Inner Models from Extended Logics: Part 2
aa r X i v : . [ m a t h . L O ] J u l Inner Models from Extended Logics: Part 2 ∗ Juliette Kennedy † Helsinki Menachem Magidor ‡ JerusalemJouko V¨a¨an¨anen § Helsinki and AmsterdamJuly 22, 2020
Abstract
We introduce a new inner model C ( aa ) arising from stationary logic.We show that assuming a proper class of Woodin cardinals, or alternativelyMM ++ , the regular uncountable cardinals of V are measurable in the innermodel C ( aa ) , the theory of C ( aa ) is (set) forcing absolute, and C ( aa ) sat-isfies CH. We introduce an auxiliary concept that we call club determinacy,which simplifies the construction of C ( aa ) greatly but may have also inde-pendent interest. Based on club determinacy, we introduce the concept ofaa-mouse which we use to prove CH and other properties of the inner model C ( aa ) . This is the second part of a two-part paper on inner models obtained by means ofextended logics. A generally acknowledged weakness of G¨odel’s in many ways ∗ The authors are grateful to Ralph Schindler, John Steel, Philip Welch and Hugh Woodin forcomments on the results presented here. † Research partially supported by grant 40734 of the Academy of Finland. ‡ Research supported by the Simons Foundation and the Israel Science Foundation grant684/17. § Research supported by the Simons Foundation, the Faculty of Science of the University ofHelsinki, and grant 322795 of the Academy of Finland. L is that it cannot support large cardinals, beyond such “small”large cardinals as inaccessible, Mahlo, and weakly compact cardinals. In the so-called Inner Model Program inner models are built for bigger and bigger largecardinals, reaching currently as far as a Woodin limit of Woodin cardinals. Thesemodels resemble G¨odel’s L in that deep fine-structure can be established for themleading, among other things, to canonical proofs of CH, ✸ , ✷ , etc. in thoseinner models. While these so-called fine-structural inner models are extremelyuseful in almost all areas of modern set theory, it cannot be denied that they arebuilt somewhat “opportunistically”, by assuming a large cardinal and building acarefully crafted model around it. With our new inner models we look for a morecanonical inner model construction which would still have desirable properties.But what should one expect from a canonical inner model? First of all wepropose that we should expect robustness . We have in mind three meanings ofrobustness: (1) Stability of the model under changes in the definition (in the fixeduniverse of set theory). (2) Robustness across universes of set theory, stabilityunder forcing extensions. (3) The theory of the model (or an important part ofit) should be invariant under forcing extensions. A second quality we propose toexpect from a canonical inner model is completeness in the sense that canonicaldefinable objects should be included. A litmus test of this would be closure undersharps or other canonical operations.The first part [7] of this two-part paper dealt mainly with some general ques-tions concerning inner models obtained from extended logics, and more specifi-cally the inner model C ∗ defined by means of the cofinality quantifier [12]. In thissecond part we focus on the a priori bigger inner model C ( aa ) defined by meansof the stationary logic [1]. Note that L ⊆ C ∗ ⊆ C ( aa ) ⊆ HOD . (1)The main results about C ∗ in [7] were that under the assumption of a properclass of Woodin cardinals, the theory of C ∗ is set forcing absolute, uncountablecardinals > ω of V are weakly compact in C ∗ (and ω is Mahlo), and the theoryof C ∗ is independent of the cofinality used. Moreover, C ∗ is closed under sharps.We were not able to solve the problem of CH in C ∗ although we showed, assumingthree Woodin cardinals and a measurable above them, that for a cone of reals r the relativized inner model C ∗ [ r ] satisfies CH.Here we show that if there is a proper class of Woodin cardinals, then uncount-able cardinals of V are measurable in C ( aa ) , and the theory of the model C ( aa ) is invariant under set forcing. This raises naturally the question of the truth-value2f CH in C ( aa ) . We show, assuming a proper class of Woodin cardinals, or al-ternatively MM ++ , that C ( aa ) satisfies CH. Again, we point out that C ( aa ) isclosed under sharps. We also consider some variants of C ( aa ) .The models C ∗ and C ( aa ) arise from general considerations involving suchbasic set-theoretical concepts as cofinality and stationarity. It is quite remarkablethat we can achieve the level of robustness that these models manifest. It shouldcome as no news that we have to make set-theoretical assumptions before we canobtain robustness results for C ∗ and C ( aa ) . For example, if V = L , then bothmodels are simply identical to L . Our assumptions are either large cardinal axiomsor forcing axioms.There are two new tools that we develop for the proofs of the results men-tioned. The first tool is club determinacy which simplifies stationary logic consid-erably in our construction. Roughly speaking, club determinacy says that everystationary definable set of countable subsets of C ( aa ) contains a club. The secondtool is the concept of an aa-mouse . Roughly speaking, an aa-mouse consists ofa transitive set together with a theory formulated in stationary logic. Intuitivelyspeaking, the transitive set satisfies the theory-part, but this is not true in general.For example, it is not true if the transitive set is countable. The major part of thispaper is devoted to proving club determinacy under large cardinal assumptions, orMM ++ , and to developing the theory of aa-mice and, what we call, aa-ultrapowersof aa-mice.We feel that there are a wealth of questions worth studying about the new innermodels. At the end of the paper we list some such questions. Notation: If κ is a cardinal and M a set, we denote the set of subsets M ofcardinality < κ by P κ ( M ) . We use bold face a , b , x etc for finite sequences. ∀ x ϕ is short for ∀ x . . . ∀ x n ϕ and aa s ϕ is short for aa s . . . aa s n ϕ . If h is a functionand x ⊆ dom( f ) , then we use h [ x ] to denote the set { h ( y ) : y ∈ x } . H ( µ ) is theset of sets of hereditary cardinality less than µ . Let us recall that a set S of countable subsets of a set M is said to be closedunbounded (club) if for every countable s ⊆ M there is s ′ ∈ S such that s ⊆ s ′ ,and for every { s n : n < ω } ⊆ S such that ∀ n ( s n ⊆ s n +1 ) the set S n s n is in S , orequivalently, S is the set of countable subsets of M closed under a countable setof functions. The set S is stationary if it meets every club set of countable subsets3f M . Stationary logic is the extension of first order logic by the following secondorder quantifier:
Definition 2.1. If a is a finite sequence of elements of M and t is a finite sequenceof countable subsets of M , then we define M | = aa sϕ ( s, t , a ) if and only if { A ∈ P ω ( M ) : M | = ϕ ( A, t , a ) } contains a club of countablesubsets of M . We denote ¬ aa s ¬ ϕ by stat sϕ . The extension of first order logicby the quantifier aa is denoted L ( aa ) .This quantifier was introduced in [12] and studied extensively in [1]. The ideais that rather than asking whether there is some countable set A satisfying ϕ ( A ) , orwhether all countable sets A satisfy ϕ ( A ) , we ask whether most A satisfy ϕ ( A ) .The second order “some/all” quantifiers are generally believed to be too strong togive rise to interesting model theory, but the “most” quantifier has turned out tobe better behaved. There is a complete axiomatization, a Compactness Theoremin countable vocabularies, and a Downward L¨owenheim-Skolem Theorem downto ℵ for countable theories (i.e. every countable consistent theory has a model ofcardinality ℵ ).Some examples of the expressive power of stationary logic are the following:We can express “ ϕ ( · ) is countable” with aa s ∀ y ( ϕ ( y ) → s ( y )) . If we have alinear order ϕ ( · , · ) , we can express the proposition that it has cofinality ω with aa s ∀ x ∃ y ( ϕ ( x, y ) ∧ s ( y )) . We can express the proposition that ϕ ( · , · ) is ℵ -likewith ∀ x aa s ∀ y ( ϕ ( y, x ) → s ( y )) . The set { α < κ : cf( α ) = ω } is L ( aa ) -definable on ( κ, < ) by means of aa s (sup( s ) = α ) . The property of a set A ⊆{ α < κ : cf( α ) = ω } of being stationary is definable in L ( aa ) by meansof stat s (sup( s ) ∈ A ) . Finally, we can express the proposition that an ℵ -likelinear order ϕ ( · , · ) contains a closed unbounded subset (i.e. a copy of ω ) with aa s (sup( s ) ∈ dom( ϕ )) .The axioms of the logic L ( aa ) are [1]: ( A aa sϕ ( s ) ↔ aa tϕ ( t )( A ¬ aa s ( ⊥ )( A aa s ( x ∈ s ) , aa t ( s ⊆ t )( A
3) ( aa sϕ ∧ aa sψ ) → aa s ( ϕ ∧ ψ )( A aa s ( ϕ → ψ ) → ( aa sϕ → aa sψ )( A ∀ x aa sϕ ( x, s ) → aa s ∀ x ∈ sϕ ( x, s ) . (2)4hese are complete in the sense that any countable L ( aa ) -theory consistent withthem has a model of cardinality ℵ . Intuitively, (A1) says that ∅ is not club. (A2)says that the set of countable sets having a fixed element as an element, as wellas the set of countable sets containing a fixed countable set as a subset, are club.(A3) and (A4) simply say that the club-filter (of definable sets) is a filter. Finally(A5) is a formulation of Fodor’s Lemma.Suppose A is a stationary subset of a regular κ > ω such that ∀ α ∈ A (cf( α ) = ω ) . The ω -club filter F ω ( A ) is the set of subsets of A which contain a club ofsubsets of A . Note that F ω ( A ) is < κ -closed. The property of B ⊆ κ of belongingto F ω ( A ) is definable from A in L ( aa ) by means of aa s (sup( s ) ∈ A → sup( s ) ∈ B ) .More generally, if δ is an uncountable cardinal such that δ = δ <δ , we considerthe quantifier aa δ with the following meaning: If a is a finite sequence of elementsof M and t is a finite sequence of subsets of M of cardinality < δ , then we define M | = aa sϕ ( s, t , a ) if and only if { s ∈ P δ ( M ) : M | = ϕ ( s, t , a ) } contains a club of subsets of M ofcardinality < δ . It is proved in [11] that a sentence of L ( aa ) has a model if andonly if it has a model when aa is interpreted as aa δ . C ( aa ) Following the idea pursued in [7], we introduce a new inner model C ( aa ) ob-tained in the same way as G¨odel’s constructible hierarchy L , but replacing in thedefinition first order logic by the logic L ( aa ) . The general construction is asfollows: Definition 3.1. [7] Suppose L ∗ is a logic. If M is a set, let Def L ∗ ( M ) denote theset of all sets of the form X = { a ∈ M : ( M, ∈ ) | = ϕ ( a, b ) } , where ϕ ( x, y ) isan arbitrary formula of the logic L ∗ and b ∈ M . We define the hierarchy ( L ′ α ) of sets constructible using L ∗ as follows: L ′ = ∅ L ′ α +1 = Def L ∗ ( L ′ α ) L ′ ν = S α<ν L ′ α for limit ν .We use C ( L ∗ ) to denote the class S α L ′ α .5n the special case that L ∗ is L ( aa ) , we denote C ( L ( aa )) by C ( aa ) . We also consider the inner model C ( aa δ ) i.e C ( L ( aa δ )) . Since the quantifier Q cf ω ,which gives rise to the inner model C ∗ ( = C ( L ( Q cf ω )) ) is definable in L ( aa ) , wehave the trivial relations of (1). Lemma 3.2. C ( aa ) is a model of ZF C .Proof.
