Thin Projective Equivalence Relations and Inner Models Philipp - PowerPoint PPT Presentation

Thin Projective Equivalence Relations and Inner Models Philipp Schlicht University of Münster, Germany Logic Colloquium Wrocław, 14-19 July 2007

Definition. An equivalence relation E ⊆ ω ω × ω ω is called thin if there is no perfect set of pairwise inequivalent reals. Question. How does an inner model look like, if for any thin projective equivalence relation, every equivalence class has a representative in the inner model? Thin Projective Equivalence Relations 2

Theorem. (Hjorth 1993) Assume x # exists for every x ∈ ω ω . Then the following statements are equivalent for an inner model M : 1. For all thin Π 1 2 ( z ) equivalence relations with z ∈ M , every equivalence class has a representative in M 2. ω M 1 = ω V 1 and M ≺ Σ 1 3 V Thin Projective Equivalence Relations 3

Theorem. (Hjorth, Schindler, Schlicht 2006) Assume 2 n ) holds and M † 1 2 n − 2 ( x ) exists for every x ∈ ω ω . Det (∆ � Then the following statements are equivalent for an inner model M : 1. For all thin Π 1 2 n ( z ) equivalence relations with z ∈ M , every equivalence class has a representative in M 2. T M 2 n − 1 = T V 2 n − 1 and M ≺ Σ 1 2 n +1 V where T 2 n − 1 is the tree from a Π 1 2 n − 1 scale. Thin Projective Equivalence Relations 4

We prove that (2) implies (1). Assume that n = 2 and E is a thin Π 1 4 equivalence relation. Suppose x ∈ ω ω . We have to find x ′ ∈ ω ω ∩ M with ( x, x ′ ) ∈ E . Thin Projective Equivalence Relations 5

Since E is Π 1 1 4 , its complement is δ 3 -Suslin via a tree � computed from T 3 . A theorem of Harrington and Shelah proves that there is a formula ϕ ∈ L ∞ , 0 ∩ L α [ T 3 ] with • ϕ ( x ) • ∀ y ( ϕ ( y ) ⇒ ( x, y ) ∈ E ) where α is least such that L α [ T 3 ] � KP . The language L ∞ , 0 is built from atomic formulas n ∈ x and n / ∈ x by infinitary conjunctions and disjunctions, so that L ∞ , 0 formulas describe a real. Since ϕ ∈ L α [ T 3 ] , there is y ∈ ω ω ∩ M such that ϕ is definable from T 3 , y by a term t ϕ in any transitive model of KP containing T 3 and y . Thin Projective Equivalence Relations 6

Idea of proof: Try to write ∃ xϕ ( x ) as a Σ 1 5 statement. For this purpose, reconstruct T 3 in an iterate of M † 2 ( x, y ) , so that you can compute ϕ = t ϕ ( y, T 3 ) in the iterate. Then you can express ϕ ( x ) in M † 2 ( x, y ) . Here M † 2 ( x, y ) is the smallest ( ω 1 + 1) -iterable premouse built over ( x, y ) with 2 Woodin cardinals and a measurable cardinal above. Let γ < δ < κ such that M † 2 ( x, y ) � γ, δ are Woodin cardinals and κ is measurable. Thin Projective Equivalence Relations 7

Let V be countable with x, y ∈ V and π : V → Σ 100 V elementary. Let M = π − 1 ” M , T 3 = π − 1 ( T 3 ) , etc. By forming Skolem hulls, in M we can construct substructures X 0 ≺ X 1 ≺ ... ≺ M † 2 ( x, y ) and ordinals γ 0 < γ 1 < ... < γ , δ 0 ≤ δ 1 ≤ ... ≤ δ , κ 0 ≤ κ 1 ≤ ... ≤ κ , with M † 2 ( x,y ) 1. V X i γ i = V for all i ∈ ω γ i 2. X i � γ i < δ i are both Woodin cardinals and κ i > δ i is measurable 3. sup i ∈ ω γ i = γ Then each X i is ω 1 -iterable. Thin Projective Equivalence Relations 8

