Chang’s Conjecture, generic elementary embeddings and inner models for huge cardinals Matt Foreman Kyoto October 25-27, 2010 These lectures summarize and organize the material appearing in: Smoke and Mirrors: Combinatorial properties of small cardinals equiconsistent with huge cardinals. Advances in Mathematics .Volume 222, Issue 2, 1 October 2009, Pages 565-595 • Much of the early history of logic in general and model theory in particular was tied up with understanding the expressive power of first and second order logic (and their variants). • One distinguishing feature of first order logic is the Downwards Lowenheim- Skolem Theorem . • Tremendous effort was put into generalizing the downwards Lowenheim-Skolem theorem so that the elementary substructure had some second order properties • The coarsest second order properties had to do with cardinality; in this dis- cussion we consider various more subtle second order properties. Among them are being correct for the non-stationary ideal. Let L be a countable language with a distinguished unary predicate R . Then A = � A, R A , f i , R j , c k . . . � i,j,k ∈ ω is said to have type ( κ, λ ) if and only iff | A | = κ and | R A | = λ . We often write A = � κ ; λ, f i . . . � to mean a structure of type ( κ, λ ). 1
Two cardinal Transfer Theorems We write ( κ, λ ) → ( κ ′ , λ ′ ) to mean that if A is an L -structure of type ( κ, λ ) then there is a B ≡ A of type ( κ ′ , λ ′ ). Classical Results: I Vaught’s Two Cardinal Theorem : ( κ, ρ ) → ( ω 1 , ω ) for all κ > ρ ≥ ω . II The infinite gap two cardinal theorem: ( κ + γ , κ ) → ( ρ + δ , ρ ) for all infinite cardinals κ, ρ and ordinals γ, δ . Jensen developed Morasses to prove: III Jensen Gap n two cardinal theorem = ( ∀ n ∈ ω )( ∀ infinite κ, λ )(( κ + n , κ ) → ( λ + n , λ )) . L | More in the spirit of the Lowenheim-Skolem theorem For κ ≥ κ ′ and λ ≥ λ ′ , we say ( κ, λ ) → → ( κ ′ , λ ′ ) iff for all A of type ( κ, λ ) in a countable language there is an elementary substructure B ≺ A of type ( κ ′ , λ ′ ). I Classical Chang’s Conjecture ( ω 2 , ω 1 ) → → ( ω 1 , ω ). This conjecture looks obviously “set theoretical” to modern eyes: If you apply it to � L ( ω 2 ) V , ( ω 1 ) V , ∈� one gets an elementary substructure N ≺ L ω 2 . If ¯ N is the transitive collapse of N and j is the inverse of the transitive collapse map, then the embedding j yields an L -ultrafilter on crit ( j ) and hence O # . As a consequence, it is clear that Chang’s Conjecture cannot be a theorem of ZFC. II (Silver) Con (ZFC + there is an ω 1 -Erd¨ os cardinal) implies Con (ZFC + GCH + ( ω 2 , ω 1 ) → → ( ω 1 , ω ). In fact the exact consistency strength of Chang’s Conjecture has been shown to be an ω 1 -Erd¨ os cardinal. III (Kunen/Laver) Con (ZFC + there is a huge cardinal) implies that for all n ≥ 1, Con (ZFC + ( ω n +1 , ω n ) → → ( ω n , ω n − 1 )) IV (Foreman) Con (ZFC + there is a 2-huge cardinal) implies Con (ZFC + GCH +( ∀ m < n )( ω n +1 , ω n ) → → ( ω m +1 , ω m ).) 2
