CHANGS CONJECTURE FOR TRIPLES REVISITED MASAHIRO SHIOYA UNIVERSITY - PDF document

CHANG’S CONJECTURE FOR TRIPLES REVISITED MASAHIRO SHIOYA UNIVERSITY OF TSUKUBA 1

2 Chang type conjectures for pairs Suppose N = ( N ; R, · · · ) is a structure for a countable first-order language with a distinguished unary pred- icate symbol interpreted by R ⊂ N . N has type ( ν, ν ′ ) if | N | = ν and | R | = ν ′ . ( ν, ν ′ ) ։ ( µ, µ ′ ) iff ∀N of type ( ν, ν ′ ) ∃M of type ( µ, µ ′ ) s.t. M ≺ N . Originally Chang conjectured ( ω 2 , ω 1 ) ։ ( ω 1 , ω ).

3 Consistency of Chang’s original conjecture Theorem (Silver). Con(an ω 1 -Erd˝ os cardinal exists) implies Con (( ω 2 , ω 1 ) ։ ( ω 1 , ω )) . Proof. First force MA with a small poset and then collapse the ω 1 -Erd˝ os cardinal λ (say) to ω 2 by the Silver collapse S ( ω 1 , λ ). � In fact they are equiconsistent. MA can be removed and the Levy collapse works as well. (Shelah)

4 Chang’s conjecture for triples Problem. Con (( ω 3 , ω 2 , ω 1 ) ։ ( ω 2 , ω 1 , ω )) ? We consider structures with two distinguished unary predicates. ( ω 3 , ω 2 , ω 1 ) ։ ( ω 2 , ω 1 , ω ) implies ( ω 3 , ω 2 ) ։ ( ω 2 , ω 1 ). PFA implies that the Levy collapse forces ( ω 3 , ω 2 ) ̸ ։ ( ω 2 , ω 1 ). (Foreman–Magidor)

5 Consistency of Chang’s conjecture for pairs Theorem (Kunen). Con(a huge cardinal exists) implies Con (( ω 3 , ω 2 ) ։ ( ω 2 , ω 1 )) . κ is huge with target λ iff ∃ j : V → M s.t. κ = crit( j ), λ = j ( κ ), λ M ⊂ M . Proof. Construct P s.t. • P collapses κ to ω 2 , • P ∗ ˙ S ( κ, λ ) ֒ → j ( P ). The final model is given by P ∗ ˙ S ( κ, λ ). �

6 Consistency of Chang’s conjecture for triples Theorem (Foreman). Con(a 2 -huge cardinal exists) implies Con (( ω 3 , ω 2 , ω 1 ) ։ ( ω 2 , ω 1 , ω )) . κ is 2-huge iff ∃ j : V → M s.t. κ = crit( j ), j 2 ( κ ) M ⊂ M . Proof. By Kunen’s method. Complicated! �

7 Open problem It has been open for 30 years whether Con (( ω 4 , ω 3 , ω 2 , ω 1 ) ։ ( ω 3 , ω 2 , ω 1 , ω )) . Perhaps Con(3-huge) would suffice. But how?

8 A new model of Chang’s conjecture for pairs Theorem. Suppose κ is huge with target λ . Let µ < κ be regular. Then   µ ∏  ∗ ˙ S ( β, κ ) S ( κ, λ )  β ∈ [ µ,κ ) ∩ R forces κ = µ + , λ = µ ++ and ( µ ++ , µ + ) ։ ( µ + , µ ) . ∏ stands for the < µ -support product. µ R denotes the class of regular cardinals.

9 From embeddings to projections Let P, R be posets. A map π : P → R is a projection if: (1) π is order-preserving, (2) π (1 P ) = 1 R , (3) r ′ ≤ R π ( p ) → ∃ p ∗ ≤ P p s.t. π ( p ∗ ) ≤ R r ′ . If π : P → R be a projection, then we get e : R ֒ → B ( P ) by ∑ r �→ { p ∈ P : π ( p ) ≤ r } . Conversely e : R ֒ → P gives rise to a projection : P → B ( R ).

10 Term spaces Suppose ˙ S is a P -name for a poset. The term space is the “set” T ( P, ˙ s ∈ ˙ S ) = { ˙ s : ˙ s is a P -name ∧ � ˙ S } s ′ ≤ ˙ s ′ ˙ ordered by: ˙ s iff � ˙ ≤ ˙ s . As sets, P × T ( P, ˙ S ) = P ∗ ˙ S.