Let ω ω ∩ V = { y i : i ∈ ω } . We can now iterate M † 2 ( x, y ) → N 0 → N 1 → ... → N i → ... so that y i is Col ( ω, π 0 i ( γ i )) -generic over π 0 i ( X i ) , by Woodin’s genericity iteration. Let π ij : N i → N j denote the iteration maps. Let N ω = dirlim i → ω N i . Then y i is still Col ( ω, π 0 i ( γ i )) -generic over π 0 j ( X i ) for all j with i ≤ j ≤ ω . Note that sup i ∈ ω π 0 i ( γ i ) = ω V 1 . Let G be a Col ( ω, < ω V 1 ) -generic filter over N ω in V such that ω ω ∩ N ω [ G ] ⊆ ω ω ∩ V . Thin Projective Equivalence Relations 9

3 = T N ω [ G ] Claim. T V 3 Proof. It is sufficient to prove that for any Π 1 3 rank and every y i ∈ ω ω ∩ V , there is z ∈ ω ω ∩ N ω [ G ] of the same rank. Note that N ω [ G ] ≺ Σ 1 3 V since N ω [ G ] has a Woodin cardinal and a measurable above it, and is iterable. To prove this, fix y i . Suppose G i is Col ( ω, π 0 i ( γ i )) -generic over π 0 ω ( X i ) with y i ∈ π 0 ω ( X i )[ G i ] . Let ˙ x be a name with x G i = y i . Let ˙ ˙ x 0 , ˙ x 1 be the corresponding names for left and right generic. We can now find a condition p ∈ G i such that ( p, p ) � ” ˙ x 0 and ˙ x 1 have the same rank”. Thin Projective Equivalence Relations 10

x H . Let H ∈ N ω [ G ] generic below p over π 0 ω ( X i ) and z = ˙ Find H ′ generic below p over both π 0 ω ( X i )[ G i ] and π 0 ω ( X i )[ H ] . Since π 0 ω ( X i )[ G i , H ′ ] and π 0 ω ( X i )[ H, H ′ ] are iterable and have a Woodin cardinal and a measurable above it, we get N ω [ G i , H ′ ] ≺ Σ 1 3 V and π 0 ω ( X i )[ G i , H ′ ] ≺ Σ 1 3 V . So these models compute the x H ′ all have the same rank correctly. Hence y i , z, ˙ rank. Thin Projective Equivalence Relations 11

Since Col ( ω, < sup γ i ) is homogeneous, we now have in V : M † • there is x ∈ ω ω such that � 2 ( x,y ) Col ( ω,< sup γ i ) t ϕ ( y, T 3 )( x ) Since M † 2 ( x, y ) is coded by a Π 1 4 ( x, y ) real, this is a Σ 1 5 statement. Hence this is true in M , let x ′ ∈ ω ω ∩ M ⊆ M witness this. Since we can again iterate M † 2 ( x, y ) to some N ′ ω to make the reals of M generic, we have T N ′ Col ( ω,<ωM 1 ) = T M = T V ω 3 3 3 Since ϕ = t ϕ ( y, T V 3 ) , then ϕ ( x ′ ) holds. Hence ( x, x ′ ) ∈ E . Thin Projective Equivalence Relations 12

Leo Harrington, Saharon Shelah: Counting equivalence classes of co- κ -Souslin equivalence relations, Logic Colloquium 1982, North-Holland 1982 Greg Hjorth: Thin equivalence relations and effective decompositions, Journal of Symbolic Logic, Vol. 58, No. 4, Dec. 1993 Greg Hjorth: Some applications of coarse inner model theory, Journal of Symbolic Logic, Vol. 62, No. 2, June 1997 John R. Steel: Projectively well-ordered inner models, Annals of Pure and Applied Logic 74, 1995 Thin Projective Equivalence Relations 13