Reflection principles on [ λ ] ω Toshimichi Usuba Nagoya University Oct. 26, 2010 RIMS Set Theory Workshop 2010, Kyoto
Weak Reflection Principle ✓ ✏ Definition 1. X : non-empty set. S ⊆ [ X ] ω is stationary (in [ X ] ω ) ⇒ For every function f : <ω X → X there is x ∈ S such ⇐ that f “ <ω x ⊆ x . ✒ ✑ ✓ ✏ Definition 2. λ : cardinal ≥ ω 2 . WRP( λ ) For every stationary S ⊆ [ λ ] ω , there is X ⊆ λ such ⇐ ⇒ that | X | = ω 1 ⊆ X and S reflects at X , i.e., S ∩ [ X ] ω is stationary in [ X ] ω . WRP ⇐ ⇒ WRP( λ ) for every λ ≥ ω 2 . ✒ ✑ 1
Weak Reflection Principle There are many useful consequences from WRP( λ ): ✓ ✏ Fact 3 (Foreman-Magidor-Shelah, Shelah, Todorcevic) . WRP( ω 2 ) ⇒ 2 ω ≤ ω 2 . ➀ ➁ WRP ⇒ NS ω 1 is presaturated. ➂ WRP ⇒ SCH . ✒ ✑ 2
Reflection Principle ✓ ✏ Definition 4. λ : regular cardinal ≥ ω 2 . RP( λ ) ⇐ ⇒ For every stationary S ⊆ [ λ ] ω , there is X ⊆ λ such that | X | = ω 1 ⊆ X , cf(sup( X )) = ω 1 , and S reflects at X . ✒ ✑ Martin’s Maximum ⇒ RP( λ ) ⇒ WRP( λ ). 3
Fodor-type Reflection Principle S : set of ordinals ⃗ c = ⟨ c α : α ∈ S ⟩ is a ladder on S if for every α ∈ S , c α ⊆ α is unbounded in α and ot( c α ) = cf( α ). ✓ ✏ Definition 5 (Fuchino-Juh´ asz-Soukup-Szentmikl´ ossy-U.) . λ ≥ ω 2 : regular FRP( λ ) ⇐ ⇒ For every stationary E ⊆ { α < λ : cf( α ) = ω } and every ladder ⃗ c = { c α : α ∈ E } , there is I ⊆ E such that: ➀ | I | = ω 1 = cf(sup( I )) . ➁ For every g : I → λ with g ( α ) ∈ c α , there is ξ < λ with { α ∈ I : g ( α ) = ξ } stationary in sup( I ) . FRP ⇐ ⇒ FRP( λ ) for every λ ≥ ω 2 . ✒ ✑ 4
Fodor-type Reflection Principle ✓ ✏ Fact 6 (F.-J.-So.-Sz.-U., Fuchino-Soukup-Sakai-U.) . FRP For every locally countably compact topo- ⇐ ⇒ logical space X , if every subspace of X with size ω 1 is metrizable, then X itself is also metrizable. ✒ ✑ 5
Semi-Stationary Reflection principle ✓ ✏ Definition 7 (Shelah) . X : a set with ω 1 ⊆ X . S ⊆ [ X ] ω is semi-stationary (in [ X ] ω ) if the set { x ∈ [ λ ] ω : ∃ a ∈ S ( a ⊆ x ∧ a ∩ ω 1 = x ∩ ω 1 ) } is stationary. ✒ ✑ ✓ ✏ Definition 8 (Shelah) . λ : cardinal ≥ ω 2 . SSR( λ ) ⇐ ⇒ For every semi-stationary (or stationary) S ⊆ [ λ ] ω , there is X ⊆ λ such that | X | = ω 1 ⊆ X and S ∩ [ X ] ω is semi- stationary in [ X ] ω . SSR ⇐ ⇒ SSR( λ ) for every λ ≥ ω 2 . ✒ ✑ 6
Semi-Stationary Reflection principle ✓ ✏ Fact 9 (Shelah) . SSR ⇐ ⇒ every ω 1 -stationary preserving forcing notion is semi-proper. ✒ ✑ ✓ ✏ Fact 10 (F.-M.-Sh., Sakai, Todorcevic) . ➀ SSR ⇒ NS ω 1 is precipitous. ➁ SSR ⇒ strong Chang’s conjecture. ➂ SSR( ω 2 ) ⇐ ⇒ WRP( ω 2 ) . ✒ ✑ 7
� � Implications It is easy to check that: RP( λ ) ⇒ WRP( λ ). RP( λ ) ⇒ FRP( λ ). WRP( λ ) ⇒ SSR( λ ). RP( λ ) WRP( λ ) � SSR( λ ) FRP( λ ) 8
