Strong Löwenheim-Skolem Theorem of stationary logics, game reflection principles and generically supercompact cardinals Sakaé 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 ) http://fuchino.ddo.jp/index.html (2019 年 7 月 28 日 (15:08 JST) version) 2019 年 6 月 17 日 ( 於 16th Asian Logic Conference) 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/nur-sultan2019-06-pf.pdf
The solution of the Continuum Problem gen. suparcomapct card. (2/11)
A The solution of the Continuum Problem gen. suparcomapct card. (2/11) ◮ The continuum is either ℵ 1 or ℵ 2 or very large.
A The solution of the Continuum Problem gen. suparcomapct card. (2/11) ◮ The continuum is either ℵ 1 or ℵ 2 or very large. ⊲ Provided that a sufficiently strong and reasonable reflection principle should hold. ◮ The continuum is either ℵ 1 or ℵ 2 or very large. ⊲ Provided that a Laver-generically supercompact cardinal should exist. Under a Laver-generically supercompact cardinal, in each of the three scenarios, the respective reflection principle in the sense of above also holds.
The results in the following slides ... gen. suparcomapct card. (3/11) are going to appear in joint papers with André Ottenbereit Maschio Rodriques and Hiroshi Sakai: [1] Sakaé Fuchino, André Ottenbereit Maschio Rodriques and Hiroshi Sakai, Strong downward Löwenheim-Skolem theorems for stationary logics, I, submitted. http://fuchino.ddo.jp/papers/SDLS-x.pdf [2] Sakaé Fuchino, André Ottenbereit Maschio Rodriques and Hiroshi Sakai, Strong downward Löwenheim-Skolem theorems for stationary logics, II — reflection down to the continuum, pre-preprint. http://fuchino.ddo.jp/papers/SDLS-II-x.pdf [3] Sakaé Fuchino, André Ottenbereit Maschio Rodriques and Hiroshi Sakai, Strong downward Löwenheim-Skolem theorems for stationary logics, III — more on Laver-generically large cardinals, in preparation.
The size of the continuum gen. suparcomapct card. (4/11) ◮ The size of the continuum is either ℵ 1 or ℵ 2 or very large. ⊲ provided that a "reasonable" and sufficiently strong reflection principle should hold.
The size of the continuum (1/2) gen. suparcomapct card. (5/11) ◮ The size of the continuum is either ℵ 1 or ℵ 2 or very large. ⊲ provided that a "reasonable" and sufficiently strong reflection principle should hold. SDLS ( L ℵ 0 Theorem 1. stat , < ℵ 2 ) implies CH . Proof ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ Actually SDLS ( L ℵ 0 stat , < ℵ 2 ) is equivalent with Sean Cox’s Diagonal Reflection Principle for internal clubness + CH. ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ stat , < 2 ℵ 0 ) implies 2 ℵ 0 = ℵ 2 . SDLS − ( L ℵ 0 Theorem 2. (a) ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ Proof SDLS − ( L ℵ 0 stat , < ℵ 2 ) is equivalent to Diagonal Reflection (b) ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ (c) SDLS − ( L ℵ 0 stat , < 2 ℵ 0 ) is Principle for internal clubness ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ equivalent to SDLS − ( L ℵ 0 stat , < ℵ 2 ) + ¬ CH . Proof stat , < 2 ℵ 0 ) implies 2 ℵ 0 is very large SDLS int + ( L PKL Theorem 3. ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ (e.g. weakly Mahlo, weakly hyper Mahlo, etc.) Proof
The size of the continuum (2/2) gen. suparcomapct card. (6/11) ◮ The size of the continuum is either ℵ 1 or ℵ 2 or very large! ⊲ provided that a strong variant of generic large cardinal exists.
The size of the continuum (2/2) gen. suparcomapct card. (6/11) ◮ The size of the continuum is either ℵ 1 or ℵ 2 or very large! ⊲ provided that a strong variant of generic large cardinal exists. Theorem 1. If there exists a Laver-generically supercompact ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ cardinal κ for σ -closed p.o.s, then κ = ℵ 2 and CH holds. More- ✿✿✿✿✿✿✿✿ over MA + ℵ 1 ( σ -closed ) holds. Thus SDLS ( L ℵ 0 stat , < ℵ 2 ) also holds. Theorem 2. If there exists a Laver-generically supercompact car- dinal κ for proper p.o.s, then κ = ℵ 2 = 2 ℵ 0 . Moreover PFA + ℵ 1 holds. Thus SDLS − ( L ℵ 0 stat , < 2 ℵ 0 ) also holds. Theorem 3. If there exists a Laver generically supercompact car- dinal κ for c.c.c. p.o.s, then κ ≤ 2 ℵ 0 and κ is very large (for all regular λ ≥ κ , there is a σ -saturated normal ideal over P κ ( λ ) ). Moreover MA + µ ( ccc , < κ ) for all µ < κ and SDLS int + ( L PKL stat , < κ ) hold.
