SakaéFuchino ALöwenheim-SkolemTheoremforavariationofstationarylogicwhichimpliesthatthecontinuumisverylarge

A Löwenheim-Skolem Theorem for a variation of stationary logic

which implies that the continuum is very large

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3月23日(17:10 EDT) version) 2019年3月22日(於Logic Workshop, CUNY) This presentation is typeset bypL^ATEXwithbeamerclass.

The most up-to-date version of these slides is downloadable as

http://fuchino.ddo.jp/slides/newyork2019-03-pf.pdf

The results in the following slides ... Löwenheim-Skolem (2/8) are going to appear in a joint paper 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 generic large cardinals and some

applications, in preparation.

The size of the continuum Löwenheim-Skolem (3/8)

◮ The size of the continuum is either ℵ ¹ or ℵ ² or very large!

The size of the continuum (1/2) Löwenheim-Skolem (4/8)

◮ 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.

Theorem 1.

✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿

SDLS(L

, < ℵ

) implies CH.

Actually SDLS(L

, < ℵ

) is equivalent with Sean Cox’s

✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿

Diagonal Reflection Principle for internal clubness + CH.

Theorem 2. (a)

✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿

SDLS

(L

, < 2

) implies 2

= ℵ

.

(b)

SDLS

(L

, < ℵ

) is equivalent to Diagonal Reflection

✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿

Principle for internal clubness (c) SDLS

(L

, < 2

) is equivalent to SDLS

(L

, < ℵ

) + ¬CH.

Theorem 3.

✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿

SDLS

(L

, < 2

) implies 2

is very large

(e.g. weakly Mahlo, weakly hyper Mahlo, etc.)

The size of the continuum (2/2) Löwenheim-Skolem (5/8)

◮ 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

and CH holds. More- over MA

(σ-closed) holds. Thus SDLS(L

, < ℵ

) follows.

Theorem 2. If there exists a Laver-generically supercompact car- dinal κ for proper p.o.s, then κ = ℵ

= 2

. Moreover PFA

holds. Thus SDLS

(L

, < 2

) follows.

Theorem 3. If there exists a Laver generically supercompact cardi-

nal κ for c.c.c. p.o.s, then κ ≤ 2

and κ is very large (e.g. weakly

Mahlo, weakly hyper Mahlo, etc.) Moreover MA

(ccc , < κ) for

all µ < κ and SDLS

(L

, < κ) hold.

Consistency of Laver-generically supercompact cardinals Löwenheim-Skolem (6/8) 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).

Conclusion Löwenheim-Skolem (7/8)

◮ 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.

⊲ These “reasonable” reflection principle in terms of Löwenheim

Skolem Theorem are consequences of respective axioms of generic

supercompactness.

Some more background and open problems Löwenheim-Skolem (8/8)

◮ By a slight modification of B. König’s results, the implication of SDLS(L

, ≤ ℵ

) from the existence of Laver-generically supercompact cardinal can be interpolated by a Game Reflection Principle which by itself characterizes the usual version of generically supercompactness of ℵ

.

⊲ Do there exist some Game Reflection Principles which play similar role in the other two scenerios in the trichotomy?

⊲ Does (Strong) Laver-generially supercompactness of κ for c.c.c.

p.o.s imply κ = 2

?

Thank you for your attention.

Thank you for your attention.

Thank you for your attention.

Thank you for your attention.

SDLS

+

(L

, < 2

) implies 2

is very large.

◮ For a regular cardinal κ and a cardinal λ ≥ κ, S ⊆ P

(λ) is said to be 2-stationary if, for any stationary T ⊆ P

(λ), there is an a ∈ S s.t. | κ ∩ a | is a regular uncountable cardinal and T ∩ P

(a) is stationary in P

(a). A regular cardinal κ has the 2-stationarity property if P

(λ) is 2-stationary (as a subset of itself) for all λ ≥ κ.

Lemma 1. For a regular uncountable κ, SDLS

(L

, < κ) implies that κ is 2-stationary.