The claim follows from the general results of [7].In the course of this paper we will see that C ( aa ) is in many ways a fairlyrobust inner model, at least if there are big enough large cardinals.It is important to keep in mind that the quantifier aa s in the construction of C ( aa ) asks whether there is a club in V of countable sets s in V with someproperty. Thus, although we obtain an inner model C ( aa ) , we let the quantifier aa “reach out” to V . Thus C ( aa ) knows certain facts about V but it may not beable to have witnesses to corroborate those facts. The whole point of using L ( aa ) in the definition of C ( aa ) is that L ( aa ) provides some information about V butnot too much.The countable levels L ′ α , α < ω , bring nothing new. They are the same as therespective levels of the constructible hierarchy, as the aa -quantifier is eliminablein countable models.Note that S = { α < κ : cf V ( α ) = ω } ∈ C ( aa ) . The property of A ⊆ S of being stationary (in V ) is definable in C ( aa ) , as is the property of containinga club. Thus, if A ∈ C ( aa ) , then the “trace” of the ω -club filter of V on A ,namely ( F ω ( A )) V ∩ C ( aa ) , is in C ( aa ) . One of the main results of this paper isthat ( F ω ( κ )) V ∩ C ( aa ) is a normal ultrafilter on κ , whenever κ > ω is regular,assuming large cardinals.Many natural questions about C ( aa ) immediately suggest themselves: • Does it satisfy CH? • Does it have large cardinals? • How absolute is it? • Is its theory forcing absolute? • How is it related to other known inner models such as L , HOD, etc?6e will provide some answers in this paper but many natural questions remainalso unanswered. We shall prove C ( aa ) | = CH from large cardinal assumptions,but let us immediately observe that ZFC alone does not limit the cardinality of thecontinuum in C ( aa ) to neither ℵ nor to ≤ ℵ . This is in sharp contrast to thecase of C ∗ (see [7]) where the continuum is always ≤ ℵ of V . Below ℵ refersto ℵ of V . Theorem 3.3.
Con ( ZF ) implies Con ( | R ∩ C ( aa ) | ≥ ℵ ) .Proof. Assume V = L . Let S ⊆ ω be a non-reflecting stationary set of ordinalsof cofinality ω with fat (i.e. for every club C ⊆ ω , S ∩ C contains closed setsof ordinals of arbitrarily large order types below ω ) complement, consisting ofordinals of cofinality ω . Let S α , α < ω , be a partitioning of S into disjointstationary sets. Let us now work in a generic extension obtained by adding Cohenreals r α , α < ω . The sets S α are still stationary, because the forcing is CCC. Let A be the set { ω · α + n : n ∈ r α , α < ω } . Let E be the union of the sets S α ,where α ∈ A . Let us move to a forcing extension obtained by forcing a club D through the fat stationary set ω \ E . This forcing does not add bounded subsetsof ω , whence ω does not change. If α ∈ A , then S α ∩ D = ∅ and S α is thereforenon-stationary after the forcing. On the other hand, If α / ∈ A , then S α ∩ E = ∅ and shooting a club through ω \ E preserves the stationarity of S α . Hence forany α ∈ ω , α ∈ A if and only if S α is non-stationary. Hence A ∈ C ( aa ) .Now r α = { n : ω · α + n ∈ A } . Hence each r α is in C ( aa ) , and therefore | R ∩ C ( aa ) | ≥ ℵ .The role of ℵ in the above theorem is not crucial, but just an example. It canbe replaced by any cardinal. Since ZF C ⊢ | R ∩ C ∗ | ≤ ℵ [7], we obtain: Corollary 3.4.
Con ( ZF ) implies Con ( C ∗ = C ( aa )) . We do not know whether sufficiently large cardinals imply C ∗ = C ( aa ) , butit seems like a natural conjecture.We can use the proof of Theorem 3.3 to prove the consistency of the non-absoluteness of C ( aa ) in the sense that inside C ( aa ) the C ( aa ) may look differ-ent than from outside: Proposition 3.5.
Con ( ZF ) implies Con ( C ( aa ) C ( aa ) = C ( aa )) .Proof. We proceed as in the proof of Theorem 3.3. Assume V = L . Let S n , n < ω , be a partitioning of ω into disjoint stationary sets. Let us then work7n a generic extension obtained by adding a Cohen real r . The sets S α are stillstationary. Let E be the union of S n , where n ∈ r . Let us move to a forcingextension V [ G ] obtained by forcing a club D through the stationary set ω \ E .Now n ∈ r if and only if S n is non-stationary. Hence r ∈ C ( aa ) , whence L ( r ) ⊆ C ( aa ) and C ( aa ) = L . One can prove by induction on the construction ( L ′ α ) of C ( aa ) in V [ G ] that a subset S ∈ L ( r ) of ω is stationary in V [ G ] if and only if itis included (up to a non-stationary set) in some S n , n / ∈ r . Hence C ( aa ) ⊆ L ( r ) ,and, in consequence, we obtain C ( aa ) = L ( r ) . But C ( aa ) L ( r ) = L . Hence V [ G ] | = C ( aa ) C ( aa ) = C ( aa ) .If x ∈ C ( aa ) and x exists, then x ∈ C ( aa ) . This is proved as for C ∗ in [7].If L µ exists, then L ν ⊆ C ∗ for some ν , and hence L ν ⊆ C ( aa ) for some ν . Wedo not know whether L µ , where µ is a measure on the smallest possible ordinal,is contained in C ( aa ) . We introduce the useful auxiliary concept of club determinacy and show that C ( aa ) satisfies it, assuming large cardinals or MM ++ . Roughly speaking, clubdeterminacy says that definable sets of ordinals of cofinality ω in C ( aa ) eithercontain a club or their complement contains a club. This simplifies the structureof C ( aa ) as we do not have any definable stationary co-stationary sets. The mainresults of the later sections are heavily based on this. Definition 4.1 ([3]) . A first order structure M is club determined if M | = ∀ x [ aa sϕ ( x , s, t ) ∨ aa s ¬ ϕ ( x , s, t )] , where ϕ ( x , s, t ) is any formula in L ( aa ) and t is a finite sequence of countablesubsets of M .On a club determined structure the quantifier stat (“stationarily many”) and aa (“club many”) coincide on definable sets. The truth of aa sϕ ( s, b ) in a structure M can be written in the form of a two-person perfect information zero-sum game G ( ϕ, M , b ) : the players alternate to pick elements a , a , . . . from M . After ω moves Player II wins if s = { a , a , . . . } satisfies ϕ ( s, b ) in M . A structure M isclub determined if and only if the game G ( ϕ, M , b ) is determined for all formulas ϕ and all parameters b . Hence the name.8here are several results in [3] suggesting that club determined structures havea ‘better’ model theory than arbitrary structures . For a start, every consistentfirst order theory has a club determined model. Moreover, every club determineduncountable model has an L ( aa ) -elementary submodel of cardinality ℵ , whilefor arbitrary structures this cannot be proved in ZFC. It fails if V = L ([2]), butholds if we assume PFA ++ (see Theorem 4.17 below). Lemma 4.2.
If a first order structure M is club determined, then M | = ∀ x [ aa s ϕ ( x , s , t ) ∨ aa s ¬ ϕ ( x , s , t )] , (3) where ϕ ( x , s , t ) is any formula in L ( aa ) and t is a finite sequence of countablesubsets of M .Proof. Suppose ϕ ( x , s , . . . , s n , t ) is a formula in L ( aa ) , t is a finite sequenceof countable subsets of M , and x is a finite sequence of elements of M . We useinduction on n . If n = 1 , the claim is true by assumption. Suppose then n > and M | = ¬ aa s aa s . . . aa s n ϕ ( x , s , . . . , s n , t ) . By the assumption (3), M | = aa s ¬ aa s . . . aa s n ϕ ( x , s , . . . , s n , t ) , whence by the Induction Hypothesis, M | = aa s aa s . . . aa s n ¬ ϕ ( x , s , . . . , s n , t ) . Definition 4.3.
We say that the inner model C ( aa ) is club determined , or that club determinacy holds, if every level ( L ′ α , ∈ ) in the construction of C ( aa ) isclub determined as a first order structure.Intuitively speaking, if C ( aa ) is club determined, its definition is more robust—the quantifier aa is more lax than it would be otherwise, and in consequence, C ( aa ) is a little easier to compute.We consider club determinacy also with the quantifier aa interpreted as aa δ .We say that ( N, ∈ ) satisfies δ -club determinacy if it satisfies club determinacywith aa replaced by aa δ .The main technical result of this paper is says that if there are a proper class ofWoodin cardinals, then C ( aa ) is club determined (Theorem 4.12). We prove the In [3] the name “finitely determinate” is used. ++ (Theorem 4.20).In view of Theorem 5.1 below some large cardinal (in V or in an inner model)assumption is necessary for club determinacy. Of course, a proper class of mea-surable cardinals is a much weaker assumption than a proper class of Woodincardinals, and we do not know the exact large cardinal assumption needed here. We are going to prove club determinacy in two cases. The first case is a properclass of Woodin cardinals. This will be the topic of the current section. In thenext section we use the assumption MM ++ . MM ++ is consistent relative to asupercompact cardinal [4] and implies the consistency of a proper class of Woodincardinals.Suppose δ is a Woodin cardinal. We use P <δ to denote the stationary towerforcing at δ and Q <δ to denote the corresponding countable stationary tower forc-ing. For details concerning the stationary tower we refer to [8].Here is a sketch of the proof of club determinacy. We look at the earliest stageat which club determinacy fails for C ( aa ) . Let us suppose it fails because a set S = { s ∈ P ω ( L ′ α ) : L ′ α | = ϕ ( a , s, t ) } (4)is stationary co-stationary, where a is a finite sequence of elements of L ′ α and t is afinite sequence of countable subsets of L ′ α . We may assume that α and ϕ ( s, x ) areminimal for which this happens. We show with a separate argument that we canassume w.l.o.g. that | α | = ℵ and δ = ω . Let now δ be a Woodin cardinal. Weforce with Q <δ and obtain the associated generic embedding j : V → M ⊆ V [ G ] .In M the set j ( S ) is the set of s ∈ P ω ( L ′ α ) such that L ′ β | = ϕ ( j ( a ) , s, j ( t )) , where β = j ( α ) . We use the minimality of α and ϕ to argue that ( L ′ β ) M is the β th level,which we denote L ∗ β , of the hierarchy of C ( aa δ ) in V . We also show that j ( a ) isan element a ∗ of V , and j ( t ) is an element t ∗ of V . Finally, we argue that j ( S ) is, up to a club, in V because it is the set of subsets, smaller than δ , of L ∗ β whichsatisfy ϕ ( a ∗ , s, t ∗ ) in L ∗ β . This shows, essentially, that j ( S ) is independent of thegeneric G . But we can choose the generic so that includes S or so that it includesthe complement of S , because both are stationary. In the former case ω ∈ j ( S ) and in the latter case ω / ∈ j ( S ) . This contradicts the fact that j ( S ) is independentof G .The following general fact about forcing will be used below:10 emma 4.4. Suppose δ is a regular cardinal, P is a forcing notion such that | P | = δ , and G is P -generic. If δ is still a regular cardinal in V [ G ] , then for all N ∈ V , every club of ( P δ ( N )) V is stationary in ( P δ ( N )) V [ G ] .Proof. Without loss of generality, N is an ordinal β . Let C be a club in ( P δ ( N )) V .Suppose τ is a forcing term for an algebra on β . Let µ be a big enough regularcardinal. We build in V a chain M α , α < δ , of elementary substructures of H ( µ ) V of cardinality < δ in such a way that P , τ, β ∈ M , M α ∈ C , M ν = S α<ν M α for limit ν , and P ⊆ S α<δ M α . Let G be P -generic. Since δ is regular in V [ G ] ,we can construct, in V [ G ] , an ordinal γ < δ such that if D ⊆ P is a dense set in M γ , then D ∩ G ∩ M γ = ∅ . Now M γ ∩ β ∈ V is closed under the value of τ in V [ G ] .The main technical tool is proving the club determinacy is the following resultabout preservation of stationarity in the forcing Q <λ : Proposition 4.5.
Suppose that λ is Woodin and G is Q <λ generic over V . If S ⊆ λ and S ∈ V is stationary in V then S is stationary in V [ G ] .Proof. Suppose that S is not stationary in V [ G ] . Let τ be a Q <λ term for a clubsubset of λ forced to be disjoint from S . To simplify notation we assume that themaximal condition forces that τ ∩ S = ∅ . For every α < λ let D α be a maximalanti-chain of conditions which force some ordinal > α in to τ . For every α < λ let F α be the function defined on D α such that F α ( q ) is the minimal ordinal above α which is forced into τ by q . Let N be an elementary substructure of H ( κ ) for abig enough κ such that h D α : α < λ i and other relevant elements of the proof arein N . Also we require that N ∩ λ is an ordinal δ ∈ S and that V δ ⊆ N . Clearly V δ is closed under F α for every α < δ .We use the following definition : Definition 4.6.