Chang’s Conjecture shows up in many situations For example: κ → [ µ ] <ω ρ + is equivalent to ( κ, ρ + ) → → ( µ, ρ ) . An example of a family of deeper results is: (Foreman) Assume the CH. Then: ( ω 3 , ω 2 ) → → ( ω 2 , ω 1 ) is equivalent to ω 3 → [ ω 2 ] ω ω 2 . Note the infinite exponent. Ideas from Chang’s Conjecture properties turned out to be crucial for semi-proper forcing type arguments–e.g. “antichain catching”. What second order properties can you ask for in an elementary substructure? To investigate this let us give a more modern looking reformulation of Chang’s Conjecture: Proposition 1 ( ω n +2 , ω n +1 ) → → ( ω n +1 , ω n ) iff (for all) (there exists) θ ≫ ω n +2 and an N ≺ � H ( θ ) , ∈ ∆ � such that if π : N → ¯ N is the transitive collapse then • π ↾ ω n = id and • π ( ω n +2 ) = ω n +1 . ⊢ ( ⇐ ) Suppose that the CC fails. let A = � ω n +2 , ω n +1 , f i , R j , c k � i,j,k ∈ ω be the ∆- least counterexample. Suppose that N ≺ A is as in the hypothesis. Then: A ∈ N so N ∩ ω n +2 is closed under Skolem functions for A . In particular N ∩ ω n +2 ≺ A . But | N ∩ ω n +2 | = ω n +1 and N ∩ ω n +1 = ω n . This is a contradiction. ( ⇒ ) Let A = � ω n +2 , ∈ , f i � be a fully Skolemized structure such that for all z ≺ A we know that sk H ( θ ) ( z ) ∩ ω n +2 = z . By the CC we know there is a z ≺ A such that the type of z is ( ω n +1 , ω n ). Let N 0 = sk H ( θ ) ( z ). Then N 0 ≺ H ( θ ). Let N = sk H ( θ ) ( N 0 ∪ ω n ). We claim that: 3
• sup( N ∩ ω n +1 ) = sup( N 0 ∩ ω n +1 ) • sup( N ∩ ω n +2 ) = sup( N 0 ∩ ω n +2 ). To see this, let τ : N 0 × ω n → ω n +1 be a Skolem function. Since ∆ is in the language, for � x ∈ N , the function τ ( � x, · ) : ω n → ω n +1 is definable in N 0 . In particular, sup( τ ( � x, · )“ ω n ) ∈ N 0 . Since � N ∩ ω n +1 = { τ ( � x, · )“ ω n : τ is a Skolem function and � x ∈ N 0 } , we see that sup( N 0 ∩ ω n +1 ) = sup( N ∩ ω n +1 ). The result for ω n +2 is seen similarly. Let π : N → ¯ N be the transitive collapse. Since ω n ⊆ N , π ↾ ω n is the identity map. To see that π ( ω n +2 ) = ω n +1 we need to see that N ∩ ω n +2 has order type ω n +1 . Note that the order type is at least ω n +1 by the choice of z . Let α ∈ N ∩ ω n +2 . Then there is a bijection f : ω n +1 → α that lies in N . Hence f : N ∩ ω n +1 → N ∩ α is a bijection. In particular, | N ∩ α | = | N ∩ ω n +1 | ; thus | N ∩ α | < ω n +1 . ⊣ In the Smoke and Mirrors paper, the following more involved extension is proved: Proposition 2 Let λ ≤ κ ≪ θ be cardinals with λ and θ regular and with cf ( κ ) ≥ λ . Let A be a structure expanding � H ( θ ) , ∈ , ∆ , { κ, λ }� and N 0 ≺ A . Let N 1 = sk A ( N 0 ∪ sup( N 0 ∩ λ )) and let ρ = | sup( N 0 ∩ λ ) | . Suppose that either: 1. The GCH holds or 2. ξ ⊆ N 0 and κ ≤ λ + ξ . Then N 1 ∩ λ = sup( N 0 ∩ λ ) and | N 1 ∩ κ | = | N 0 ∩ κ | · ρ. Note: I don’t know if the GCH is relevant or necessary. Can we ask for more properties of ¯ N ? For example: can we ask that ( ω 3 , ω 2 ) → → ( ω 2 , ω 1 ) is witnessed by an N ≺ H ( θ ) and N ∩ ω 3 ∈ ¯ N ? Digression on NS ideals Let S ⊆ P ( X ). Then S is stationary iff for all A = � X, f i � i ∈ ω , there is a z ≺ A such that z ∈ S . (In some places this is called “weakly stationary”.) This is a good generalization of “stationary” in other contexts: 4