11 Basic lemma of term spaces Lemma (Laver). The identity map id : P × T ( P, ˙ S ) → P ∗ ˙ S is a projection. Using the lemma we will get a projection j ( P ) → P ∗ ˙ S ( κ, λ ) , where µ ∏ P = S ( β, κ ) . β ∈ [ µ,κ ) ∩ R

12 The Silver collapse (with slight modification) Suppose κ < λ are regular cardinals with λ inaccessible. The Silver collapse S ( κ, λ ) is the set of s : δ × d → λ such that • δ < κ , d ⊂ [ κ, λ ) is a set of κ -closed cardinals of size ≤ κ and • s ( α, γ ) < γ for every ( α, γ ) ∈ δ × d . ordered by reverse inclusion: s ′ ≤ s iff s ′ ⊃ s . A cardinal γ is κ - closed if γ <κ = γ . S ( κ, λ ) has nice properties of the original Silver collapse.

13 Identifying the term space Main Lemma. Suppose P has κ -cc and size ≤ κ . Then S ( κ, λ ) is isomorphic to a dense subset of T ( P, ˙ S ( κ, λ )) . Corollary. Suppose P has κ -cc and size ≤ κ . Then there is a projection of the following form: id × i : P × S ( κ, λ ) → P ∗ ˙ S ( κ, λ ) . Results should hold for suitable modifications of other canonical collapses as well.

14 Proof sketch The dense set is D = { ˙ s : ∃ δ < κ ∃ d ⊂ [ κ, λ ) � dom ˙ s = δ × d } . Define i : s ∈ S ( κ, λ ) �→ ˙ s ∈ D by s = dom s ∧ ˙ � dom ˙ s ( α, γ ) = τ ( s ( α, γ )) . Here P -names τ ( ξ ) are chosen so that for every κ -closed γ { τ ( ξ ) : ξ < γ } is a 1-1 enumeration of all P -names ˙ α s.t. � ˙ α < γ .

15 Master conditions (Extending elementary embeddings) Suppose • j : V → M is elementary, • ϕ : j ( P ) → P is a projection. A condition p ∗ ∈ j ( P ) is a master condition for ( j and) ϕ if p ≤ p ∗ ¯ ∀ ¯ p ≤ j ( ϕ (¯ p )) . If ¯ G ⊂ j ( P ) is generic and contains a master condition for ϕ , then ( j ◦ ϕ )“ ¯ G ⊂ ¯ G and j can be extended to j : V [ ϕ [ ¯ G ]] → M [ ¯ G ] in V [ ¯ G ]. ( ϕ [ ¯ G ] = the filter over P generated by ϕ “ ¯ G .)

16 Getting master conditions Lemma (Kunen). Suppose • j : V → M witnesses that κ is huge with target λ , • P ⊂ V κ has κ -cc, • π : j ( P ) → P ∗ ˙ S ( κ, λ ) is a projection, • 1 j ( P ) a master condition for π − : j ( P ) → P . Then there is a master condition s ∗ ) (1 j ( P ) , ˙ for π + : j ( P ∗ ˙ S ( κ, λ )) → P ∗ ˙ S ( κ, λ ) . Proof. Define a j ( P )-name ˙ X by ˙ X = { ( j ( ˙ p ) ≤ (1 P , ˙ s ) } , s ) , ¯ p ) : π (¯ and let ∪ ˙ s ∗ = � ˙ X. �

17 Proof for a new model Let j : V → M witness that κ is huge with target λ . Let µ ∏ P = S ( β, κ ) . β ∈ [ µ,κ ) ∩ R We claim that P ∗ ˙ S ( κ, λ ) works. Define a projection π : j ( P ) → P ∗ ˙ S ( κ, λ ) by µ µ µ ∼ ∏ − − − − → ∏ S ( β, λ ) × ∏ j ( P ) = S ( β, λ ) S ( β, λ ) β ∈ [ µ,λ ) β ∈ [ µ,κ ) β ∈ [ κ,λ )  � ( Q rs κ ) × pr κ  ( ) µ ∏ S ( β, κ ) × S ( κ, λ ) β ∈ [ µ,κ )   � id × i ( ) µ ∗ ˙ ∏ S ( β, κ ) S ( κ, λ ) . β ∈ [ µ,κ ) By Kunen’s lemma there is a master condition for π + : j ( P ∗ ˙ S ( κ, λ )) → P ∗ ˙ S ( κ, λ ) s ∗ ) , (1 j ( P ) , ˙ below which j can be extended to S ( κ,λ ) → M j ( P ∗ ˙ j : V P ∗ ˙ S ( κ,λ )) .