Let D be a maximal anti-chain in Q λ . We say that X ∈ P ω ( V λ ) catches D below ρ if there is q ∈ D ∩ X ∩ V ρ such that X ∩ ∪ q ∈ q .The following definition is a modification to Q <λ of definition 2.5.1 of [8]. Definition 4.7.
Let D be a maximal anti-chain in Q <λ . We say that D is semiproperat ρ if for every X ≺ V ρ +2 , X countable, there is a countable Y ≺ V ρ +2 such that Y catches D below ρ and Y end extends X below ρ (i.e. if α ∈ ( Y − X ) ∩ ρ then α ≥ sup( X ∩ ρ ) ).The following fact follows immediately from the modification to Q <λ of the-orem 2.5.9 of [8]: 11 laim. For every D α there are unboundedly many inaccessible cardinals γ < λ such that D α is semiproper at γ .By N ≺ H ( κ ) and N ∩ λ = δ it follows that for every α < δ , there areunboundedly many γ < δ such that D α is semiproper at γ .In the following arguments we assume that V δ +2 is also endowed with a fixedwell order. Lemma 4.8.
For every countable X ≺ V δ +2 such that h D α : α < δ i ∈ X there isa countable Y ≺ V δ +2 such that X ⊆ Y and for every α ∈ Y ∩ δ , Y catches D α below δ .Proof. We define by induction an increasing sequence h X n : n < ω i of countableelementary substructures of V δ +2 where X = X , a sequence h α n : n < ω i ofordinals less than δ such that α n ∈ X n , and an increasing sequence h γ n : n < ω i , γ n ∈ X n , such that D α n is semiproper at γ n . By dovetailing we make sure thatfor every n < ω and α ∈ X n ∩ δ there is k such that α = α k . Also we keepthe inductive assumption that X n +1 catches D α n below γ n and that it is an endextension of X n below γ n . So it follows that X n +1 continues to catch D α k below γ k for all k < n .Given X n . Pick α n ∈ X n so as to continue our dovetailing process. Let γ n bean element of X n above γ n − such that D α n is semiproper at γ n . Such a γ n existsin X n since X n is an elementary substructure of V δ +2 and D α n ∈ X n . (Recall that h D α : α < λ i ∈ X ⊆ X n .)Since X n ≺ V δ +2 , we have R = X n ∩ V γ n +2 ≺ V γ n +2 , hence there is acountable Z ≺ V γ n +2 such that R ⊆ Z , Z is an end extension of R below γ n , and Z catches D α n below γ n . We define X n +1 to be all the elements of V δ +2 whichare definable in V δ +2 from X n ∪ Z . Clearly X n +1 ≺ V δ +2 . Claim. X n +1 ∩ V γ n = Z ∩ V γ n . Proof.
Clearly Z ∩ V γ n ⊆ X n +1 ∩ V γ n . For the other direction let a ∈ X n +1 ∩ V γ n .Then a is definable from some elements ~b of X n and an element c of Z by aformula ϕ ( x,~b, c ) . (It is enough to consider a single element c of Z , since Z is closed under forming finite sequences.) Consider the following function h : V γ n → V γ n . We let h ( y ) to be the unique element d of V γ n satisfying ϕ ( d,~b, y ) ifthere is such a unique element, and otherwise. Now h ∈ V δ +2 and h is definablein V δ +2 from ~b . Moreover, it is a function from V γ n to V γ n . So h ∈ X n ∩ V γ n +2 .So h ∈ Z . Clearly a = h ( c ) . Hence a = h ( c ) ∈ Z .12ontinuing the proof of Lemma 4.8, it follows from the claim that X n +1 endextends X n below γ n and catches D α n From our inductive assumptions it followsthat X n +1 catches D α k below γ k for all k ≤ n . Now, if we define Y = ∪ n X n ,then Y satisfies the requirements of the lemma.We continue the proof of Proposition 4.5 with the following: Claim.
The set T = { X ∈ P ω ( V δ +1 ) : X ≺ V δ +1 , X catches D α below δ for every α ∈ X ∩ δ . } (5)is stationary on P ω ( V δ +1 ) . Proof.
Assume otherwise, then there is a function g : V δ +1 → V δ +1 such thatevery countable X ⊆ V δ +1 which is closed under g is not in T . F can be codedas an element of V δ +2 . (We use g also for the code in V δ +2 .) Let X be a countableelementary substructure of V δ +2 containing g and the sequences h D α ∩ V δ : α < δ i .By the above lemma, there is a countable X ⊆ Y ≺ V δ +2 such that Y catches D α below δ for every α ∈ Y . It is obvious that Y ∩ V δ +1 ∈ T , but g ∈ Y so Y ∩ V δ +1 is closed under the function g , which is a contradiction.We can now finish the proof of Proposition 4.5. By the above claim T ∈ Q <λ .We claim that T (cid:13) δ ∈ τ . Suppose that T ′ ≤ T such that T ′ forces that α < δ is abound for τ ∩ λ . We can assume without loss of generality that for every Z ∈ T ′ α ∈ Z . Since T ′ ≤ T we can also assume that every Z ∈ T ′ catches D α below δ .For Z ∈ T ′ let G ( Z ) ∈ Z ∩ D α witness the fact that Z catches D α . By Fodor’slemma there is q ∈ D α such that the set T ′′ = { Z ∈ T ′ : G ( Z ) = q } is stationaryin ∪ T ′ . But T ′′ forces T ′′ ≤ q . But q forces some ordinal above α to be in τ ∩ δ .We have a contradiction. Proposition 4.9.
For every set A in V , if S ⊆ P δ ( A ) is stationary in V , then it isstationary in V [ G ] .Proof. This is proved as Proposition 4.5.Let ( L ′ α ) be the hierarchy of C ( aa ) in V . Let ( L ∗ α ) be the hierarchy of C ( aa δ ) in V . We will compare these two inner models, or rather C ( aa δ ) and the image of C ( aa ) under a generic ultrapower embedding. We use A | = δ ϕ to denote A | = ϕ when we think of ϕ as a sentence of L ( aa δ ) rather than of L ( aa ) .Suppose now δ is a Woodin cardinal. Let G be Q <δ -generic and j : V → M ⊆ V [ G ] the generic ultrapower embedding. Let ( L ′′ α ) be the hierarchy of C ( aa ) in13 . As a part of the proof that C ( aa ) satisfies club determinacy we show that C ( aa ) of M is actually C ( aa δ ) of V (see Proposition 5.2). We show this by alevel by level analysis of the two aa-hierarchies ( L ′′ α ) and ( L ∗ α ) .In the subsequent proofs we will use parameters from V although we are deal-ing also with M . The lemma below shows that while j is certainly not definablein V , it maps relevant parameters to V . First we need an auxiliary result. Thefollowing result is a widely known folklore result, but we include a sketch of theproof for the reader’s convenience: Lemma 4.10.
Assume x exists for every x ⊆ ω . Then δ = sup { (( ℵ V ) + ) L [ x ] : x ⊆ ω } . (6) Proof.
Kunen’s proof of the result of Martin ([9]) to the effect that every wellfounded Σ relation has rank < ω , shows that the rank is actually less than (( ℵ V ) + ) L [ x ] , where x is the real parameter of the Σ -definition. This gives onedirection of (6). For the other direction, suppose x is a real and η = (( ℵ V ) + ) L [ x ] .Every ordinal less than η is definable in L [ x ] from some x -indiscernibles ≤ ℵ V .This gives the other direction and finishes the proof of (6). We can define a re-lation between n -tuples of reals coding the indiscernibles and the formula. Thisrelation is ∆ using x as a parameter. The rank of the relation is η and therefore η < δ . Lemma 4.11.
Assume δ = ω and δ is a Woodin cardinal. Then j ↾ ω ∈ V . Inparticular, if s is a countable subset of ω , then j ( s ) ∈ V . Moreover, there is t ∈ V such that (cid:13) Q <δ j (ˇ s ) = ˇ t .Proof. Let, by Lemma 4.10, g ∈ V be a function on ω such that for all α < ω , g ( α ) is a subset of ω such that α < (( ℵ V ) + ) L [ g ( α )] . Let f ( α ) ∈ L [ g ( α )] be aone-one function α → ℵ V . Now j ( f ( α )) ∈ L [ j ( g ( α ))] = L [ g ( α )] and therefore j ( f ( α )) ∈ V . To see that j ↾ ω ∈ V it suffices to note that we can compute it in V : j ↾ ω = [ α<ω j ↾ f ( α ) − [ ℵ V ]) = j ( f ( α )) − ↾ j ( ℵ V ) = j ( f ( α )) − ↾ δ. Thus j ↾ ω ∈ V . Theorem 4.12.