• it coincides with the usual definition of a “stationary set S ⊆ κ ” for subsets of κ , • it coincides with the usual definition of a “stationary set S ⊆ P κ ( λ )” for subsets of P κ ( λ ). With this language we see that all “Chang Conjectures” are simply statements that certain sets are stationary. For example ( ω n +2 , ω n +1 ) → → ( ω n +1 , ω n ) is equivalent to { z ∈ [ ω n +2 ] ω n +1 : | z ∩ ω n +1 | = ω n } is stationary . Let us see what happens to the non-stationary ideal under elementary substruc- tures: Suppose that N ≺ H ( θ ) and | N ∩ ω i | = ω i − 1 for i = 2 , 3 , 4, and assume that ω 1 ⊆ N . Let A ′ ⊆ [ ω 4 ] ω 3 . Let π : N → ¯ N be the transitive collapse. Then π ( A ′ ) = A for some A ⊆ [ ω 3 ] ω 2 . Moreover, π ( NS ↾ A ′ ) ⊆ ( NS ↾ A ) ∩ ¯ N. Definition 3 N is correct for NS ↾ A iff A ∈ N and there is are A ′ , I ′ ∈ N such that π ( I ′ ↾ A ′ ) = ( NS ↾ A ) ∩ ¯ N. (We write π N for the transitive collapse map of N . The set A ′ is mentioned only to make sure that there is a set A ′ ∈ N such that π N ( A ′ ) = A .) Note: Being correct about NS ↾ A is a thickness property (or closure property) of ¯ N . Canonically well-ordered stationary sets Proposition 4 (Baumgartner) Let M, N ≺ H ( θ ) . Suppose that • sup( M ∩ ω n +2 ) = sup( N ∩ ω n +2 ) ∈ cof ( > ω ) , • N ∩ ω n +1 = M ∩ ω n +1 . 5
Then M ∩ ω n +2 = N ∩ ω n +2 . To use the Baumgartner proposition, we need the following result: Proposition 5 (Foreman-Magidor) For n ∈ ω and N ≺ H ( θ ) : if N ∈ [ ω n +2 ] ω n +1 , N ∩ ω n +1 ∈ ω n +1 , then cof ( N ∩ ω n +1 ) = ω n . Putting these two propositions together we get the following: Corollary 6 Let p : [ ω n +2 ] ω n +1 → ω n +1 × ω n +2 be defined by z �→ ( z ∩ ω n +1 , sup( z ∩ ω n +2 )) . Then p is 1-1 on the collection of z such that: 1. sk H ( θ ) ( z ) ∩ ω n +2 = z , 2. z ∩ ω n +1 ∈ ω n +1 . Upshot: Relative to a closed unbounded set restricted to { z ∈ [ ω n +2 ] ω n +1 : z ∩ ω n +1 ∈ ω n +1 } , p is 1-1. Definition 7 A set A ⊆ { z ∈ [ ω n +2 ] ω n +1 : z ∩ ω n +1 ∈ ω n +1 } is ordinary if p is 1-1 on A . We note that ordinary sets have canonical and absolute well-orderings. Strong Chang Reflection We will say that Strong Chang reflection holds for ( ω n +3 , ω n ) iff for all large enough θ there is an ordinary A ∈ [ ω n +2 ] ω n +1 and an O ′ such that for some N ≺ � H ( θ ) , ∈ , ∆ , A, O ′ � we have: 1. N ∩ ω n +2 ∈ A and | N ∩ ω n +3 | = ω n +2 , 2. π N ( O ′ ) = N ∩ ω n +2 , 3. N is correct for NS ↾ A . Informally SCR says that the collection of N that are correct about NS ↾ A and whose transitive collapse contains N ∩ ω n +2 is stationary and canonically well-ordered. 6
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