reﬂection principles formulated as löwenheim-skolem ...reﬂection principles formulated as...

Reflection Principles formulated asLöwenheim-Skolem Theorems for stationary logics

and the Continuum Problem

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 1月 23日 (10:24 GMT) version)

2019年 1月 22日 (於 Logic Seminar, Bristol, England)

This presentation is typeset by pLATEX with beamer class.

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

http://fuchino.ddo.jp/slides/bristol2019-slides-pf.pdf

http://fuchino.ddo.jp/index.html

http://fuchino.ddo.jp/slides/bristol2019-slides-pf.pdf

Löwenheim-Skolem Theorems on stationary logics Downward Löwenheim-Skolem (2/22)

◮ A part of the following considerations will appear in a joint paperwith Hiroshi Sakai and André Ottenbreit.

Löwenheim-Skolem Theorems on stationary logics Downward Löwenheim-Skolem (3/22)

◮ The logics:

Lℵ0,II denotes the second order logic with the interpretation of thesecond order variables such that they run over countablesubsets of the underlining set of the considered structure.The logic permits quantification ∃X , ∀X over second ordervariables and the logical predicate x ε X where x is a first ordervariable and X a second order variable.

Lℵ0 is the logic as above but without the quantification over secondorder variables.

Lℵ0,IIstat is the logic Lℵ0,II with the new quantifier stat X where the

semantics A |= stat X ϕ(X , ...) is defined by“{U ∈ [A]ℵ0 : A |= ϕ(U, ...)} is stationary in [A]ℵ0”.

Lℵ0stat is the logic Lℵ0,II

stat without second order quantifiers ∃X , ∀X .

Löwenheim-Skolem Theorems on stationary logics (2/4)Downward Löwenheim-Skolem (4/22)

◮ 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, forany L-formula ϕ = ϕ(x0, ..., xm−1,X0, ...,Xn−1), a0, ..., am−1 ∈ B

and U0, ...,Un−1 ∈ [B]ℵ0 we have A |= ϕ(a0, ..., am−1,U0, ...,Un−1)⇔ B |= ϕ(a0, ..., am−1,U0, ...,Un−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 Löwenheim-Skolemproperty for L:

SDLS−(L, < κ) : For any structure A of countable signature, there is asubstructure B of of A of cardinality < κ s.t. B ≺−

L A.

SDLS(L, < κ) : For any structure A of countable signature, there is asubstructure B of of A of cardinality < κ s.t. B ≺L A.


◮ For “the reflection down to < ℵ2” we obtain the following 8principles:

SDLS−(Lℵ0 , < ℵ2), SDLS−(Lℵ0,II , < ℵ2), SDLS−(Lℵ0stat , < ℵ2),

SDLS−(Lℵ0,IIstat , < ℵ2), SDLS(Lℵ0 , < ℵ2), SDLS(Lℵ0,II , < ℵ2),

SDLS(Lℵ0stat , < ℵ2), SDLS(Lℵ0,II

stat , < ℵ2).

Lemma 1. SDLS−(Lℵ0 , < ℵ2) follows from the usual DownwardLöwenheim Skolem Theorem and hence it holds in ZFC.

Observation 2. (This was mentioned in a tutorial by M. Magidor,Barcelona 2016 [Magidor, 2016])

SDLS−(Lℵ0stat , < ℵ2) implies the

✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿

Fodor-type Reflection Principle .

Actually SDLS−(Lℵ0stat , < ℵ2) implies

✿✿✿✿✿

RPIC .


◮ The other 7 principles are also nicely characterized:

Theorem 1.The following are equivalent: (a) CH;(b) SDLS(Lℵ0 , < ℵ2); (c) SDLS−(Lℵ0,II , < ℵ2);(d) SDLS(Lℵ0,II , < ℵ2). Proof

Theorem 2.The following are equivalent: (a) Diagonal Reflec-tion Principle for internally clubness (in the sense of [Cox, 2012]),(b) SDLS−(Lℵ0

stat , < ℵ2). More ...

Theorem 3.The following are equivalent: (a) Diagonal ReflectionPrinciple for internally clubness (in the sense of [Cox, 2012]) +CH,(b) SDLS−(Lℵ0

stat , < ℵ2) + CH;

(c) SDLS−(Lℵ0,IIstat , < ℵ2);

(d) SDLS(Lℵ0stat , < ℵ2);

(e) SDLS(Lℵ0,IIstat , < ℵ2).