If there is a proper class of Woodin cardinals, then C ( aa ) is clubdetermined. roof. Suppose α is the smallest ordinal for which L ′ α fails to satisfy club deter-minay. We can collapse | α | to ℵ without changing C ( aa ) . Hence we may assumew.l.o.g. | α | = ℵ . By a result of Shelah we can, starting from a Woodin cardinal,force the ω -saturation of the non-stationary ideal on ω with semi-proper forcing.This forcing does not change C ( aa ) up to the level α + 1 . Since we have also ameasurable cardinal, we may conclude that δ = ω ([15, Theorem 3.17]). Hencewe may assume, w.l.o.g. δ = ω . By Lemma 4.10 there is a real x and f ∈ L [ x ] such that f : L ′ α → ω V is a bijection. Suppose f is the least in the canonicalwell-order of L [ x ] . Let δ > α be a Woodin cardinal and j : V → M ⊆ V [ G ] asabove. Let β = j ( α ) . Now g = j ( f ) : L ′′ β → δ is a bijection and the L [ x ] -leastsuch. Hence g ∈ V .Suppose ϕ witnesses the failure of club determinacy at L ′ α , i.e. there is a set P = { s ∈ P ω ( L ′ α ) : L ′ α | = ϕ ( a , s, t ) } , where a ∈ L ′ α and t is a countable subset of L ′ α , which is stationary co-stationary.We may assume that ϕ is minimal for such a counter-example to exists.By the elementarity of j , j ( P ) is a counter-example to club determinacy of L ′′ β in M in the sense that j ( P ) = { s ∈ ( P δ ( L ′′ β )) M : ( L ′′ β | = ϕ ( j ( a ) , s, j ( t ))) M } is stationary co-stationary in M . Moreover, β is minimal for this to happen in M and ϕ is minimal for this to happen at L ′′ β in M .Let us say that L ∗ ξ satisfies weak δ -club determinacy if it satisfies club deter-minacy with the restriction that the parameters ( t in Definition 4.1) have to be in M . Lemma 4.13. L ∗ β = L ′′ β and weak δ -club determinacy holds for L ∗ β in V .Proof. We use induction on γ ≤ β to prove the following claim: Claim. L ∗ γ = L ′′ γ and weak δ -club determinacy holds for L ∗ γ in V .Let us suppose the claim holds for (and below) γ . We prove the claim for γ +1 .Let N = L ∗ γ = L ′′ γ . Thus N ∈ V ∩ M . Moreover, club determinacy holds for N in M and weak δ -club determinacy holds for N in V . We show by induction on ψ ∈ L ( aa ) that for finite sequences x in N : { a ∈ N : ( N | = δ ψ ( x , a )) V } = { a ∈ N : ( N | = ψ ( x , a )) M } . L ∗ γ +1 = L ′′ γ +1 . It suffices to prove the induction step for the aa -quantifier i.e. that for all x in N and all finite sequences t in ( P δ ( N )) V ∩ ( P ω ( N )) M the equality A = B holds, where A = { a ∈ N : ( N | = δ aa sθ ( x , s, t , a )) V } B = { a ∈ N : ( N | = aa sθ ( x , s, t , a )) M } , assuming we know for all s in ( P δ ( N )) V ∩ ( P ω ( N )) M : { a ∈ N : ( N | = δ θ ( x , s, t , a )) V } = { a ∈ N : ( N | = θ ( x , s, t , a )) M } . (7)Suppose a ∈ A i.e. V | = “ { s ∈ P δ ( N ) : N | = δ θ ( x , s, t , a ) } contains a club H ” . If a / ∈ B , then since club determinacy holds for N in M , M | = “ { s ∈ P ω ( N ) : N | = θ ( x , s, t , a ) } is disjoint from a club K ” . The set K is still a club in V [ G ] , as M ω ⊆ M . By Lemma 4.4 H is stationaryin V [ G ] . Therefore there is some s in K ∩ H . Note that s ∈ M ∩ V . Then onthe one hand V | = “ N | = δ θ ( x , s, t , a )” , since s ∈ H , and on the other hand M | = “ N = θ ( x , s, t , a )” , since s ∈ K , contrary to (7).For the converse, suppose a ∈ B i.e. M | = “ { s ∈ P ω ( N ) : N | = θ ( x , s, t , a ) } contains a club K ” . The set K is still a club in V [ G ] , as M ω ⊆ M . If a / ∈ A , then by weak δ -clubdeterminacy of N in V , V | = “ { s ∈ P δ ( N ) : N | = δ θ ( x , s, t , a ) } is disjoint from a club H ” . By Lemma 4.4 H is stationary in V [ G ] . Therefore there is some s in H ∩ K .Note that s ∈ M ∩ V . Then on the one hand V | = “ N = δ θ ( x , s, t , a )” , since s ∈ H , and on the other hand M | = “ N | = θ ( x , s, t , a )” , since s ∈ K . We have acontradiction with (7).This ends the proof that L ∗ γ +1 = L ′′ γ +1 . Now we prove weak δ -club determi-nacy for L ∗ γ +1 . Suppose S = { s ∈ P δ ( L ∗ γ +1 ) : L ∗ γ +1 | = δ θ ( x , s, t , a ) } , x ∈ L ∗ γ +1 and t ∈ P δ ( L ∗ γ +1 ) ∩ M , is stationary co-stationary in V . A proofsimilar to the proof of L ∗ γ +1 = L ′′ γ +1 above shows that S ∩ ( P ω ( L ′′ γ +1 )) M = T, (8)where T = { s ∈ ( P ω ( L ′′ γ +1 )) M : ( L ′′ γ +1 | = δ θ ( x , s, t , a )) M } . By Proposition 4.9 S is stationary co-stationary in V [ G ] . Recall that L ′′ γ +1 satisfiesclub determinacy in M . By club determinacy either T or P ω ( L ′′ γ +1 ) \ T containsa club in M . Suppose T contains a club K . Now K is a club also in V [ G ] , whencethere is s ∈ K \ S . Now s ∈ K and s ∈ ( P ω ( L ′′ γ +1 )) M but s / ∈ S , a contradictionwith (8). If P ω ( L ′′ γ +1 ) \ T contains a club, we get a contradiction in the same way.For limit ν ≤ β the proof of L ∗ ν = L ′′ ν and of weak δ -determinacy of L ∗ ν follows from the above proof.Since j ( a ) ∈ L ′′ β = L ∗ β , j ( a ) ∈ V ∩ M and, moreover, there is c ∈ V ∩ M such that (cid:13) Q <δ j (ˇ a ) = ˇ c . By Lemma 4.11, j ( t ) ∈ V ∩ M and there is u ∈ V ∩ M such that (cid:13) Q <δ j (ˇ t ) = ˇ u . Lemma 4.14.
For s ∈ P δ ( L ∗ β ) ∩ M : ( L ′′ β | = ϕ ( c , s, u )) M ⇐⇒ L ∗ β | = δ ϕ ( c , s, u ) . Proof.
As Lemma 4.13.Let us denote Q = { s ∈ P δ ( L ∗ β ) : L ∗ β | = δ ϕ ( c , s, u ) } . Note that Q is defined in V from parameters in V and therefore Q is in V . NowLemma 4.14 implies Q = j ( P ) . Let S ∈ V be the stationary co-stationary set { γ < ω : f − [ γ ] ∈ P } . It follows from g ∈ V that j ( S ) ∈ V and, moreover,there is T ∈ V such that (cid:13) Q <δ j ( ˇ S ) = ˇ T .Let G be Q <δ -generic, and j the associated embedding, such that S ∈ G .Let G be Q <δ -generic, and j the associated embedding, such that ω \ S ∈ G .Then ω ∈ j ( S ) = T and ω / ∈ j ( S ) = T , a contradiction. Corollary 4.15.
Suppose there is a supercompact cardinal. Then club determi-nacy holds.Proof.
Suppose κ is supercompact. Let α be the least such that L ′ α is not clubdetermined. Since V κ ≺ V , we can assume α < κ . Since κ is a Woodin cardinal,we can proceed as above. 17 .2 Club determinacy from MM ++ The proof of club determinacy from MM ++ can be proved following the samepattern as above from a proper class of Woodin cardinals. However, there is alsoa ‘soft’ argument which allows us to use the derivation of club determinacy froma proper class of Woodin cardinals, to make the same conclusion from MM ++ .Let us recall: Definition 4.16.
Martin’s Maximum ++ (MM ++ ) is the statement that for everystationary set preserving forcing P , any sequence h D α : α < ω i of dense opensubsets of P and any P -terms τ α , α < ω , such that P ⊢ τ α is a stationary subset of ω there is a filter F ⊆ P which meets every D α and for which each set { ξ : ∃ p ∈ F ( p (cid:13) ξ ∈ τ α ) } is stationary in ω . The ordinary Martin’s Maximum MM isMM ++ without the condition involving the terms τ α . PFA ++ is the same statementas MM ++ but assuming the forcing is proper.MM ++ is consistent relative to the consistency of a supercompact cardinal [4],and implies, among other things, that ℵ = ℵ and that the non-stationary idealon ω is saturated. Of course, MM ++ implies PFA ++ .The following Downward L¨owenheim-Skolem-Tarski Theorem for stationarylogic is well-known but we present a proof for the convenience of the reader. Wewill use this theorem later. Theorem 4.17 (Folklore) . Assume
PFA ++ . If A is a model with vocabulary τ and X ⊆ A such that | τ | + | X | ≤ ℵ , then there is an L ( aa ) -elementary submodel B of A such that X ⊆ B and | B | ≤ ℵ .Proof. Let P be the usual collapsing forcing which adds a generic F : ω → A by conditions which are countable partial functions ω → A . The forcing P iscountably closed and therefore proper. Moreover, P (cid:13) “A subset S of P ω ( ˇ A ) is stationary if and only if { α < ω : ˙ F [ α ] ∈ S } is stationary in ω ” . (9)Let D ( α ) be the set of conditions p with α ∈ dom( p ) . For every x ∈ X , let D ( x ) be the set of conditions p with x ∈ rng( p ) . Let θ n,mk ( s , . . . , s n , x , . . . , x m ) ,where n, m, k < ω , enumerate all L ( aa ) -formulas with free set-variables s , . . . , s n x , . . . , x n . We denote δ , . . . , δ n by δ , and η , . . . , η n by η .Let D ( k, n, δ , η ) be the set of conditions p such that δ , η ∈ dom( p ) , andthere is β ∈ dom( p ) such that p (cid:13) “ ˇ A | = ∃ yθ k ( y, ˙ F [ ˇ δ ] , . . . , ˙ F [ ˇ δ n ] , ˙ F ( ˇ η ) , . . . , ˙ F ( ˇ η m )) → ˇ A | = θ k ( ˙ F ( ˇ β ) , ˙ F [ ˇ δ ] , . . . , ˙ F [ ˇ δ n ] , ˙ F ( ˇ η ) , . . . , ˙ F ( ˇ η m ))” . Suppose p is a condition such that β , δ , η ∈ dom( p ) , and p (cid:13) “ ˇ A | = aa sθ k ( s, ˙ F [ ˇ δ ] , . . . , ˙ F [ ˇ δ n ] , ˙ F ( ˇ η ) , . . . , ˙ F ( ˇ η m )) . Then there is q p = q p ( j, n, δ , η ) ≤ p and a club C p = C p ( j, n, δ , η ) ⊆ ω (in V )such that q p (cid:13) ∀ α ∈ ˇ C p (“ ˇ A | = θ k ( ˙ F [ α ] , ˙ F [ ˇ δ ] , . . . , ˙ F [ ˇ δ n ] , ˙ F ( ˇ η ) , . . . , ˙ F ( ˇ η m ))”) . (10)Let D ( k, n, β, δ , η ) be the set of q p for p ∈ P .The P -term σ k,n, δ , η is defined as follows: Suppose p ∈ P . If p (cid:13) “ A | = stat sθ ξ ( s, ˙ F [ ˇ δ ] , . . . , ˙ F [ ˇ δ n ] , ˙ F ( ˇ η ) , . . . , ˙ F ( ˇ η m ))” , then (recall (9)) p forces σ k,n, δ , η to be a name for the set of β < ω such that A | = θ k ( F [ β ] , F [ δ ] , . . . , F [ δ k ] , F ( η ) , . . . , F ( η n )) . Otherwise p forces σ k,n, δ , η to be ω . Note that now every condition forces σ k,n, δ , η to be stationary.By PFA ++ there is a filter G ⊆ P such that G meets each D ( α ) , D ( x ) , D ( k, n, δ , η ) and D ( k, n, β, δ , η ) , where k, n, m < ω , x ∈ X , α, β , δ , η arecountable ordinals, and in addition, all the sets σ Gk,n, δ , η are stationary. Clearly, F G is a total function ω → A . Let B be the unique submodel of A of cardinality ≤ ℵ with universe F G [ ω ] . Since G meets the dense sets D ( k, n, δ , η ) , it is easyto see that B is closed under the interpretations of the function symbols of τ andcontains interpretations of the constant symbols of τ , so B is really a submodel of A . Moreover, since G meets each D ( x ) , X ⊆ B .To show that B is an L ( aa ) -elementary submodel of A we use induction on ϕ ( s , . . . , s n , t , . . . , t m ) ∈ L ( aa ) to prove: If δ , η < ω , then the following areequivalent: 19Tr1) B | = ϕ ( F G [ δ ) , ..., F G [ δ n ] , F G ( η ) , ..., F G ( η m )) (Tr2) A | = ϕ ( F G [ δ ) , ..., F G [ δ n ] , F G ( η ) , ..., F G ( η m )) .The case of the existential quantifier follows from the fact that G meets thedense set D ( k, n, δ , η ) with the appropriate k . Suppose ϕ ( s , . . . , s n , t , . . . , t m ) is the formula aa sθ k ( s, s , . . . , s n , t , . . . , t m ) . The implication (Tr2) → (Tr1) fol-lows from the Induction Hypothesis and the definition of D ( k, n, β, δ , η ) . Theimplication (Tr1) → (Tr2) follows from the Induction Hypothesis and the fact that σ Gr,n, δ , η is stationary, where r is such that ¬ θ k = θ r . Lemma 4.18.
Assume MM ++ . Suppose α is the least α such that club determi-nacy fails for L ′ α . Then | α | ≤ ℵ V .Proof. By Theorem 4.17 there is an L ( aa ) -elementary submodel A of ( L ′ α , ∈ ) ofcardinality ≤ ℵ containing the parameters of the failure of club determinacy. Let ( M ∈ ) ∼ = A be the transitive collapse and γ = M ∩ On . It is easy to prove byinduction on β ≤ γ that ( L ′ β ) M = L ′ β . Hence M = L ′ γ . Also, again since A is an L ( aa ) -elementary submodel of ( L ′ α , ∈ ) , club determinacy fails at L ′ γ . Hence bythe minimality of α , necessarily | α | ≤ ℵ . Proposition 4.19 ([6]) . Assuming PFA, there is, for every set X , an inner modelwith a proper class of Woodin cardinals, containing X .Proof. We modify Theorem 0.1 of [6] as follows. Suppose X is an arbitrary setof ordinals, e.g. X ⊆ δ . Let an X - mouse be a mouse as in [6] except that themouse is assumed to contain X and, moreover, it is required that all the extenderson the coherent sequence have the critical point above δ . With this modificationthe proof of Theorem 0.1 in [6] gives the result that if ✷ ( κ ) and ✷ κ + fail forsome κ > δ , then there is inner model with a proper class of Woodin cardinalscontaining X . Theorem 4.20.