Lemma 2. Suppose that κ is a regular uncountable cardinal.

(1) If κ is 2-stationary then κ is a limit cardinal.

(2) For any λ ≥ κ, 2-stationary S ⊆ P

(λ), and any stationary T ⊆ P

(λ), there are stationarily many r ∈ S s.t. T ∩ P

(r) is stationary.

(3) If κ is 2-stationary then κ is a weakly Mahlo cardinal.

もどる

SDLS

(L

, < 2

) is equivalent to SDLS

(L

, < ℵ

) + ¬ CH .

◮ If SDLS

(L

, < 2

) holds then 2

= ℵ

by (a). Thus, it follows that SDLS

(L

, < ℵ

) + ¬CH holds.

◮ Suppose SDLS

(L

, < ℵ

) holds. Then we have 2

≤ ℵ

by a theorem of Todorčević already mentioned. Thus, if 2

> ℵ

in addition, we have 2

= ℵ

. Thus SDLS

(L

, < 2

) follows.

もどる

Baumgartner’s Theorem

Theorem 1 (J.E. Baumgartner). Let ω < κ < λ and κ be regular. Then any club subset of [λ]

has cardinality ≥ λ

.

もどる

SDLS

(L

, < κ) for κ > ℵ

implies κ > 2

.

◮ SDLS

(L

, < ℵ

) implies 2

≤ ℵ

: it is easy to see that SDLS

(L

, < ℵ

) implies the reflection principle RP(ω

) in [millennium-book]. RP(ω

) implies 2

≤ ℵ

(Todorčević). Thus we have κ > ℵ

≥ 2

.

◮ Thus we may assume that SDLS

(L

, < ℵ

) does not hold.

Hence there is a structure A s.t., for any B ≺

(L^ℵstat⁰)

A , we have k B k ≥ ℵ

. Let λ = k A k . W.l.o.g., we may assume that the underlying set of A is = λ. Let A

= hH(λ

), λ, ...

|{z}

, ∈i.

◮ By SDLS

(L

, < κ), there is M ∈ [H(λ

)]

s.t.

A

↾ M ≺

(L^ℵstat⁰)

A

. It follows that A ↾ (λ ∩ M ) ≺

(L^ℵstat⁰)

A . By the choice of A , we have | M | ≥ | λ ∩ M | ≥ ℵ

.

◮ By elementarity, there is C ⊆ [M ]

∩ M which is a club in [M]

. By

a theorem of Baumgartner , it follows that

κ > | M | ≥ | C | ≥ 2

.

もどる

SDLS

(L

, < 2

) implies 2

= ℵ

.

Proposition 1. SDLS

(L

, < κ) for κ > ℵ

implies κ > 2

.

Proof

◮ Suppose that SDLS

(L

, < 2

) holds. Then 2

≤ ℵ

by the Proposition 1.

◮ SDLS

(L

, < ℵ

) does not hold since

“there are uncountably many x s.t. ...”

is expressible in L

. Thus, 2

> ℵ

.

もどる

SDLS (L

, < ℵ

) implies CH .

◮ Suppose that A = hH(ω

), ∈i and Let B ∈ [H(ω

)]

be s.t.

A ↾ B ≺

A . Then for any U ∈ [B]

we have A | = “ ∃x ∀y (y ∈ x ↔ y ε U)” .

◮ By elementarity we also have B | = “ ∃x ∀y (y ∈ x ↔ y ε U)” .

⊲ It follows that U ∈ B . Thus [B]

⊆ B and 2

≤ | B | ≤ ℵ

.

もどる

Strong Downward Löwneheim-Skolem Theorem for stationary logic

⊲ L

is a weak second order logic with monadic second-order variables X , Y etc. which run over the countable subsets of the underlying set of a structure. The logic has only the weak second order quantifier “stat ” and its dual “aa” (but not the second-order existential (or universal) quantifiers) with the interpretation:

A | = stat X ϕ(..., X ) :⇔

{U ∈ [A]

: A | = ϕ(..., U )} is a stationary subset of [A]

.