18 Toward Chang’s conjecture for triples Theorem (Foreman). Suppose that κ is 2 -huge. Let µ < κ be regular. Then κ = µ + and ( µ +++ , µ ++ , µ + ) ։ ( µ ++ , µ + , µ ) in some forcing extension. Proof. Let j : V → M witness that κ is 2-huge, λ = j ( κ ) and θ = j ( λ ). We claim that P ( κ ) ∗ ˙ Q ( κ, λ ) ∗ ˙ S ( λ, θ ) works. j maps the above poset to P ( λ ) ∗ ˙ Q ( λ, θ ) ∗ ˙ S ( θ, j ( θ )) . Claim 1. There is a projection: P ( λ ) → P ( κ ) ∗ ˙ Q ( κ, λ ) . Claim 2. P ( λ ) forces that there is a projection: S ( λ, θ ) P ( κ ) ∗ ˙ Q ( λ, θ )) → ˙ ˙ Q ( κ,λ ) . Claim 3. There is a master condition for the projection: j ( P ( κ ) ∗ ˙ Q ( κ, λ )) → P ( κ ) ∗ ˙ Q ( κ, λ ) . �

19 Generalization 1 Suppose • π : P → R is a projection, • ˙ S is an R -name for a poset. Let P ⋆ π ˙ S (or P ⋆ ˙ S ) be the set P × T ( R, ˙ S ) ordered by: ( p ′ , ˙ s ′ ) ≤ ( p, ˙ s ) iff p ′ ≤ P p ∧ π ( p ′ ) � R ˙ s ′ ˙ ≤ ˙ s. If P = R and π = id, P ⋆ ˙ S = P ∗ ˙ S as posets.

20 Laver type lemma 1 Lemma. Suppose • π : P → R is a projection, • ˙ S is an R -name for a poset. Then id : P × T ( R, ˙ S ) → P ⋆ π ˙ S is a projection. Corollary. Suppose further R has κ -cc and size ≤ κ . Then there is a projection of the following form: id × i : P × S ( κ, λ ) → P ⋆ ˙ S ( κ, λ ) R .

21 Generalization 2 Suppose X, Y are disjoint sets of ordinals, and for β ∈ X ∪ Y • π β : P → R β is a projection and • ˙ S β is an R β -name for a poset. Let   κ E ∏ ˙ ∏ ˙ P ⋆ S β × S β   β ∈ X β ∈ Y be the set   κ E ∏ T ( R β , ˙ ∏ T ( R β , ˙ P × S β ) × S β )   β ∈ X β ∈ Y ordered by: ( p ′ , q ′ ) ≤ ( p, q ) iff p ′ ≤ P p ∧ dom q ′ ⊃ dom q ∧ ∀ β ∈ dom q π β ( p ′ ) � β q ′ ( β ) ˙ ≤ β q ( β ) . ∏ stands for the Easton support product. E � β denotes the forcing relation w.r.t. R β .

22 Laver type lemma 2 Lemma.     κ E κ E ∏ T ( R β , ˙ ∏ T ( R β , ˙ ∏ ˙ ∏ ˙  → P ⋆ id : P × S β ) × S β ) S β × S β    β ∈ X β ∈ Y β ∈ X β ∈ Y is a projection. Corollary. Suppose further for β ∈ X ∪ Y • κ β ∈ [ κ, λ ) ∩ R , • R β has κ -cc and size ≤ κ . Then there is a projection of the form ∏ id × i β β ∈ X ∪ Y from   κ E ∏ ∏ P × S ( κ β , λ ) × S ( κ β , λ )   β ∈ X β ∈ Y to   κ E ∏ S ( κ β , λ ) R β × ˙ ∏ ˙ S ( κ β , λ ) R β  . P ⋆  β ∈ X β ∈ Y