Assuming MM ++ , club determinacy holds for C ( aa ) .Proof. Suppose club determinacy fails at L ′ α and α is minimal. By Lemma 4.18, | α | ≤ ℵ . Let X contain everything we need for the failure of club determinacy,e.g. X = V ω . By Proposition 4.19 there is an inner model M with a proper classof Woodin cardinals such that M contains X . By the choice of X , M fails tosatisfy club determinacy. But this contradicts Theorem 4.12.20 Applications of club determinacy
We give three types of applications of club determinacy. The first is the immediateconsequence that uncountable cardinals are measurable in C ( aa ) . Our large car-dinal assumption in the proof of club determinacy was a proper class of Woodincardinals, so we are far from an optimal result. Our second application is theforcing absoluteness of the theory of C ( aa ) . Here we assume a proper class ofWoodin cardinals and use club determinacy merely as a tool in the proof. Ourthird and more substantial application is a proof of CH in C ( aa ) , using clubdeterminacy. Recall that, assuming a proper class of Woodin cardinals, uncountable cardinalsare Mahlo in C ∗ , and even weakly compact above ℵ . In [7] we were not ableto prove that there are measurable cardinals in C ∗ under any assumption, evenconsistently. For the conceivably bigger inner model C ( aa ) we establish now themeasurability of all uncountable regular cardinals. As it turns out, the proof is animmediate consequence of club determinacy. Theorem 5.1.
Suppose C ( aa ) is club determined. Then every regular κ ≥ ℵ ismeasurable in C ( aa ) .Proof. Let F aa = F ω ( { α < κ : cf( α ) = ω } ) ∩ C ( aa ) . This is a normal filter on κ in C ( aa ) . Suppose X ⊆ κ is in C ( aa ) . There is α > κ such that X = { β < κ : L ′ α | = ϕ ( a, b ) } for some b ∈ L ′ α . Since L ′ α in club determined, L ′ α | = aa s ∃ x ( x = sup( s ) ∧ x < κ ∧ ϕ ( x, b )) or L ′ α | = aa s ¬∃ x ( x = sup( s ) ∧ x < κ ∧ ϕ ( x, b )) . In the first case X ∈ F aa . In the second case κ \ X ∈ F aa .It remains open whether club determinacy, or some reasonable stronger as-sumption, implies that there are higher measurable cardinals in C ( aa ) . By Corol-lary 5.25 below, we cannot hope to have Woodin cardinals in C ( aa ) as a con-sequence of some large cardinal assumptions. It remains open what happens tosingular cardinals. Are they regular, or even large cardinals in C ( aa ) ?21 .2 Forcing absoluteness The first order theory of L ( R ) is absolute under set forcing, assuming a properclass of Woodin cardinals. With a stronger assumption the same is true of theChang model C ( L ω ω ) . We can prove the absoluteness of C ( aa ) under set forc-ing assuming a proper class of Woodin cardinals. Proposition 5.2.
Suppose club-determinacy holds, δ is Woodin, G ⊆ Q <δ isgeneric and M is the associated generic ultrapower. Then C ( aa ) M = C ( aa δ ) V .Proof. We assume club determinacy. By elementarity, also C ( aa ) M satisfiesclub determinacy. Now we show by induction on α and on the L ( aa ) -formula ϕ ( s, z , y ) that if we denote N = ( L ′ α ) M and assume, as part of the InductionHypothesis, N = ( L ∗ α ) V , then for every b ∈ N and t ∈ P δ ( N ) ∩ V ∩ M thefollowing are equivalent: (A1) ( N | = δ aa sϕ ( s, t , b )) V . (A2) ( N | = aa sϕ ( s, t , b )) M .Suppose first (A1). Let C be a club in V of sets s ∈ P δ ( N ) satisfying ( N | = δ ϕ ( s, t , b )) V . By the Induction Hypothesis for every s ∈ C ∩ M , ( N | = ϕ ( s, t , b )) M . By (4.9), C is still stationary in V [ G ] , hence C ∩ M is stationary in M . Since M satisfies club determinacy there must be in M a club of s satisfying ( N | = ϕ ( s, t , b )) M , and (A2) follows.For the other direction: suppose (A2) i.e. in M there is a club D of countablesets s such that ( N | = ϕ ( s, t , b )) M . Suppose that (A1) fails. Hence the set S of s ∈ P δ ( N ) in V that satisfy ( N | = δ ¬ ϕ ( s, t , b )) V is stationary in V . By (4.9) itis stationary in V [ G ] . Let s ∈ D ∩ S . Now s ∈ P δ ( N ) ∩ V ∩ M , so we have acontradiction with the Induction Hypothesis. Theorem 5.3.
Suppose there are a proper class of Woodin cardinals. Then thefirst order theory of C ( aa ) is (set) forcing absolute.Proof. Suppose P is a forcing notion and δ be a Woodin cardinal > | P | . Let j : V → M be the associated elementary embedding. By Proposition 5.2 we canargue C ( aa ) ≡ ( C ( aa )) M = C ( aa δ ) . On the other hand, let H ⊆ P be generic over V . Then δ is still Woodin in V [ H ] , so we have the associated elementary embedding j ′ : V [ H ] → M ′ . By22roposition 5.2 we can again argue ( C ( aa )) V [ H ] ≡ ( C ( aa )) M ′ = ( C ( aa δ )) V [ H ] . Finally, we may observe that ( C ( aa δ )) V [ H ] = C ( aa δ ) , since | P | < δ . Hence ( C ( aa )) V [ H ] ≡ C ( aa ) . The fact (Theorem 5.3) that under large cardinal hypotheses the theory of C ( aa ) is forcing absolute, strongly suggest that we should be able to determine the truthvalue of the Continuum Hypothesis in C ( aa ) . Indeed, we now use club determi-nacy to prove the Continuum Hypothesis in C ( aa ) (Theorem 5.23 below). Theproof uses the auxiliary concepts of an aa-mouse and an aa-ultrapower, whichhave hopefully also other uses in the study of C ( aa ) . For example we use thembelow also to prove ✸ in C ( aa ) . Our method does not seem to yield GCH in C ( aa ) , although we have at the moment no reason to believe that GCH would beviolated in C ( aa ) . Our previous paper [ ] gives the consistency of the failure ofCH in C ∗ relative to the consistency of ZFC. This result extends to C ( aa ) (seeTheorem 3.3 above). Convention:
In the rest of this Section we assume club determinacy.
Our proof uses a new inner model concept which we call aa-premouse. Roughlyspeaking, an aa-premouse is a pair ( M, T ) , where M is a model and T is an L ( aa ) -theory. Here M can very well by countable and in countable domains the aa -quantifier is eliminable, so in general we do not assume M to be a modelof T . Rather, M is a model that has potential to become a model of T . Wefulfil this potential by building an ω V -chain of elementary extensions of M withthe idea that in the limit the theory T is really true. For this purpose we definean ultrapower construction—called the aa-ultrapower—for aa-premice. It allowsus to iterate a well-chosen countable aa-premouse (iterable aa-premice are calledaa-mice) to a big uncountable aa-premouse ( M ′ , T ′ ) where T ′ is actually truein M ′ . We use the concepts of aa-premouse and aa-ultrapower to prove CH in C ( aa ) . The proof is reminiscent of Silver’s proof of GCH in L µ [13]. Since we23ssume club determinacy, ω V is actually a measurable cardinal in C ( aa ) . Thusfrom the point of view of C ( aa ) we start with a countable premouse and iterateit a measurable cardinal times until we end up with a model of the size of themeasurable cardinal. Definition 5.4. An aa-premouse is a pair ( M, T ) , where1. M is a structure with some given vocabulary τ containing the binary pred-icate symbols ∈ and ≺ as well as possibly some unary predicate symbols.In addition we shall use constant symbols c a (not included in τ ) for a ∈ M .The interpretation of c a is a . (When no confusion arises, we use M to de-note also the domain of the structure M )2. T is a complete consistent (w.r.t. the axioms (2)) L ( aa ) -theory in the vo-cabulary τ ∪ { c a : a ∈ M } .3. T contains the first order diagram of M in the vocabulary τ ∪ { c a : a ∈ M } .4. If ∃ xϕ ( x, c a , . . . , c a n ) is in T , then there is a ∈ M such that the sentence ϕ ( c a , c a , . . . , c a n ) is in T .5. T contains the L ( aa ) -sentences ∀ y ( aa s ∃ xϕ ( s, x, y ) → aa s ∃ x ( ϕ ( s , x, y ) ∧ ∀ z ( z ≺ x → ¬ ϕ ( s, z, y )))) for all ϕ ( s, x, y ) in L ( aa ) .Condition 5 simply says that if we can find for a club of s an x such that ϕ ( s, x, y ) , then for a club of s we can find a ≺ - minimal x such that ϕ ( s, x, y ) .This assumption allows us to have, in a sense, definable Skolem-functions.We say that the aa-premouse ( M, T ) is club determined if for all L ( aa ) -formulas ϕ ( s, x ) and all a , . . . , a n ∈ M : aa sϕ ( s, c a , . . . , c a n ) ∈ T or aa s ¬ ϕ ( s, c a , . . . , c a n ) ∈ T. Definition 5.5.
Suppose ( M, T ) is an aa-premouse with vocabulary τ . A mapping π : M → M ′ is called an elementary embedding of ( M, T ) into ( M ′ , T ′ ) , insymbols π : ( M, T ) → ( M ′ , T ′ ) , π is an elementary embedding (in the usual sense) M → M ′ ↾ τ and for all ϕ ( x a , . . . , x a n ) ∈ L ( aa ) with vocabulary τ and all a , . . . , a n ∈ M , ϕ ( c a , . . . , c a n ) ∈ T ⇐⇒ ϕ ( c π ( a ) , . . . , c π ( a n ) ) ∈ T ′ . If π is the identity mapping, we call ( M, T ) an elementary substructure of ( M ′ , T ′ ) and write ( M, T ) ( M ′ , T ′ ) . Example 5.6.
The canonical example of an aa-premouse is the pair N = (( L ′ α , ∈ , ≺ ) , Th L ( aa ) (( L ′ α , ∈ , ≺ ))) , where Th L ( aa ) (( L ′ α , ∈ , ≺ )) is the set of sentences of L ( aa ) true in ( L ′ α , ∈ , ≺ ) with constants from L ′ α , and ≺ is the canonical well-order of L ′ α . Note that theaa-quantifier is interpreted in V , not in L ′ α . Note also that N ∈ C ( aa ) . Since weassume club determinacy, this aa-premouse is club determined. We obtain otherexamples of aa-premice by taking elementary substructures M = (( M, ∈ , ≺ ) , T ) N . (11)Since N ∈ C ( aa ) , we can take such an M also inside C ( aa ) , which is im-portant for our proof of the CH in C ( aa ) . We write N in shortened notation as ( L ′ α , Th L ( aa ) ( L ′ α )) and M as ( M, T ) , when no confusion can arise. Lemma 5.7.