� � � � � � � � � Implications RP( λ ) ⇒ WRP( λ ). RP( λ ) ⇒ FRP( λ ). WRP( λ ) ⇒ SSR( λ ). Question: What about other directions? Especially does WRP( λ ) ⇒ RP( λ )? ? ? RP( λ ) WRP( λ ) SSR( λ ) ? ? ? ? FRP( λ ) 9
� � � � � � � � Facts ✓ ✏ Fact 11 (Sakai) . SSR ̸⇒ WRP( ω 3 ) . Fact 12 (F.-J.-So.-Sz.-U.) . FRP ̸⇒ WRP( ω 2 ) . ✒ ✑ ? � ⧸ RP( λ ) WRP( λ ) SSR( λ ) ⧸ ⧸ ? ? FRP( λ ) 10
� � � � � � � Facts about WRP and RP ✓ ✏ Fact 13 (K¨ onig-Larson-Yoshinobu) . Under GCH, WRP( ω n ) ⇒ RP( ω n ) for every 2 ≤ n < ω . ✒ ✑ ✓ ✏ Fact 14 (Krueger) . Con(ZFC + ∃ κ + -supercompact cardinal κ ) ⇒ Con(ZFC + WRP( ω 2 ) + ¬ RP( ω 2 ) ) ✒ ✑ � ⧸ � ⧸ RP( λ ) WRP( λ ) SSR( λ ) ⧸ ⧸ ? ? FRP( λ ) 11
� � � � � � � � Theorem 1 ✓ ✏ Theorem 1. Suppose that there exists a weakly compact cardinal. Then there exists a forcing extension in which the following hold: ➀ WRP( ω 2 ) . ➁ ¬ FRP( ω 2 ) . ✒ ✑ � ⧸ � ⧸ RP( λ ) WRP( λ ) SSR( λ ) ⧸ ⧸ ⧸ ⧸ ⧸ FRP( λ ) 12
Theorem 1 ✓ ✏ Fact 15 (Shelah, Velickovic) . Con(ZFC + ∃ weakly compact cardinal) ⇒ Con(ZFC + WRP( ω 2 ) ) ⇐ ✒ ✑ ✓ ✏ Corollary 16. ➀ Con(ZFC + ∃ weakly compact cardinal) ⇐ ⇒ Con(ZFC + WRP( ω 2 ) + ¬ RP( ω 2 )) . ➁ WRP( ω 2 ) ̸⇒ FRP( ω 2 ) ✒ ✑ 13
Theorem 2 ✓ ✏ Theorem 17. Suppose that there exists a supercompact cardinal. Then there exists a forcing extension in which the following hold: ➀ SSR . ➁ ¬ FRP( ω 2 ) . ✒ ✑ Hence SSR does not imply FRP( ω 2 ). 14
Stationary set induced by ladder For i < 2, let S 2 i = { α < ω 2 : cf( α ) = ω i } \ ( ω 1 + 1). Fix a surjection π α : ω 1 → α for ω 1 < α < ω 2 and let C ∗ be the set of all x ∈ [ ω 2 ] ω such that: sup( x ) > ω 1 x ∩ ω 1 ∈ ω 1 and sup( x ) / ∈ x . ∀ α ∈ x, π α “( x ∩ ω 1 ) = x ∩ α . C ∗ forms a club in [ ω 2 ] ω . ✓ ✏ c be the set of c on S 2 0 , let S ⃗ Definition 18. For a ladder ⃗ all x ∈ C ∗ with c sup( x ) ⊆ x . ✒ ✑ 15
Basic properties of S ⃗ c ✓ ✏ c and [ ω 2 ] ω \ S ⃗ c are stationary. Lemma 19. S ⃗ ✒ ✑ ✓ ✏ Lemma 20. [ ω 2 ] ω \ S ⃗ c does not reflect at any α ∈ S 2 0 . ✒ ✑ Idea of Proof of theorems: Collapse a weakly compact κ to ω 2 . c non-reflecting at any α ∈ S 2 By forcing, we make S ⃗ 1 . Using a iteration of club shootings, destroy the sta- tionarity of non-reflecting subset of S ⃗ c . (Hence every c is reflecting). stationary subset of S ⃗ Using the weak compactness of κ , show that every sta- tionary subset of [ ω 2 ] ω \ S ⃗ c reflects at α ∈ S 2 1 . 16
Basic properties of S ⃗ c ✓ ✏ Lemma 21. For x, y ∈ S ⃗ c , if x ∩ ω 1 = y ∩ ω 1 and sup( x ) = sup( y ) then x = y . ✒ ✑ ✓ ✏ Lemma 22. θ : large regular cardinal, M ≺ H θ : a countable model with c : x ⊆ M ∩ ω 2 } . C ∗ ,⃗ c ∈ M . X = { x ∈ S ⃗ ➀ X is countable. ➁ For x ∈ X , if x ∩ ω 1 = M ∩ ω 1 then x = M ∩ α for some α ∈ ( M ∩ ω 2 ) ∪ { ω 2 } . If S ∈ M is non-reflecting subset of S ⃗ c , x ∈ S ∩ X , and ➂ x ∩ ω 1 = M ∩ ω 1 , then x = M ∩ ω 2 . ✒ ✑ 17
