are done. Suppose otherwise, and fix a condition p ≤ p , p from P α . Let G ⊆ P α be any filter generic over V , containing p . p , p ∈ G ∩ P Nα , so ˙ q [ G ∩ P Nα ] ⊥ ˙ q [ G ∩ P Nα ] in model V [ G ∩ P Nα ] , and in V [ G ] as well. Since p and G were arbitrary, it follows that ( p , ˙ q ) ⊥ ( p , ˙ q ) in P α +1 .• lim α. Follows from the induction hypothesis, since supports are finite.9.• α = 0 . Clear.• α + 1 . Assume ( p , ˙ q ) ≤ ( p , ˙ q ) in P Nα +1 . From the induction hypothesis we know,that p ≤ p in P α , and p (cid:13) ˙ q ≤ ˙ q in the sense of P Nα . We must show that theassertion p (cid:13) ˙ q ≤ ˙ q holds also in the sense of P α . If ˙ Q Nα = { } it is trivial. Otherwise ˙ Q Nα = ˙ Q α . In thatcase ˙ q and ˙ q are P Nα -names, and the (cid:13) relation for them is the same in P Nα as in P α .• lim α. Follows from the induction hypothesis, since supports are finite.3.• α = 0 . Clear.• α + 1 . The proof is exactly the same, as in the paragraph after Lemma 1, so we needto prove the conclusion of Lemma 1 in the current setting. But this is trivial, once werecall that P Nα (cid:13) ˙ Q Nα = { } , or P Nα (cid:13) ˙ Q Nα = ˙ Q α . • lim α . Let { p n | n < ω } be a maximal antichain in P Nα , and p ∈ P α . There is some γ < α such that p ∈ P γ . { p n ↾ γ | n < ω } might not be an antichain in P Nγ , howevereach condition in P Nγ is compatible with some p n ↾ γ . We can refine { p n ↾ γ | n < ω } to an antichain in P Nγ , and this antichain will remain maximal in P γ by the inductionhypothesis. Therefore { p n ↾ γ | n < ω } intersects every condition in P γ , and inparticular some p n ↾ γ is compatible with p in P γ . But then p n is compatible with p in P α . Proof of Theorem 7.
Let V | = ZF C + GCH + Slice κ and let P be the forcing defined in the beginning of the Section. Suppose that { V α | α < κ } witnesses Slice κ in V , and G ⊆ P is generic over V . We aim to shows that the sequence V [ G ∩ P V α ] witnesses Slice κ in V [ G ] . For this we need to show two things1. If F ∈ <κ , then F ∈ V [ G ∩ P V α ] for some α < κ .2. None of the models V [ G ∩ P V α ] contains all sequences from <κ .Concerning . assume that P κ (cid:13) ˙ F ∈ δ , for some ordinal δ < κ . We can assumethat ˙ F = { ( α, α n , p αn ) | α < δ, n < ω } , and all conditions p αn are simple. By Lemma 4each condition p αn is definable with a parameter E αn ∈ κ ω . The set { E αn | α < δ, n < ω } isdefinable from a sequence of length < κ , so it belongs to some model V α , and so ˙ F ∈ V α .Therefore ˙ F is a P V α κ -name, and it follows that ˙ F [ G ] = ˙ F [ G ∩ P V α ] ∈ V [ G ∩ P V α ] . Concerning . fix a sequence F ∈ <κ \ V α . There exists a representation of the Cohenforcing, say C F , from which the sequence F is definable and | C F | < κ . Since F / ∈ V α , itfollows that for some γ < κ P V α γ (cid:13) ˙ Q V α γ = { } , and P γ (cid:13) ˙ Q γ = C F . Therefore we can find a complete embedding of C F into the quotient forcing C F ֒ → P κ / ( P V α κ ∩ G ) , x P γ ⌢ (1 P γ , x ) ⌢ P κ \ ( γ +1) . This shows that V [ G ] contains a Cohen real over V [ P V α ∩ G ] . Corollary 1.
The following theories are consistent
ZF C + M A ω + Slice ω +”2 ω = ω ” ,ZF C + M A ω + Slice ω +”2 ω = ω ” ,ZF C + M A ω + Slice ω +”2 ω = ω ” . We proved that
M A ω and Slice ω are not compatible. It should be expected that for anyregular cardinal κ M A κ = ⇒ ¬ Slice κ . What about singular κ ?In Theorem 7 the assumption that κ is regular looks redundant. If this is the case, wewould have a method for producing numerous models of ZFC with the Martin’s number m issingular. Up to my knowledge, for this moment only one such model is known, with m = ℵ ω References [1] U. Avraham, S. Shelah,
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