The aa-premouse ( M, T ) of Example 5.6 is club determined.Proof. The claim follows from the fact that if aa sϕ ( s, c a , . . . , c a n ) ∈ Th L ( aa ) ( L ′ α ) , where a , . . . , a n ∈ M , then by (11), aa sϕ ( s, c a , . . . , c a n ) ∈ T. To simplify notation we use a to denote c a , . . . , c a n . If aa s ∃ xϕ ( s, x, a ) ∈ T ,we use the term f ϕ ( s,x, a ) ( s ) to denote the ≺ -minimal x intuitively satisfying ϕ ( s, x, a ) , i.e. we work in aconservative extension of T , denoted also T , which contains: aa s ∃ xϕ ( s, x, a ) → aa s ( ϕ ( s, f ϕ ( s,x, a ) ( s ) , a ) ∧∀ z ( z ≺ f ϕ ( s,x, a ) ( s ) → ¬ ϕ ( s, z, a ))) . In particular, the sentence aa s ∃ xϕ ( s, x, a ) → aa sϕ ( s, f ϕ ( s,x, a ) ( s ) , a ) is in T . 25 .3.2 The aa-ultrapower We define now what we call the aa-ultrapower ( M ∗ , T ∗ ) of an aa-premouse ( M , T ) . We do not use an ultrafilter for the construction, but rather the family F of L ( aa ) -definable sets which contain a club of countable subsets of M . Since weassume club determinacy, this family behaves sufficiently like an ultrafilter. Thus,intuitively we define M ∗ = def M P ω ( M ) / F , where P ω ( M ) is computed in V . However, we cannot define M ∗ in this way, atleast if we want to build M ∗ inside C ( aa ) . We certainly cannot count on P ω ( M ) being in C ( aa ) , even though M ∈ C ( aa ) , and even though we can define sets in C ( aa ) by reference to clubs in P ω ( M ) .In order to decide CH in C ( aa ) , we want to build M ∗ in C ( aa ) and thereforewe modify the usual ultraproduct construction in a special way. Instead of defining M ∗ as the set of equivalence classes of definable functions f : P ω ( M ) → M , wedefine M ∗ as the set of equivalence classes of L ( aa ) -formulas ϕ ( s, x ) that definefunctions f : P ω ( M ) → M .Let us now go into the details. Suppose ( M, T ) is an aa-premouse withvocabulary τ . Let M ′ be the set of ϕ ( s, x, a ) in L ( aa ) where a ∈ M and aa s ∃ xϕ ( s, x, a ) ∈ T . Define in M ′ : ϕ ( s, x, a ) ∼ ϕ ′ ( s, x, a ′ ) if and only if aa s ( f ϕ ( s,x, a ) ( s ) = f ϕ ′ ( s,x, a ′ ) ( s )) ∈ T. Note that ∼ is an equivalence relation in M ′ . Moreover, if (1) R ∈ τ , (2) thesentence aa sR ( f ϕ ( s,x, a ) ( s ) , . . . , f ϕ n ( s,x, a n ) ( s )) is in T , and (3) ϕ i ( s, x, a i ) ∼ ϕ ′ i ( s, x, a ′ i ) for i = 1 , . . . , n , then we may easilyconclude that aa sR ( f ϕ ′ ( s,x, a ′ ) ( s ) , . . . , f ϕ ′ n ( s,x, a ′ n ) ( s )) is in T . Thus we can formthe set M ∗ of equivalence classes [ ϕ ( s, x, a )] of ∼ and define in M ∗ the interpre-tation of the predicates R ∈ τ as follows: R M ∗ ([ ϕ ( s, x, a )] , . . . , [ ϕ n ( s, x, a n )]) ⇐⇒ aa sR ( f ϕ ( s,x, a ) ( s ) , . . . , f ϕ n ( s,x, a n ) ( s )) ∈ T. canonical embedding j : M → M ∗ is defined by j ( a ) = [ x = a ] . It is clearlyone-one and satisfies M | = R ( a , . . . , a n ) ⇐⇒ R ( a , . . . , a n ) ∈ T ⇐⇒ aa sR ( f x = a ( s ) , . . . , f x = a n ( s )) ∈ T ⇐⇒ M ∗ | = R ([ x = a ] , . . . , [ x = a n ]) ⇐⇒ M ∗ | = R ( j ( a ) , . . . , j ( a n )) for R ∈ τ . Definition 5.8.
Let τ ∗ = τ ∪ { P ∗ } , where P ∗ is a new unary predicate symbol.We make M ∗ a τ ∗ -structure by defining ( P ∗ ) M ∗ = { j ( a ) : a ∈ M } . We let T ∗ consist of ψ ( P ∗ , [ ϕ ( s, x, a )] , . . . , [ ϕ n ( s, x, a )]) , where ψ ( s, x , . . . , x n ) is a τ -formula of L ( aa ) , and aa sψ ( s, f ϕ ( s,x, a ) ( s ) , . . . , f ϕ n ( s,x, a ) ( s )) ∈ T. Lemma 5.9.
The pair ( M ∗ , T ∗ ) is an aa-premouse with vocabulary τ ∗ . It is clubdetermined if ( M, T ) is. The mapping j is an elementary embedding ( M, T ) → ( M ∗ , T ∗ ) (in the sense of Definition 5.5).Proof. T ∗ is consistent: Suppose ϕ i ( P ∗ , [ ϕ ( t, x, a )] , . . . , [ ϕ n ( t, x, a )]) , i = 1 , . . . , m ,is a finite set of sentences in T ∗ such that m ^ i =1 ϕ i ( P ∗ , [ ϕ ( s, x, a )] , . . . , [ ϕ n ( s, x, a )]) ⊢ ⊥ . (12)By the definition of T ∗ , for i = 1 , . . . , m m ^ i =1 aa tϕ i ( t, f ϕ ( t,x, a ) ( t ) , . . . , f ϕ n ( t,x, a ) ( t )) ∈ T, whence aa t m ^ i =1 ϕ i ( t, f ϕ ( t,x, a ) ( t ) , . . . , f ϕ n ( t,x, a ) ( t )) ∈ T. It can be shown by induction on L ( aa ) -proofs that (12) implies aa t m ^ i =1 ϕ i ( t, f ϕ ( t,x, a ) ( t ) , . . . , f ϕ n ( t,x, a ) ( t )) ⊢ aa s ⊥ , aa s ⊥ ∈ T , contrary to the consistency of T . T ∗ is complete: This follows immediately from the club determinacy of T . T ∗ satisfies club determinacy: By Lemma 4.2, aa t aa sϕ ( t, f ϕ ( s,x, a ) ( t ) , . . . , f ϕ n ( s,x, a ) ( t )) ∈ T or aa t aa s ¬ ϕ ( t, f ϕ ( s,x, a ) ( t ) , . . . , f ϕ n ( s,x, a ) ( t )) ∈ T. Hence aa sϕ ( P ∗ , [ ϕ ( s, x, a )] , . . . , [ ϕ n ( s, x, a )]) ∈ T ∗ or aa s ¬ ϕ ( P ∗ , [ ϕ ( s, x, a )] , . . . , [ ϕ n ( s, x, a )]) ∈ T ∗ . Thus T ∗ satisfies club determinacy.Finally, j is elementary, because ϕ ( a , . . . , a n ) ∈ T ⇐⇒ aa sϕ ( f x = a ( s ) , . . . , f x = a n ( s )) ∈ T ⇐⇒ ϕ ([ x = a ] , . . . , [ x = a n ]) ∈ T ∗ ⇐⇒ ϕ ( j ( a ) , . . . , j ( a n )) ∈ T ∗ . Lemma 5.10. j [ M ] = M ∗ .Proof. We consider [ ¬ x ∈ s ] ∈ M ∗ . Suppose [ ¬ x ∈ s ] = j ( a ) for some a ∈ M , i.e. [ ¬ x ∈ s ] = [ x = a ] . Then aa s ( f ¬ x ∈ s ( s ) = f x = a ( s )) ∈ T , whence aa s ( a s ) ∈ T . But by Axiom (A2) of stationary logic aa s ( a ∈ s ) ∈ T , acontradiction. Lemma 5.11. ψ ( P ∗ , [ ϕ ( s, x, a )] , . . . , [ ϕ n ( s, x, a )]) ∈ T ∗ if and only if aa sψ ( s, f ϕ ( s,x, a ) ( s ) , . . . , f ϕ n ( s,x, a ) ( s )) ∈ T. Proof.
The “if”-part is true by definition. For the “only if”-part, suppose aa sψ ( s, f ϕ ( s,x, a ) ( s ) , . . . , f ϕ n ( s,x, a ) ( s )) T. Then by club determinacy aa s ¬ ψ ( s, f ϕ ( s,x, a ) ( s ) , . . . , f ϕ n ( s,x, a ) ( s )) ∈ T, whence ¬ ψ ( P ∗ , [ ϕ ( s, x, a )] , . . . , [ ϕ n ( s, x, a )]) ∈ T ∗ , contrary to the consistency of T ∗ . 28 emma 5.12. aa s ∀ x ( P ∗ ( x ) → s ( x )) ∈ T ∗ i.e. P ∗ is “countable” in the senseof T ∗ .Proof. Clearly, aa t aa s ∀ x ( t ( x ) → s ( x )) ∈ T . Hence by definition 5.8, aa s ∀ x ( P ∗ ( x ) → s ( x )) ∈ T ∗ . The following lemma summarises the basic property of ultrapowers. In ourcase it holds trivially because we have built T ∗ in a way which makes it true. Theusual Ło´s Lemma talks about truth in the ultrapower vs. truth in the model. In thecase of an aa-premouse ( M, T ) we cannot talk about truth in M , but just aboutwhat the theory T says. Lemma 5.13 (Ło´s Lemma) . Suppose ( M, T ) is a club determined aa-premouseand ( M ∗ , T ∗ ) is its aa-ultrapower. The following holds for L ( aa ) -formulas ψ ( x , . . . , x n ) in the vocabulary τ ∗ : aa sψ ( f ϕ ( s,x, a ) ( s ) , . . . , f ϕ n ( s,x, a n ) ( s )) ∈ T if and only if ψ ([ ϕ ( s, x, a )] , . . . , [ ϕ n ( s, x, a n )]) ∈ T ∗ . Proof.
The claim follows trivially from Lemma 5.11.
Lemma 5.14. If ∃ xψ ( P ∗ , x, [ ϕ ( s, x, a )] , . . . , [ ϕ n ( s, x, a n )]) ∈ T ∗ , then there is b ∈ M ∗ such that ψ ( P ∗ , b, [ ϕ ( s, x, a )] , . . . , [ ϕ n ( s, x, a n )]) ∈ T ∗ .Proof. Suppose ∃ xψ ( P ∗ , x, [ ϕ ( s, x, a )] , . . . , [ ϕ n ( s, x, a )]) ∈ T ∗ . By the defini-tion of T ∗ , aa s ∃ xψ ( s, x, f ϕ ( s,x, a ) ( s ) , . . . , f ϕ n ( s,x, a ) ( s )) ∈ T. Hence aa sψ ( s, f ψ ′ ( s,x, a ) ( s ) , f ϕ ( s,x, a ) ( s ) , . . . , f ϕ n ( s,x, a ) ( s )) ∈ T, where ψ ′ ( s, x, a ) is the formula ψ ( s, x, f ϕ ( s,x, a ) ( s ) , . . . , f ϕ n ( s,x, a ) ( s )) . Hence ψ ( P ∗ , [ ψ ′ ( s, x, a )] , [ ϕ ( s, x, a )] , . . . , [ ϕ n ( s, x, a )]) ∈ T ∗ . Definition 5.15 (The aa-ultrapower) . We call the aa-premouse ( M ∗ , T ∗ ) the aa-ultrapower of ( M, T ) . 29 emma 5.16. Suppose ( M, T ) is a countable premouse with vocabulary τ and π : ( M, T ) → ( N, Th L ( aa ) ( N )) is an elementary embedding, where N is an expansion of L ′ α to a τ -structure and cf V ( α ) > ω . There are P + ⊆ L ′ α and an elementary π ∗ : (( M ∗ , P ∗ ) , T ∗ ) → (( N, P + ) , Th L ( aa ) (( N, P + ))) such that π ∗ ( j ( a )) = π ( a ) for all a ∈ M .Proof. Suppose [ ϕ ( s, x, a )] ∈ M ∗ . Then aa s ∃ xϕ ( s, x, a ) ∈ T , whence N | = aa s ∃ xϕ ( s, x, π ( a )) . Let C a ,ϕ be a club of countable subsets s of L ′ α such that N | = ∃ xϕ ( s, x, π ( a )) . Let Q be the intersection of the countably many C a ,ϕ where a ∈ M and ϕ ∈ L ( aa ) . Let us fix S ∈ Q . Thus for all a ∈ M and ϕ thereis a ≺ -least z a ,ϕ ∈ N such that N | = ϕ ( S, z a ,ϕ , π ( a )) . We let π ∗ ([ ϕ ( s, x, a )]) = z a ,ϕ and P + = S ∩ rng( π ∗ ) . Note that we do not require that P + ∈ C ( aa ) .Obviously, π ∗ ( j ( a )) = π ( a ) for all a ∈ M .Now we prove that π ∗ is elementary. Suppose R ([ ϕ ( s, x, a )] , . . . , [ ϕ m ( s, x, a )]) is atomic, R ∈ τ , and M ∗ | = R ([ ϕ ( s, x, a )] , . . . , [ ϕ m ( s, x, a )]) . Thus aa sR ( f ϕ ( s,x, a ) ( s ) , . . . , f ϕ n ( s,x, a ) ( s )) ∈ T, whence N | = aa sR ( f ϕ ( s,x,π ( a )) ( s ) , . . . , f ϕ n ( s,x,π ( a )) ( s )) . Hence N | = R ( π ∗ ([ ϕ ( s , x, π ( a ))]) , . . . , π ∗ ([ ϕ m ( s , x, π ( a ))])) . For the other direction one can use club determinacy.Suppose then j ( a ) = [ x = a ] ∈ P ∗ . π ∗ ( j ( a )) = π ( a ) . N | = aa s ( π ( a ) ∈ s ) .Hence π ( a ) ∈ S . Hence, π ∗ ( j ( a )) ∈ P + . Conversely, suppose z ∈ P + .Let z = π ∗ ( y ) , where y = [ ϕ ( s, x, a )] . We show that y ∈ P ∗ . We have N | = aa s ∃ x ( ϕ ( s, x, a ) ∧ x ∈ s ) . Hence there is b ∈ N such that N | = aa s ∃ x ( ϕ ( s, x, a ) ∧ x = c b ) . Since ( M, T ) is a premouse, there is such a b in M . It follows that y = j ( b ) .The other cases are easy. Lemma 5.17.