Downward Löwenheim-Skolem (7/22)

Fodor-type Reflection Principle (FRP) Semi-stationary Reflection (SSR)

Axiom R = RPIU

Rado Conjecture (RC)

RPIC

MA+(σ-closed)SDLS−(Lℵ0stat, <ℵ2)

MA+ω1(σ-closed) MM

MM+ω1

SDLS (Lℵ0,IIstat , <ℵ2)

(Strong) Game Reflection Principle (GRP)

Sakaé Fuchino

Sakaé Fuchino

Game Reflection Principle Downward Löwenheim-Skolem (8/22)

◮ The Game Reflection Principle (GRP) of Bernhard König (StrongGame Reflection Principle in his terminology in [König, 2004]) isdefined using the following notion of infinite games:

For any uncountable set A and A ⊆ ω1>A, Gω1>A(A) is the game of

length ω1 for Players I and II. A match in Gω1>A(A) looks like the

following:

I a0 a1 a2 · · · aξ · · ·

II b0 b1 b2 · · · bξ · · ·(ξ < ω1)

where aξ, bξ ∈ A for ξ < ω1.II wins this match if 〈aξ, bξ : ξ < ω1〉 ∈ [A] where 〈aξ, bξ : ξ < ω1〉is the sequence f ∈ ω1A s.t. f (2ξ) = aξ and f (2ξ + 1) = bξ for allξ < ω1 and [A] = {f ∈ ω1A : f ↾ α ∈ A for all α < ω1}.

Game Reflection Principle (2/2) Downward Löwenheim-Skolem (9/22)

GRP: For all uncountable set A and ω1-club C ⊆ [A]ℵ1 , if the playerII has no winning strategy in G

ω1>A(A), there is B ∈ C s.t. IIhas no winning strategy in G

ω1>B(A ∩ ω1>B).

Theorem 1. ([König, 2004]) (a) GRP implies CH.

(b) GRP implies Rado’s Conjecture.

(c) GRP is forced by starting from a supercompact κ and collapsingit to ℵ2 by the standard σ-closed collapsing poset.

Theorem 2. GRP implies the Diagonal Reflection Principle forinternally closedness.

Corollary 3. GRP implies SDLS(Lℵ0,IIstat , <ℵ2).



Axiom R = RPIU


RPIC



MM+ω1



Sakaé Fuchino

Sakaé Fuchino



Axiom R = RPIU


RPIC



MM+ω1



CH follows.

The continuum can be “arbitrary” large.

2ℵ0 = ℵ2 follows.

2ℵ0 ≤ ℵ2

Sakaé Fuchino

Sakaé Fuchino

SDLS with large continuum Downward Löwenheim-Skolem (12/22)

Proposition 1. SDLS−(Lℵ0stat , <ℵ2) implies 2ℵ0 ≤ ℵ2.

Proof. SDLS−(Lℵ0stat , <ℵ2) implies RPIC which is known to imply

2ℵ0 ≤ ℵ2. �

Proposition 2. SDLS−(Lℵ0stat , < 2ℵ0) implies 2ℵ0 ≤ ℵ2.

Proof. Assume that SDLS−(Lℵ0stat , < κ) holds for κ = 2ℵ0 > ℵ2. By

Proposition 1, there is a structure A s.t., for any substructure B ofA, B ≺−

(Lℵ0stat)

A implies ‖B‖ ≥ ℵ2. Let |A | = λ w.l.o.g. and let

A∗ = 〈H(λ+), λ, ...

︸︷︷︸

=A

,∈〉.

Note thatA∗ |= aa X∃x∀y (y ε X ↔ y ∈ x)

︸︷︷︸:=ϕ

.

Let U ∈ [H(λ)]<κ be s.t.A∗ ↾ U ≺−

(Lℵ0stat)

A∗. By the choice of A, we have |U | ≥ ℵ2. By ϕ,

there is a club C ⊆ [U]ℵ0 ∩ U. By [Baumgartner-Taylor], it followsthat κ > |U | ≥ | C | ≥ 2ℵ0 = κ. A contradiction. �

SDLS with large continuum (2/3) Downward Löwenheim-Skolem (13/22)

◮ There are some known instances of (mathematically significant)reflection down to < 2ℵ0 for very large 2ℵ0 .

⊲ Both of the following assertions hold in the model obtained byadding strongly compact many Cohen reals.

