Saka e Fuchino ( ) Graduate School of System Informatics, Kobe - PowerPoint PPT Presentation

Reflection principles formulated as L¨ owenheim-Skolem Theorems Saka´ e Fuchino ( 渕野 昌 ) Graduate School of System Informatics, Kobe University, Japan ( 神戸大学大学院 システム情報学研究科 ) Reflections on http://fuchino.ddo.jp/index.html [Reflections on Set-Theoretic Reflection] in celebration of Joan Bagaria’s 60th birthday (2018 年 11 月 24 日 (15:45 CET) version) 2018 年 11 月 18 日 ( 於 Sant Bernat, Montseny, Catalonia) This presentation is typeset by pL A T EX with beamer class. The most up-to-date version of these slides is downloadable as http://fuchino.ddo.jp/slides/joan60-2018-fuchino-slides-pf.pdf

L¨ owenheim-Skolem Theorems on stationary logics L¨ owenheim-Skolem Theorems (2/17) ◮ A part of the following considerations will appear in a joint paper with Hiroshi Sakai and Andr´ e Ottenbreit.

L¨ owenheim-Skolem Theorems on stationary logics L¨ owenheim-Skolem Theorems (3/17) ◮ The logics: L ℵ 0 , II denotes the second order logic with the interpretation of the second order variables such that they run over countable subsets of the underlining set of the considered structure. The logic permits quantification ∃ X , ∀ X over second order variables and the logical predicate x ε X where x is a first order variable and X a second order variable. L ℵ 0 is the logic as above but without the quantification over second order variables. is the logic L ℵ 0 , II with the new quantifier stat X where the L ℵ 0 , II stat semantics A | = stat X ϕ ( X , ... ) is defined by “ { U ∈ [ A ] ℵ 0 : A | = ϕ ( U , ... ) } is stationary in [ A ] ℵ 0 ”. L ℵ 0 stat is the logic L ℵ 0 , II stat without second order quantifiers ∃ X , ∀ X .

L¨ owenheim-Skolem Theorems on stationary logics (2/4) L¨ owenheim-Skolem Theorems (4/17) ◮ Let L be one of the logics defined in the previous slide. ⊲ For a structure A and its substructure B , we write B ≺ L A if, for any L -formula ϕ = ϕ ( x 0 , ..., x m − 1 , X 0 , ..., X n − 1 ), a 0 , ..., a m − 1 ∈ B and U 0 , ..., U n − 1 ∈ [ B ] ℵ 0 we have A | = ϕ ( a 0 , ..., a m − 1 , U 0 , ..., U n − 1 ) ⇔ B | = ϕ ( a 0 , ..., a m − 1 , U 0 , ..., U n − 1 ). ⊲ B ≺ L − A is defined similarly except we only consider L -formulas without any free second order variables. ◮ We define the following strong Downward L¨ owenheim-Skolem property for L : SDLS − ( L , < κ ) : For any structure A of countable signature, there is a substructure B of of A of cardinality < κ s.t. B ≺ L − A . SDLS( L , < κ ) : For any structure A of countable signature, there is a substructure B of of A of cardinality < κ s.t. B ≺ L A .

L¨ owenheim-Skolem Theorems on stationary logics (3/4) L¨ owenheim-Skolem Theorems (5/17) ◮ For “the reflection down to < ℵ 2 ” we obtain the following principles: SDLS − ( L ℵ 0 , < ℵ 2 ), SDLS − ( L ℵ 0 , II , < ℵ 2 ), SDLS − ( L ℵ 0 stat , < ℵ 2 ), SDLS − ( L ℵ 0 , II stat , < ℵ 2 ), SDLS( L ℵ 0 , < ℵ 2 ), SDLS( L ℵ 0 , II , < ℵ 2 ), SDLS( L ℵ 0 stat , < ℵ 2 ), SDLS( L ℵ 0 , II stat , < ℵ 2 ). Lemma 1. SDLS − ( L ℵ 0 , < ℵ 2 ) follows from the usual Downward L¨ owenheim Skolem Theorem and hence it holds in ZFC . Observation 2. (This observation was mentioned in a tutorial by M. Magidor, Barcelona 2016 [Magidor, 2016]) SDLS − ( L ℵ 0 stat , < ℵ 2 ) implies the ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ Fodor-type Reflection Principle . Actually SDLS − ( L ℵ 0 stat , < ℵ 2 ) implies ✿✿✿✿✿ RP IC .

L¨ owenheim-Skolem Theorems on stationary logics (4/4) L¨ owenheim-Skolem Theorems (6/17) ◮ The 7 other statements are also quite tractable: Theorem 1. The following are equivalent: (a) CH ; (b) SDLS( L ℵ 0 , < ℵ 2 ) ; (c) SDLS − ( L ℵ 0 , II , < ℵ 2 ) ; (d) SDLS( L ℵ 0 , II , < ℵ 2 ) . Proof Theorem 2. The following are equivalent: (a) Diagonal Reflec- tion Principle for internally clubness (in the sense of [Cox, 2012]), (b) SDLS − ( L ℵ 0 stat , < ℵ 2 ) . More ... Theorem 3. The following are equivalent: (a) Diagonal Reflection Principle for internally clubness (in the sense of [Cox, 2012]) + CH , (b) CH and SDLS − ( L ℵ 0 stat , < ℵ 2 ) ; (c) SDLS − ( L ℵ 0 , II stat , < ℵ 2 ) ; (d) SDLS( L ℵ 0 stat , < ℵ 2 ) ; (e) SDLS( L ℵ 0 , II stat , < ℵ 2 ) .