Suppose (( M β , T β ) , j βγ ) , β ≤ γ ≤ δ , where δ = ∪ δ ≤ ω , is adirected system of aa -premice. Let τ β be the vocabulary of ( M β , T β ) . Supposethe mappings π β : ( M β , T β ) → ( N β , Th L ( aa ) ( N β )) , β < δ , are elementary, here N β is some expansion of L ′ α to a τ β -structure, such that N β = N γ ↾ τ β and π β ( x ) = π γ ( j βγ ( x )) for all x ∈ M β and all β < γ < δ . Then there is an expansion N δ of L ′ α to a τ δ -structure and an elementary π δ : ( M δ , T δ ) → ( N δ , Th L ( aa ) ( N δ )) such that N β = N δ ↾ τ β and π β ( x ) = π δ ( j βδ ( x )) for all x ∈ M β and all β < δ .Proof. Obvious.Suppose ( M, T ) has vocabulary τ . Let (( M α , T α ) , j αβ ) , α ≤ β ≤ ω , be adirected system of aa -premice such that ( M , T ) = ( M, T )( M α +1 , T α +1 ) = ( M ∗ α , T ∗ α )( M ν , T ν ) = lim −→ ( M α , T α ) , limit ν .The vocabulary τ α of M α consists of τ and the unary predicate symbols P β , β <α . The mappings j αβ of the directed system are elementary embeddings j αβ :( M α , T α ) → ( M β ↾ τ α , T β ) , α < β ≤ ω , such that j αα +1 is always the canonicalembedding. Lemma 5.18.
The interpretations of the unary predicates { P α : α < ω } in M ω form a club in P ω ( M ω ) .Proof. Let C = { P M ω α : α < ω } . Since we take direct limits at limits, thissequence is continuous. By Lemma 5.10 it is properly increasing. Any countablesubset s of M ω is in the i αω -image of M α for some α . Since M α +1 = M ∗ α , theset s is included in P M ω α +1 . Definition 5.19.
We call the aa-premice ( M α , T α ) iterates of ( M, T ) . An aa-premouse ( M, T ) is an aa-mouse if its iterates ( M α , T α ) , α < ω , are all well-founded. In this case we say that the aa-premouse ( M, T ) is iterable . Lemma 5.20.
Suppose ( M, T ) ( L ′ α , Th L ( aa ) ( L ′ α )) , where cf( α ) > ω . Then ( M, T ) is a club determined aa-mouse.Proof. If ( M, T ) is as in Example 5.6, we may use Lemmas 5.16 and 5.17 induc-tively to build π β : ( M β , T β ) → ( N β , Th L ( aa ) ( N β )) for all β ≤ ω , where each N α is an expansion of L ′ β , with the consequence thateach M β is well-founded. 31 roposition 5.21. Suppose ( M, T ) is a countable aa-mouse and ( M ω , T ω ) is its ω ’st iterate. Then aa s ϕ ( s , a ) ∈ T ω ⇐⇒ M ω | = aa s ϕ ( s , a ) . Proof.
We prove the claim by induction on ϕ ( a ) . Let β be the least β such that a ∈ M β .1. Atomic ϕ ( s , a ) : Suppose aa s . . . aa s n ϕ ( s , a ) ∈ T ω . By club determinacy, aa s . . . aa s n ϕ ( s , a ) ∈ T β , whence aa s . . . aa s n ϕ ( s , a ) ∈ T γ for β ≤ γ < ω .Therefore aa s . . . aa s n ϕ ( P γ , s , . . . , s n , a ) ∈ T γ +1 for β ≤ γ < ω . Further-more, by construction, ϕ ( P γ , . . . , P γ n , a ) ∈ T ω for β ≤ γ < . . . < γ n < ω .Since ϕ ( s , a ) is atomic, M ω | = ϕ ( P γ , . . . , P γ n , a ) for all β ≤ γ < . . . < γ n <ω . By Lemma 5.18, M ω | = aa s ϕ ( s , a ) . Conversely, if aa s ϕ ( s , a ) / ∈ T β , then aa s ¬ ϕ ( s , a ) ∈ T β and we can argue as above.2. Conjunction: Suppose aa s ( ϕ ( s , a ) ∧ ψ ( s , a )) ∈ T ω . Then aa s ϕ ( s , a ) ∈ T ω and aa s ψ ( s , a ) ∈ T ω . By Induction Hypothesis, M ω | = aa s ( ϕ ( s , a ) ∧ ψ ( s , a )) .3. Negation: Easy, by club determinacy.4. Existential quantifier: Suppose aa s ∃ xϕ ( s , x, a ) ∈ T β . Then aa s ϕ ( s , f ϕ ( s ,x, a ) ( s ) , a ) ∈ T β . By the Induction Hypothesis, M ω | = aa s ϕ ( s , f ϕ ( s ,x, a ) ( s ) , a ) . Hence we have M ω | = aa s ∃ xϕ ( s , x, a ) . The converse follows by Case 3.5. The aa -quantifier: We show aa s aa t ϕ ( s , t , a ) ∈ T ω if and only if M ω | = aa s aa t ϕ ( s , t , a ) . This follows from the Induction Hypothesis, since we can com-bine s and t into one finite sequence of variables. C ( aa ) Lemma 5.22.
Suppose ( M , T ) ≺ ( L ′ α , Th L ( aa ) ( L ′ α )) , α limit, and club deter-minacy holds. Then ( M ω , T ω ) does not have new reals over those in ( M , T ) .Proof. Suppose r is a real in M ω . Let ξ < ω such that r ∈ M ξ +1 \ M ξ . Then r = [ ϕ ( s, x, a )] for some ϕ ( s, x, y ) ∈ L ( aa ) and some a ∈ M ξ such that aa s ∃ xϕ ( s, x, a ) ∈ T ξ . In particular, M ω | = aa s ∃ xϕ ( s, x, a ) . Since M ω | =“[ ϕ ( s, x, a )] ⊆ ω ” , the sentence ∃ x ( x ⊆ ω ∧ ∀ n ( n ∈ [ ϕ ( s, x, a )] ↔ n ∈ x )) is in32 ω , whence aa s ∃ x ( x ⊆ ω ∧ ∀ n ( n ∈ f ϕ ( s,x, a ) ( s ) ↔ n ∈ x )) is in T ξ and there-fore true in L ′ α . So there is a club of sets s such that L ′ α | = ∃ x ( x ⊆ ω ∧ ∀ n ( n ∈ f ϕ ( s,x, a ) ( s ) ↔ n ∈ x )) . Since L ′ α has only countably many reals (a consequenceof club determinacy, see Theorem 5.1), this club is divided into countably manyparts according to the x ⊆ ω such that L ′ α | = ∀ n ( n ∈ f ϕ ( s ,x, a ) ( s ) ↔ n ∈ x ) . Oneof those parts is stationary and therefore, by club determinacy, contains a club.Hence ∃ x aa s ( x ⊆ ω ∧ ∀ n ( n ∈ f ϕ ( s,x, a ) ( s ) ↔ n ∈ x )) is in T ξ . Since ( M ξ , T ξ ) isan aa-mouse, there is b ∈ M ξ such that aa s ( b ⊆ ω ∧ ∀ n ( n ∈ f ϕ ( s,x, a ) ( s ) ↔ n ∈ b )) is in T ξ . Hence aa s ( b ⊆ ω ∧ ∀ n ( n ∈ f ϕ ( s,x, a ) ( s ) ↔ n ∈ b )) is true in L ′ α , andtherefore r = b , a contradiction.We are now ready to prove the main result of this section: Theorem 5.23.
If club determinacy holds, then CH holds in C ( aa ) .Proof. Since we assume club determinacy, there are only countably many reals in C ( aa ) , but we show that there are, in the sense of C ( aa ) , only ℵ C ( aa )1 many. Theordinal ℵ C ( aa )1 is in our case a countable ordinal in the sense of V .Suppose L ′ α is a stage where a new real r of C ( aa ) is constructed, i.e. r ∈ L ′ α +1 \ L ′ α . (13)We show that L ′ α ∩ ω is countable in C ( aa ) . It follows that C ( aa ) ∩ ω hascardinality ℵ in C ( aa ) . Hence C ( aa ) | = CH .We can collapse | α | to ℵ without changing C ( aa ) or C ( aa ) ∩ ω . Also clubdeterminacy is preserved in this forcing, because the forcing is countably closed.Therefore we can assume, w.l.o.g., that | α | = ℵ .Let ( M, T ) ∈ C ( aa ) be countable in C ( aa ) such that { r, α, L ′ α } ⊆ ( M, T ) ( L ′ℵ V , Th L ( aa ) ( L ′ℵ V )) . (14)So now r ∈ ( L ′ α +1 ) M \ ( L ′ α ) M . (15)The idea of the rest of the proof is the following. We iterate ( M, T ) until we obtain ( M ω , T ω ) . We show that ( L ′ α ) M ω = L ′ α , whence L ′ α ∩ ω ⊆ M ω . Lemma 5.22implies M ω ∩ ω ⊆ M. It follows that L ′ α ∩ ω is countable, as we wished todemonstrate.In more details, let ( M ξ , T ξ ) , i ξη , ξ < η ≤ ω be the iteration of ( M, T ) asdefined above. The models M ξ are well-founded and collapse to transitive models N ξ with elementary embeddings j ξη : N ξ → N η , as in Figure 1.33 i i ξξ +1 M → M → M . . . → M ξ → M ξ +1 . . . M ω π ↓ π ↓ π ↓ π ξ ↓ π ξ +1 ↓ π ω ↓ N → N → N . . . → N ξ → N ξ +1 . . . N ω j j j ξξ +1 Figure 1: The iteration.By Lemma 5.22, no new reals are generated in the iteration. By Lemma 5.21,the model N ω is correct about the aa -quantifier. Thus if the hierarchy L ′ β of the C ( aa ) -model is built inside N ω , then we obtain for all β ∈ N ω the result: L ′ β = ( L ′ β ) N ω . (16)Combining this with (13) and (15) yields, r = j ω ( π ( r )) ∈ j ω ( π (( L ′ α +1 ) M )) \ j ω ( π (( L ′ α ) M )= ( L ′ j ω ( π ( α ))+1 ) N ω \ ( L ′ j ω π ( α )) ) N ω = L ′ j ω ( π ( α ))+1 \ L ′ j ω π ( α )) . By (13), j ω ( π ( α )) = α ∈ N ω and further by (16), j ω ( π ( L ′ α )) = L ′ α . Thusall the reals of L ′ α are in N ω and hence in M , and they are countably manyonly.The above proof shows that C ( aa ) | = 2 ℵ α = ℵ α +1 for all α ≤ ω (= ω V ) . For α < ω the above proof works, and for α = ω the claim therefore follows fromthe fact (Theorem 5.1) that ω is measurable in C ( aa ) . Theorem 5.24.