⊲ For B = hB, ...i ⊆ A , B ≺

A :⇔

B | = ϕ(a

, ..., U

, ...) ⇔ A | = ϕ(a

, ..., U

, ...) for all L

-formula ϕ = ϕ(x

, ..., X

, ...) and for all a

, ... ∈ B and for all

U

, ... ∈ [B]

.

◮ SDLS(L

, < κ) :⇔

For any structure A = hA, ...i of countable signature, there is a structure B of size < κ s.t. B ≺

stat

A .

A weakening of the Strong Downward Löwneheim-Skolem Theorem

⊲ For B = h B, ...i ⊆ A, B ≺

L^ℵstat⁰

A :⇔

B | = ϕ(a

, ...) ⇔ A | = ϕ(a

, ...) for all L

-formula ϕ = ϕ(x

, ...) without free seond-order variables and for all a

, ... ∈ B.

◮ SDLS

(L

, < κ) :⇔

For any structure A = hA, ...i of countable signature, there is a structure B of size < κ s.t. B ≺

L^ℵ0stat

A .

もどる

Strong Downward Löwneheim-Skolem Theorem for PKL logic

⊲ L

is the weak second-order logic with monadic second order variables X , Y , etc. with built-in unary predicate symbol K . The monadic seond order variables run over elements of P

(A) for a structure A = hA, K

, ...i where we denote

P

(T ) = P

(T ) = {u ⊆ T : | u | < | S |}. The logic has the unique second order quantifier “stat” (and its dual).

⊲ For B = hB, K ∩ B, ...i ⊆ A = hA, K , ...i, B ≺

A :⇔

B | =

ϕ(a

, ..., U

, ...) ⇔ A | =

ϕ(a

, ..., U

, ...) for all

L

-formula ϕ = ϕ(x

, ...) a

, ... ∈ B and U

, ... ∈ P

(B) ∩ B.

◮ SDLS

(L

, < κ) :⇔

for any regular λ ≥ κ and a structuer A = hA, K , ...i of countable signature with | A | = λ and | K | = κ. hH(λ), κ, ∈i, there is a structure B of size < κ s.t. B ≺

stat

A .

Strong Downward Löwneheim-Skolem Theorem for PKL logic (2/2)

◮ SDLS

(L

, < κ) :⇔

for any regular λ ≥ κ and a structuer A = hA, K , ...i of countable signature with | A | = λ and | K | = κ. hH(λ), κ, ∈i, there are stationarily many structures B of size < κ s.t. B ≺

stat

A .

⊲ The internal interpretation of the quantifier is defined by:

A | =

stat X ϕ(a

, ..., U

, ..., X ) :⇔

{U ∈ P

(A) ∩ A : A | =

ϕ(a

, ..., U

, ..., U )} is a stationary subset of P

(A) for a

, ...A and U

, ... ∈ P

(A) ∩ A.

もどる

Laver generically supercompact cardinals

◮ For a class P of p.o.s, a cardinal κ is a Laver-generically supercomact for P if, for all regular λ ≥ κ and P ∈ P there is Q ∈ P with Q = P ∗ R

∼

, s.t., for any (V, Q )-generic H , there are a inner model M ⊆ V[H], and an elementary embedding j : V → M s.t.

(1) crit(j ) = κ, j (κ) > λ.

(2) P, H ∈ M, (3) j

λ ∈ M .

もどる

Diagonal Reflection Principle

◮ For a regular cardinal θ > ℵ

:

DRP(θ, IC): There are stationarily many M ∈ [H((θ

)

)]

s.t.

(1) M ∩ H(θ) is

✿✿✿✿✿✿✿✿✿✿✿✿

internally club ;

(2) for all R ∈ M s.t. R is a stationary subset of [θ]

, R ∩ [θ ∩ M]

is stationary in [θ ∩ M]

.