L¨ owenheim-Skolem Theorems (7/17) (Strong) Game Reflection Principle (GRP) MM + ω 1 SDLS ( L ℵ 0 ,II stat , < ℵ 2 ) MA + ω 1 ( σ -closed ) MM MA + ( σ -closed ) SDLS − ( L ℵ 0 stat , < ℵ 2 ) Rado Conjecture (RC) RP IC Axiom R = RP IU Fodor-type Reflection Principle (FRP) Semi-stationary Reflection (SSR)

Game Reflection Principle L¨ owenheim-Skolem Theorems (8/17) ◮ The Game Reflection Principle (GRP) of Bernhard K¨ onig (Strong Game Reflection Principle in his terminology in [K¨ onig, 2004]) is defined using the following notion of infinite games: ω 1 > A ( A ) is the game of For any uncountable set A and A ⊆ ω 1 > A , G ω 1 > A ( A ) looks like the length ω 1 for Players I and II. A match in G following: · · · · · · I a 0 a 1 a 2 a ξ ( ξ < ω 1 ) · · · · · · II b 0 b 1 b 2 b ξ where a ξ , b ξ ∈ A for ξ < ω 1 . II wins this match if � a ξ , b ξ : ξ < ω 1 � ∈ [ A ] where � a ξ , b ξ : ξ < ω 1 � is the sequence f ∈ ω 1 A s.t. f (2 ξ ) = a ξ and f (2 ξ + 1) = b ξ for all ξ < ω 1 and [ A ] = { f ∈ ω 1 A : f ↾ α ∈ A for all α < ω 1 } .

Game Reflection Principle (2/2) L¨ owenheim-Skolem Theorems (9/17) GRP: For all uncountable set A and ω 1 -club C ⊆ [ A ] ℵ 1 , if the player ω 1 > A ( A ), there is B ∈ C s.t. II II has no winning strategy in G ω 1 > B ( A ∩ ω 1 > B ). has no winning strategy in G Theorem 1. ([K¨ onig, 2004]) (a) GRP implies CH . (b) GRP implies Rado’s Conjecture. (c) GRP is forced by starting from a supercompact κ and collap- sing it to ℵ 2 by the standard σ -closed collapsing poset. Theorem 2. GRP implies the Diagonal Reflection Principle for internally closedness.

L¨ owenheim-Skolem Theorems (10/17) (Strong) Game Reflection Principle (GRP) MM + ω 1 SDLS ( L ℵ 0 ,II stat , < ℵ 2 ) MA + ω 1 ( σ -closed ) MM MA + ( σ -closed ) SDLS − ( L ℵ 0 stat , < ℵ 2 ) Rado Conjecture (RC) RP IC Axiom R = RP IU Fodor-type Reflection Principle (FRP) Semi-stationary Reflection (SSR)

L¨ owenheim-Skolem Theorems (11/17) CH follows. (Strong) Game Reflection Principle (GRP) MM + ω 1 2 ℵ 0 = ℵ 2 follows. SDLS ( L ℵ 0 ,II stat , < ℵ 2 ) MA + ω 1 ( σ -closed ) MM MA + ( σ -closed ) SDLS − ( L ℵ 0 stat , < ℵ 2 ) Rado Conjecture (RC) 2 ℵ 0 ≤ ℵ 2 RP IC Axiom R = RP IU Fodor-type Reflection Principle (FRP) Semi-stationary Reflection (SSR) The continuum can be “arbitrary” large.

SDLS with large conrtinuum L¨ owenheim-Skolem Theorems (12/17) stat , < ℵ 2 ) implies 2 ℵ 0 ≤ ℵ 2 . Proposition 1. SDLS − ( L ℵ 0 Proof. SDLS − ( L ℵ 0 stat , < ℵ 2 ) implies RP IC which is known to imply 2 ℵ 0 ≤ ℵ 2 . � stat , < 2 ℵ 0 ) implies 2 ℵ 0 ≤ ℵ 2 . Proposition 2. SDLS − ( L ℵ 0 Proof. Assume that SDLS − ( L ℵ 0 stat , < κ ) holds for κ = 2 ℵ 0 > ℵ 2 . By Proposition 1, there is a structure A s.t., for any substructure B of A , B ≺ ( L ℵ 0 stat ) − A implies | B | ≥ ℵ 2 . Let | A | = λ and let A ∗ = �H ( λ + ) , λ, ... , ∈� . ���� Note that = A A ∗ | = aa X ∃ x ∀ y ( y ε X ↔ y ∈ x ) . � �� � Let U ∈ [ H ( λ )] <κ be s.t. := ϕ A ∗ ↾ U ≺ ( L ℵ 0 stat ) − A ∗ . By the choice of A , we have | U | ≥ ℵ 2 . By ϕ , there is a club C ⊆ [ U ] ℵ 0 ∩ U . By [Baumgartner-Taylor], it follows that κ > | U | ≥ | C | ≥ 2 ℵ 0 = κ . A contradiction. �

SDLS with large conrtinuum (2/3) L¨ owenheim-Skolem Theorems (13/17) ◮ There are some (consistent and noteworthy) instances of reflection down to < 2 ℵ 0 where 2 ℵ 0 is very large (e.g. weakly Mahlo and much more): Theorem 1. ([A. Dow, F.D Tall and W.A.R. Weiss 1990]) If ZFC + “there is a strongly compact cardinal” is consistent, then the following assertion is also consistent: 2 ℵ 0 is very large and, for any first countable non-metrizable topological space X , there is a subspace Y of X of cardinality < 2 ℵ 0 which is also non-metrizable. Theorem 2. If ZFC + “there is a strongly compact cardinal” is consistent, then the following assertion is also consistent: 2 ℵ 0 is very large and, for any non-special tree T , there is a non-special subtree T ′ of T of cardinality < 2 ℵ 0 .