If club determinacy holds, there is a ∆ well-ordering of the realsin C ( aa ) .Proof. We show that the canonical well-order ≺ of C ( aa ) is ∆ . The proof ofTheorem 5.23 essentially shows that for any reals x, y in C ( aa ) : x ≺ y ⇐⇒ ∃ z ⊆ ω ( z codes an aa-mouse M such that34 , y ∈ M and M | = “ x ≺ y ”) . Being a real that codes a countable aa-mouse is Π . Hence the right hand side ofthe equivalence is Σ and the claim follows. Corollary 5.25.
If club determinacy holds, there are no Woodin cardinals in C ( aa ) .Proof. The proof of Theorem 5.24 shows that, assuming club determinacy, there isa ∆ -well-ordering of the reals, this well-ordering is in C ( aa ) and ∆ in C ( aa ) .Suppose there is a Woodin cardinal in C ( aa ) . There would be a measurablecardinal above it by Theorem 5.1. A measurable cardinal above a Woodin cardinalimplies Σ -determinacy ([10]). On the other hand, Σ -determinacy implies that Σ -sets are Lebesgue measurable which contradicts the existence of a Σ -well-ordering. Theorem 5.26.
If club determinacy holds for C ( aa ) , then ✸ holds in C ( aa ) .Proof. Let ω aa denote the ω of C ( aa ) . We define S α for α < ω aa as follows:Let ( C, X ) be the ≺ C ( aa ) -minimal pair ( C, X ) , where C ⊆ α is a club and X ∩ S β = S β for β ∈ C . We then let S α = X . Suppose S = h S α : α < ω aa i isnot a ✸ -sequence in C ( aa ) . Then there are X ⊆ ω aa and a club C ∈ C ( aa ) suchthat C ⊆ ω aa and β ∈ C implies X ∩ β = S β . Let δ be minimal such that sucha pair can be found in L ′ δ . W.l.o.g., δ < ℵ V . Let ( M, T ) ∈ C ( aa ) be countablesuch that {S , δ, L ′ δ } ⊆ ( M, T ) ≺ ( L ′ℵ V , Th L ( aa ) ( L ′ℵ V )) . (17)We build models M ξ and N ξ as well as elementary mappings i αβ , j αβ and isomor-phisms π α for α < β ∈ N ω with M = M as in the proof of Theorem 5.23. Let α = M ∩ ω aa , ¯ C = C ∩ α and ¯ X = X ∩ α . Let δ ∗ = j ω ( π ( δ )) . The ordinal δ ∗ is the minimal δ ∗ such that there is a counterexample such as ( ¯ C, ¯ X ) in L ′ δ ∗ in N ω . The ordinal α is below the critical point of j , whence j ω ( h S β : β < α i ) = h S β : β < α i . Therefore, according to our definition, S α = ( ¯ C, ¯ X ) , contradicting α ∈ C . There are several variants of stationary logic. The earliest variant is based on thefollowing quantifier introduced in [12], a predecessor of the quantifier aa :35 efinition 6.1. M | = Q St xyzϕ ( x, a ) ψ ( y, z, a ) if and only if ( M , R ) , where M = { b ∈ M : M | = ϕ ( b, a ) } and R = { ( b, c ) ∈ M : M | = ψ ( b, c, a ) } isan ℵ -like linear order and the set I of initial segments of ( M , R ) with an R -supremum in M is stationary in the set D of all (countable) initial segments of M in the following sense: If J ⊆ D is unbounded in D (i.e. ∀ x ∈ D∃ y ∈ J ( x ⊆ y ) )and σ -closed in D (i.e. if x ⊆ x ⊆ . . . in J , then S n x n ∈ J ), then J ∩ I 6 = ∅ .The logic L ( Q St ) , a sublogic of L ( aa ) , is recursively axiomatizable and ℵ -compact [12]. We call this logic Shelah’s stationary logic , and denote C ( L ( Q St )) by C ( aa − ) . For example, we can say in the logic L ( Q St ) that a formula ϕ ( x ) defines a stationary (in V ) subset of ω in a transitive model M containing ω asan element as follows: M | = ∀ x ( ϕ ( x ) → x ∈ ω ) ∧ Q St xyzϕ ( x )( ϕ ( y ) ∧ ϕ ( z ) ∧ y ∈ z ) . Hence C ( aa − ) ∩ F ω ∈ C ( aa − ) and in fact the set D = C ( aa − ) ∩ F ω , where F ω is the club-filter on ω , sufficesto characterise C ( aa − ) completely: C ( aa − ) = L [ D ] . In particular, C ( aa − ) ⊆ C ( aa ) . Theorem 6.2.
If there are two Woodin cardinals, then C ( aa − ) = L [ D ] , where D = C ( aa − ) ∩ F ω is an ultrafilter in C ( aa − ) . In particular, C ( aa − ) | = GCH .Proof.
We know C ( aa − ) = L [ F ω ] . We show that D = F ω ∩ C ( aa − ) measuresevery set in C ( aa − ) . Let us assume the contrary. We take a minimal α suchthat there is a set B ⊆ ω in L ′ α (the hierarchy generating C ( aa − ) ) such that B / ∈ D and ω V \ B / ∈ D . The logic L ( aa − ) satisfies a Downward L¨owenheimSkolem Tarski Theorem down to ℵ ([12]). Hence, as in Lemma 4.18, | α | ≤ ℵ .As in the beginning of the proof of Theorem 4.12, we can assume, w.l.o.g., that δ = ω and we have still one Woodin cardinal δ left. Let G be Q <δ -generic and j : V → M ⊆ V [ G ] the generic ultrapower embedding. Let j ( α ) = β . Now j ( B ) is a stationary co-stationary subset of δ ( = ω M ) in M . Moreover, β is theminimal ordinal for which there is such a set in L ′ β in M , and the definition of j ( B ) in L ( aa − ) is minimal for such a set. As in the proof of Theorem 4.12 we cannow argue that j ( B ) ∈ V . We get a contradiction by taking two different genericsets for Q <δ , one containing B and the other containing ω \ B . Proposition 6.3. If exists, then ∈ C ( aa − ) . roof. Assume ♯ . A first order formula ϕ ( x , . . . , x n ) holds in L for an increas-ing sequence of indiscernibles below ω V if and only if there is a club C of ordi-nals < ω V such that every increasing sequence a < . . . < a n from C satisfies ϕ ( a , . . . , a n ) in L . Similarly, ϕ ( x , . . . , x n ) does not hold in L for an increas-ing sequence of indiscernibles below ω V if and only if there is a club of ordi-nals a < ω such that there ia a club of ordinals a with a < a < ω suchthat . . . such that there is a club of ordinals a n with a n − < a n < ω satisfying ¬ ϕ ( a , . . . , a n ) . From this it follows that ∈ C ( aa − ) . Theorem 6.4.
It is consistent relative to the consistency of ZFC that C ∗ * C ( aa − ) ∧ C ( aa − ) * C ∗ . Proof.
We force over L and first we add two Cohen reals r and r , to obtain V .Now we use modified Namba forcing to make cf( ℵ Ln +1 ) = ω if and only if n ∈ r .This forcing satisfies the S -condition (see [7]), and therefore will not—by [4]—kill the stationarity of any stationary subset of ω . The argument is essentiallythe same as for Namba forcing. Let the extension of V by P be V . In V wehave C ( aa − ) = L because we have not changed stationary subsets of ω . But V | = r ∈ C ∗ .Let S n , n < ω , be in L a definable sequence of disjoint stationary subsets of ω such that S n S n = ω . Working in V , we use the canonical forcing notionwhich kills the stationarity of S n if and only if n ∈ r . Let the resulting modelbe V . The cofinalities of ordinals are the same in V and V , whence ( C ∗ ) V isthe same as ( C ∗ ) V . Thus V | = r ∈ C ( aa − ) \ C ∗ . Now we argue that V | = C ( aa − ) = L ( r ) . First of all, L ( r ) ⊆ C ( aa − ) by the construction of V . Nextwe prove by induction on the construction of C ( aa − ) as a hierarchy L ′ α , α ∈ On ,that L ′ α ⊆ L ( r ) . When we consider L ′ α +1 and assume L ′ α ⊆ L ( r ) , we have todecide whether a subset S of ω , constructible from r , is stationary or not. Theset S is stationary in V if and only if it is stationary in L ( r ) and it is not includedmodulo the club filter in S n ∈ r S n . Thus L ′ α +1 ⊆ L ( r ) .In V the real r is in C ∗ \ C ( aa − ) and the real r is in C ( aa − ) \ C ∗ .The logics L ( Q cf ω ) , giving rise to C ∗ , and L ( aa − ) , giving rise to C ( aa − ) , aretwo important logics, both introduced by Shelah. Since L ( Q cf ω ) is fully compact, L ( aa − ) cannot be a sub-logic of it. On the other hand, it is well-known and easyto show that L ( Q cf ω ) is a sub-logic of L ( aa ) . Therefore it is interesting to note thefollowing corollary to the above theorem:37 orollary 6.5. It is consistent, relative to the consistency of ZFC, that L ( Q cf ω ) * L ( Q St ) and hence L ( Q St ) = L ( aa ) . We do not know whether it is consistent that L ( Q cf ω ) ⊆ L ( aa − ) or that L ( aa − ) = L ( aa ) .A modification of C ( aa − ) is the following C ( aa ) : Definition 6.6.
M | = Q St, xyzuϕ ( x, y, a ) ψ ( u, a ) if and only if M = { ( b, c ) ∈ M : M | = ϕ ( b, c, a ) } is a linear order of cofinality ω and every club of initialsegments has an element with supremum in R = { b ∈ M : M | = ψ ( b, a ) } . Theinner model C ( aa ) is defined as C ( L ( Q St, )) . Proposition 6.7.
If there are two Woodin cardinals, then C ( aa ) | = “ ℵ V is ameasurable cardinal”.Proof. The proof of this is— mutatis mutandis —as the proof for C ( aa − ) . Proposition 6.8. If † exists, then † ∈ C ( aa ) .Proof. Assume † . There is a club class of indiscernibles for the inner model L [ U ] where U is (in L [ U ] ) a normal measure on an ordinal δ . Let us choose anindiscernible α above δ of V -cofinality ω V . We can define † as follows: An in-creasing sequence of indiscernibles satisfies a given formula ϕ ( x , . . . , x n ) if andonly if there is a club C of ordinals below α such that every increasing sequence a < . . . < a n from C satisfies ϕ ( a , . . . , a n ) in L [ U ] . Similarly, ϕ ( x , . . . , x n ) does not hold in L [ U ] for an increasing sequence of indiscernibles below α if andonly if there is a club of ordinals a < α such that there is a club of ordinals a with a < a < α such that . . . such that there is a club of ordinals a n with a n − < a n < α satisfying ¬ ϕ ( a , . . . , a n ) . From this it follows that † ∈ C ( aa ) . Corollary 6.9.
If there are two Woodin cardinals, then C ( aa − ) = C ( aa ) . Thenalso the logics L ( Q St ) and L ( Q St, ) are non-equivalent.Proof. If there are two Woodin cardinals, then then † exists and C ( aa − ) does notcontain † by Theorem 6.2, while C ( aa ) does contain by Proposition 6.8.Note that it is probably possible to prove the non-equivalence of the logics L ( Q St ) and L ( Q St, ) in ZFC with a model theoretic argument using the exactdefinitions of the logics and by choosing the structures very carefully. But the non-equivalence result given by the above Corollary is quite robust in the sense thatit is not at all sensitive to the exact definitions of the logics as long as the centralseparating feature, manifested in structures of the form ( α, < ) , is respected.38 Open problems
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