Theorem 1. (Hamburger’s Problem for < 2ℵ0 [A. Dow, F.D Talland W.A.R. Weiss 1990]) The following assertion is “consistent”:

For any first countable non-metrizable topological space X , thereis a subspace Y of X of cardinality < 2ℵ0 which is also non-metrizable.

Theorem 2. (Rado Conjecture for < 2ℵ0) The following assertionis “consistent”:

For any non-special tree T , there is a non-special subtree T ′ of Tof cardinality < 2ℵ0 .

SDLS with large continuum (3/3) Downward Löwenheim-Skolem (14/22)

◮ SDLS−(Lℵ0stat ,≤ 2ℵ0) is consistent with large continuum (under a

large cardinal assumption).

◮ The weakening of SDLS−(Lℵ0stat , < 2ℵ0) [i.e. (∗)−

< 2ℵ0 ,λas below

for all regular λ ≥ 2ℵ0 ] is consistent with large continuum.

(∗)−<κ,λ: For any countable expansion A of 〈H(λ),∈〉, if

〈Sa : a ∈ H(λ)〉, is a family of stationary subsets of [H(λ)]ℵ0 ,then there is an internally stationary M ∈ [H(λ)]<κ s.t.A ↾ M ≺ A and Sa ∩ [M]ℵ0∩M (= Sa ∩M) is stationary in[M]ℵ0 , for all a ∈ M.

⊲ (∗)−<κ,λ can also be characterized as a✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿

Löwenheim Skolem Theorem .

Theorem 1. (Sakai, Ottenbreit, S.F.) Assuming the consis-tency of two supercompact cardinals, it is consistent that 2ℵ0 is“very large” (e.g. 2ℵ0 carries a σ-saturated ideal) and (∗)−

< 2ℵ0 ,λ

for all regular λ ≥ 2ℵ0 + SDLS−(Lℵ0stat ,≤ 2ℵ0).

Internal interpretation of the stationary logic Downward Löwenheim-Skolem (15/22)

◮ For a structure A = 〈A, ...〉 and the internal interpretation of Lℵ0,IIstat

formulas in A is defined by:

⊲ For Lℵ0,IIstat -formula ϕ = ϕ(x0, ...,X0, ...,X ),

a0, ... ∈ A and U0, ... ∈ [A]ℵ0 ∩ A,

A |=int stat X ϕ(a0, ...,U0, ...,X ) ⇔{U ∈ [A]ℵ0 ∩ A : A |=int ϕ(a0, ...,U0, ...,U)} is stationary in [A]ℵ0 .

A |=int ∃X ϕ(a0, ...,U0, ...,X ) ⇔there is U ∈ [A]ℵ0 ∩ A s.t. A |=int stat X ϕ(a0, ...,U0, ...,U).

◮ For structures A, B with B ⊆ A and B = 〈B , ...〉,

B ≺int

Lℵ0,IIstat

A ⇔ For any Lℵ0,IIstat -formula ϕ = ϕ(x0, ...,X0, ...,X ),

b0, ... ∈ B and U0, ... ∈ [B]ℵ0 ∩ B ,

B |=int ϕ(b0, ...,U0, ..., ) ⇔ A |=int ϕ(b0, ...,U0, ..., ).

Internal interpretation of the stationary logic (2/2) Downward Löwenheim-Skolem (16/22)

◮ SDLSint(Lℵ0,IIstat , < κ) ⇔ For any structure A of countable

signature, there is a substructure B of A of cardinality < κ s.t.B ≺int

Lℵ0,IIstat

A.

Theorem 1. For regular κ > ℵ1, the following are equivalent

(a) SDLSint(Lℵ0,IIstat , < κ);

(b) (∗)−<κ,λ for all regular λ ≥ κ.

◮ Recall that (∗)−<κ,λ was defined as✿✿✿✿

this .

PKL logic LPKL,IIstat The following is an adaptation of an idea by H.Brickhill. Downward Löwenheim-Skolem (17/22)

◮ The language of LPKL,IIstat :

⊲ built-in unary predicate K (x) (x is a first-order variable)⊲ monadic second-order variables X , Y , Z ,... interpreted as subsets of

size < |KA | of the given structure A

⊲ built-in predicate ε with x ε X for first-order variable x andsecond-order X

⊲ second-order quantification: “stat X ”, “∃X ”

Notation: PB(A) = P|B |(A) = {C ⊆ A : |C | < |B |}.