Consistency of Laver-generically supercompact cardinals gen. suparcomapct card. (7/11) Theorem 1. (1) Suppose that ZFC + “there exists a supercom- pact cardinal” is consistent. Then ZFC + “there exists a Laver- generically supercompact cardinal for σ -closed p.o.s” is consistent as well. (2) Suppose that ZFC + “there exists a superhuge cardinal” is consistent. Then ZFC + “there exists a Laver-generically super- compact cardinal for proper p.o.s” is consistent as well. (3) Suppose that ZFC + “there exists a supercompact cardinal” is consistent. Then ZFC + “there exists a strongly Laver-generically supercompact cardinal for c.c.c. p.o.s” is consistent as well. Proof. Starting from a model of ZFC with a supercompact cardinal κ (a superhuge cardinal in case of (2)), we can obtain models of respective assertions by iterating (in countable support in case of (1), (2) and in finite support in case of (3)) with respective p.o.s κ times along a Laver function (for (1) and (2) Laver function for supercompactness; for (2), Laver function for super- almost-hugeness). �
Some more background and open problems gen. suparcomapct card. (8/11) ◮ By a slight modification of a proof by B. König, the implication of SDLS ( L ℵ 0 stat , < ℵ 2 ) from the existence of Laver-generically supercompact cardinal for σ -closed p.o.s can be interpolated by a Game Reflection Principle which by itself characterizes the usual version of generic supercompactness of ℵ 2 by σ -closed p.o.s. Problem 1. Does there exist some sort of Game Reflection Principle which plays similar role in the other two scenarios in the trichotomy? Problem 2. Does (some variation of) Laver-generic supercompactness of κ for c.c.c. p.o.s imply κ = 2 ℵ 0 ? Problem 3. Is there any characterization of MA ++ (...) which would fit our context? Problem 4. What is about Laver-generic supercompactness for Cohen reals? What is about Laver-generic supercompactness for stationary preserving p.o.s?
A partial solution of Problem 2 gen. suparcomapct card. (9/11) Lemma 1. Suppose that P is a class of p.o.s containing a p.o. P which adds a new real. If κ is a Laver-generically supercompact for P , then κ ≤ 2 ℵ 0 . Proof. Let P ∈ P be s.t. any generic filter over P codes a new real. Suppose that µ < κ . We show that 2 ℵ 0 > µ . Let � a = � a ξ : ξ < µ � be a sequence of subsets of ω . It is enough to show that � a does not enumerate P ( ω ) . ◮ By Laver-generic supercompactness of κ for P , there are Q ∈ P ◦ Q , ( V , Q ) -generic H , transitive M ⊆ V [ H ] and j ⊆ M [ H ] with P ≤ with j : V ≺ → M , crit ( j ) = κ and P , H ∈ M . Since µ < κ , we have j ( � a ) = � a . ◮ Since H ∈ M where G = H ∩ P and G codes a new real not in V, we have a ) does not enumerate 2 ℵ 0 ” . M | = “ j ( � ◮ By elementarity, it follows that a does not enumerate 2 ℵ 0 ” . V | = “ � �
A partial solution of Problem 2 (2/2) gen. suparcomapct card. (10/11) Theorem 2. If κ is ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ tightly Laver-generically superhuge for ccc p.o.s, then κ = 2 ℵ 0 . Proof. Suppose that κ is tightly Laver-generically superhuge for ccc p.o.s. By Lemma 1 on the previous slide, we have 2 ℵ 0 ≥ κ . To prove 2 ℵ 0 ≤ κ , let λ ≥ κ , 2 ℵ 0 be large enough and let Q be a ccc p.o. s.t. there are ( V , Q ) -generic H and j : V ≺ → M ⊆ V [ H ] with crit ( j ) = κ , | Q | ≤ j ( κ ) > λ , H ∈ M and j ′′ j ( κ ) ∈ M . ◮ Since M | = “ j ( κ ) is regular ” by elementarity, j ( κ ) is also regular in = “ j ( κ ) ℵ 0 = j ( κ ) ” by V by the closedness of M . Thus, we have V | SCH above max { κ, 2 ℵ 0 } (available under the assumption on κ ). ◮ Since Q has the ccc and | Q | ≤ j ( κ ) , it follows that = “ 2 ℵ 0 ≤ j ( κ ) ” . Now we have ( j ( κ ) + ) M = ( j ( κ ) + ) V [ H ] by V [ H ] | = “ 2 ℵ 0 ≤ j ( κ ) ” . j ′′ j ( κ ) ∈ M . Thus M | = “ 2 ℵ 0 ≤ κ ” . ◮ By elementarity, it follows that V | �
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