Club Shooting ✓ ✏ Definition 23. For X ⊆ [ ω 2 ] ω , C ( X ) is the poset consists of all functions p such that: p : d ( p ) × d ( p ) → ω 1 for some d ( p ) ∈ [ ω 2 ] ω . ➀ ➁ For every x ∈ X , if x ⊆ d ( p ) then x is not closed under p . For p, q ∈ C ( X ) , p ≤ q ⇐ ⇒ p ⊇ q . ✒ ✑ 18
Club Shooting ✓ ✏ C ( X ) satisfies the (2 ω ) + -c.c. Lemma 24. ➀ For every a ∈ [ ω 2 ] ω , { p ∈ C ( X ) : a ⊆ d ( p ) } is dense ➁ open. If G is ( V, C ( X )) -generic and F = ∪ G , then F : ω V ➂ 2 × ω V 2 → ω 1 and there is no x ∈ X closed under F . ✒ ✑ 19
Complete forcing ✓ ✏ Definition 25 (Shelah) . Let P be a poset and θ a large regular cardinal. ≺ H θ with P ∈ M , For a countable M a sequence ⟨ p i : i < ω ⟩ is generic sequence of M ⟨ p i : i < ω ⟩ is a descending sequence of P and for ⇐ ⇒ every dense open set D ∈ M in P , there is i < ω with p i ∈ D ∩ M . T ⊆ [ λ ] ω : stationary for some λ . A poset P is T -complete every countable M ≺ H θ with P ∈ M ∩ λ ∈ T and ⇐ ⇒ every generic sequence of M has a lower bound. ✒ ✑ 20
Complete forcing ✓ ✏ Lemma 26 (Shelah) . ➀ T -complete poset is σ -Baire. For every stationary T ′ ⊆ T , T -complete poset pre- ➁ serves the stationarity of T ′ . ➂ If P is a countable support iteration of T -complete posets, then P is T -complete. ✒ ✑ 21
Non-reflecting ladder ✓ ✏ c on S 2 Definition 27. A ladder ⃗ 0 is non-reflecting ⇒ for every α ∈ S 2 1 , there is a club d ⊆ α ∩ S 2 ⇐ 0 in α such that { c β : β ∈ d } has an injective choice function. ✒ ✑ ✓ ✏ Lemma 28. If ⃗ c is a non-reflecting ladder, then FRP( ω 2 ) c and ⃗ fails. In fact S ⃗ c witnesses that ¬ FRP( ω 2 ) . ✒ ✑ 22
✓ ✏ Lemma 29. Suppose ⃗ c is a non-reflecting ladder and S ⊆ c a non-reflecting. Let θ be a large regular cardinal and S ⃗ M ≺ H θ a countable model with C ∗ ,⃗ c ∈ M . Suppose M ∩ ω 2 / ∈ S . Then for every x ∈ S , x ⊆ M ∩ ω 2 ⇐ ⇒ x ∈ M . ✒ ✑ ✓ ✏ c is c is non-reflecting ladder and S ⊆ S ⃗ Lemma 30. If ⃗ non-reflecting subset, then C ( S ) is [ ω 2 ] ω \ S ⃗ c -complete. ✒ ✑ 23
✓ ✏ Lemma 31. For every stationary E ⊆ ω 1 , the set { x ∈ C ∗ : x ∩ ω 1 , c sup( x ) ⊈ x } is stationary. ✒ ✑ ✓ ✏ c is non- c is non-reflecting ladder and S ⊆ S ⃗ Lemma 32. If ⃗ reflecting subset, then C ( S ) is ω 1 -stationary preserving. ✒ ✑ 24
✓ ✏ Lemma 33. Suppose CH. Let ⃗ c is non-reflecting ladder and α an ordinal. Suppose that P α is a countable support iteration of the ˙ Q β ’s such that for every β < α , ⊩ P β “ ˙ Q β = S β ⊆ S ⃗ c ”. Then P α satisfies C ( ˙ S β ) for some non-reflecting ˙ the ω 2 -c.c. ✒ ✑ 25
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