◮ For a regular cardinal λ > ℵ

(∗)

: For any countable expansion A ˜ of hH(λ), ∈i, if

hS

: a ∈ H(λ)i, is a family of stationary subsets of [H(λ)]

, then there is an internally club M ∈ [H(λ)]

s.t. A ˜ ↾ M ≺ A ˜ and S

∩ [M ]

is stationary in [M ]

, for all a ∈ M .

Proposition 1. TFAE: (a) The global version of Diagonal Reflec- tion Principle of S.Cox for internal clubness (i.e. DRP(θ, IC) for all regular θ > ℵ

) holds.

(b) (∗)

for all regular λ > ℵ

holds.

(c) SDLS

(L

, < ℵ

) holds.

もどる

Reflection Principles RP

◮ The following are variations of the “Reflection Principle” in [Jech, Millennium Book].

RP

For any uncountable cardinal λ, stationary S ⊆ [H(λ)]

and any countable expansion A of the structure hH(λ), ∈i, there is an

internally club M ∈ [H(λ)]

s.t. (1) A ↾ M ≺ A ; and (2) S ∩ [M]

is stationary in [M]

.

RP

For any uncountable cardinal λ, stationary S ⊆ [H(λ)]

and any countable expansion A of the structure hH(λ), ∈i, there is an

internally unbounded M ∈ [H(λ)]

s.t. (1) A ↾ M ≺ A ; and (2) S ∩ [M ]

is stationary in [M]

.

Since every internally club M is internally unbounded, we have:

Lemma 1. RP

implies RP

.

RP

is also called Axiom R in Set-Theoretic Topology.

Theorem 2. ([?]) RP

implies FRP.

もどる

Stationary subsets of [ X ]

◮ C ⊆ [X ]

is club in [X ]

if (1) for every u ∈ [X ]

, there is v ∈ C with u ⊆ v; and (2) for any countable increasing chain F in C we have S

F ∈ C .

⊲ S ⊆ [X ]

is stationary in [X ]

if S ∩ C 6= ∅ for all club C ⊆ [X ]

.

◮ A set M is internally unbounded if M ∩ [M ]

is cofinal in [M]

(w.r.t. ⊆)

⊲ A set M is internally stationary if M ∩ [M ]

is stationary in [M ]

⊲ A set M is internally club if M ∩ [M]

contains a club in [M ]

.

“ Diagonal Reflection Principle”にもどる “RP_??”にもどる

Fodor-type Reflection Principle (FRP) (FRP) For any regular κ > ω

, any stationary E ⊆ E

and any

mapping g : E → [κ]

with g(α) ⊆ α for all α ∈ E , there is γ ∈ E

s.t.

(*) for any I ∈ [γ]

closed w.r.t. g and club in γ, if hI

: α < ω

i is a filtration of I then sup(I

) ∈ E and g (sup(I

)) ⊆ I

hold for stationarily many α < ω

.

⊲ F = hI

: α < λi is a filtration of I if F is a continuously increasing

⊆-sequence of subsets of I of cardinality < | I | s.t. I = S

α<λ

I

.

◮ FRP follows from Martin’s Maximum or Rado’s Conjecture.

MA

(σ-closed) already implies FRP but PFA does not imply FRP since PFA does not imply stationary reflection of subsets of E

(Magidor, Beaudoin) which is a consequence of FRP.

◮ FRP is a large cardinal property: FRP implies the total failure of the square principle.

⊲ FRP is known to be equivalent to the reflection of uncountable

coloring number of graphs down to cardinality < ℵ

.

Proof of Fact 1

Fact 1. (A. Hajnal and I. Juhász, 1976) For any uncountable cardi- nal κ there is a non-metrizable space X of size κ s.t. all subspaces Y of X of cardinality < κ are metrizable.

Proof.

◮ Let κ

≥ κ be of cofinality ≥ κ, ω

.

⊲ The topological space (κ

+ 1, O) with

O = P(κ

) ∪ {(κ

\ x) ∪ {κ

} : x ⊆ κ

, x is bounded in κ

} is non-metrizable since the point κ

has character = cf (κ

) > ℵ

.

⊲ Any subspace of κ

+ 1 of size < κ is discrete and hence metrizable.

もどる

Proof of Fact 3

◮ It is enough to prove the following:

Lemma 1. (Folklore ?, see [?]) For a regular cardinal κ ≥ ℵ

if, there is a

✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿

non-reflectingly stationary S ⊆ E

, then there is a non

meta-lindelöf (and hence non metrizable) locally compact and locally countable topological space X of cardinality κ s.t. all subspace Y of X of cardinality < κ are metrizable.

Proof.

◮ Let I = {α + 1 : α < κ} and X = S ∪ I .

⊲ Let ha

: α ∈ Si be s.t. a

∈ [I ∩ α]

, a

is of order-type ω and cofinal in α. Let O be the topology on X introduced by letting

(1) elements of I are isolated; and

(2) {a

∪ {α} \ β : β < α} a neighborhood base of each α ∈ S.

◮ Then (X , O) is not meta-lindelöf (by Fodor’s Lemma) but each

α < κ as subspace of X is metrizable (by induction on α).

Coloring number and chromatic number of a graph

◮ For a cardinal κ ∈ Card, a graph G = hG , K i has coloring number

≤ κ if there is a well-ordering ⊑ on G s.t. for all p ∈ G the set {q ∈ G : q ⊑ p and q K p}

has cardinality < κ.

⊲ The coloring number col (G ) of a graph G is the minimal cardinal among such κ as above.

◮ The chromatic number chr(G ) of a graph G = hG , K i is the minimal cardinal κ s.t. G can be partitioned into κ pieces G = S

α<κ

G

s.t. each G

is pairwise non adjacent (independent).

⊲ For all graph G we have chr (G) ≤ col (G ).

もどる

κ-special trees

◮ For a cardinal κ, a tree T is said to be κ-special if T can be represented as a union of κ subsets T

, α < κ s.t. each T

is an antichain (i.e. pairwise incomparable set).

もどる

Stationary subset of E

◮ For a cardinal κ,

E

= {γ < κ : cf(γ) = ω}.

◮ A subset C ⊆ ξ of an ordinal ξ of uncountable cofinality, C is closed unbounded (club) in ξ if (1): C is cofinal in ξ (w.r.t. the canonical ordering of ordinals) and (2): for all η < ξ, if C ∩ η is cofinal in η then η ∈ C .

◮ S ⊆ ξ is stationary if S ∩ C 6= ∅ for all club C ⊆ ξ.

◮ A stationary S ⊆ ξ if reflectingly stationary if there is some η < ξ of uncountable cofinality s.t.S ∩ η is stationary in η. Thus:

◮ A stationary S ⊆ ξ if non reflectingly stationary if S ∩ η is non stationary for all η < ξ of uncountable cofinality.

もどる

Proof of Theorem 1.

CH ⇒ SDLS(L

, < ℵ

): For a structure A with a countable signature L and underlying set A, let θ be large enough and A ˜ = hH(θ), A, A , ∈i. where A = A

for a unary predicate symbol A and A = A

for a constant symbol A . Let B ˜ ≺ A ˜ be

s.t.| B | = ℵ

for the underlying set B of B and [B]

⊆ B.

B = A ↾ A

is then as desired.

SDLS(L

, < ℵ

) ⇒ CH: Suppose A = {ω

∪ [ω

]

, ∈}. Consider the L

-formula ϕ(X ) = ∃x∀y (y ∈ x ↔ y ε X ).

If B = hB, ...i is s.t. | B | ≤ ℵ

and B ≺

, then for C ∈ [B]

, since A | = ϕ(C ), we have B | = ϕ(C ). It dollows that [B]

⊆ B and 2

≤ (| B |)

≤ | B | = ℵ

.

もどる