◮ The semantics in internal interpretation

⊲ A |=int

∃Xstat X ϕ(a0, ...,

∈PKA (A)∩A

︷︸︸︷

U0, ...,X )⇔there is U ∈ PKA(A) ∩ A s.t. A |=int ϕ(a0, ...,U0, ...,U).{U ∈ PKA(A) ∩ A : A |=int ϕ(a0, ...,U0, ...,U)} is stationary inPKA(A).

PKL logic LPKL,IIstat (2/3) Downward Löwenheim-Skolem (18/22)

◮ SDLSint(LPKL,IIstat , < κ) ⇔ for any structure A of PLK logic with

|KA | = κ, there is B ≺int

LPKL,II

stat

A with ‖B‖ < κ.

(∗)PKL<κ,λ: For any countable expansion A of 〈H(λ), κ,∈〉 and〈Sa : a ∈ H(λ)〉 s.t. Sa is a stationary subset of Pκ(H(λ)) forall a ∈ H(λ), there is X ∈ Pκ(H(λ)) s.t. A ↾ X ≺ A andSa ∩ Pκ∩X (X ) is stationary in Pκ∩X (X ) for all a ∈ X .

Theorem 1. For a regular cardinal κ, the following are equivalent:

(a) SDLSint(LPKL,IIstat , < κ);

(b) (∗)PKL<κ,λ holds for all regular λ ≥ κ.

PKL logic LPKL,IIstat (3/3) Downward Löwenheim-Skolem (19/22)

◮ The following follow from Theorem 1 (of the last slide):

Proposition 1. (1) SDLSint(LPKL,IIstat , < κ) holds for a supercom-

pact cardinal κ.

(2) SDLSint(LPKL,IIstat , < 2ℵ0) holds in a model obtained by adding

supercompactly many reals “homogeneously” by ccc forcing.

Theorem 2. (1) SDLSint(LPKL,IIstat , < κ) implies that κ is weakly

Mahlo.

(2) SDLSint(LPKL,IIstat , < 2ℵ0) implies that the continuum is (at

least) weakly Mahlo.

Summary and open problems Downward Löwenheim-Skolem (20/22)

◮ Strong reflection statements seem to support either CH or verylarge continuum while strong forcing axioms (generic absoluteness)support 2ℵ0 = ℵ2.

◮ ◮

◮ Is there a reflection principle which furhter extends GRP ?

◮ What is the “maximal” reflection principle for reflection down to< 2ℵ0 under very large continuum?

⊲ What is the relationship between SDLSint(Lℵ0,IIstat , < 2ℵ0) and

SDLSint(LPKL,IIstat , < 2ℵ0)?

⊲ How is the Hamburger’s Problem for “< 2ℵ0” connected to thesereflection principles?

⊲ Is there a Pκλ version of Game Reflection Principle?

◮ Galvin Conjecture for “< 2ℵ0”.

◮ (The original) Hamburger’s Problem for “<ℵ2”.

References Downward Löwenheim-Skolem (21/22)

Sean Cox, The diagonal reflection principle, Proceedings of theAmerican Mathematical Society, vol.140, no.8 (2012)2893–2902.

Alan Dow, Franklin D. Tall, William A.R. Weiss, New proofs ofthe consistency of the normal Moore space conjecture II,Topology and its Applications 37, (1990), 115-129.

Sakaé Fuchino, István Juhász, Lajos Soukup, ZoltanSzentmiklóssy and Toshimichi Usuba, Fodor-type ReflectionPrinciple and reflection of metrizability and meta-Lindelöfness,Topology and its Applications Vol.157, 8 (2010), 1415–1429.

Sakaé Fuchino, Hiroshi Sakai, Lajos Soukup and ToshimichiUsuba, claim More about Fodor-type Reflection Principle,submitted.

Sakaé Fuchino, Hiroshi Sakai, Victor Torres Perez andToshimichi Usuba, Rado’s Conjecture and the Fodor-typeReflection Principle, in preparation.

http://fuchino.ddo.jp/papers/moreFRP.pdf

http://fuchino.ddo.jp/notes/RCimpliesFRP2.pdf

References (2/2) Downward Löwenheim-Skolem (22/22)

Sakaé Fuchino, André Ottenbreit Maschio Rodrigues andHiroshi Sakai, Downward Löwenheim-Skolem Theorems forstationary logics, in preparation.

Sakaé Fuchino, André Ottenbreit Maschio Rodrigues andHiroshi Sakai, Reflection of properties with uncountablecharacteristics, in preparation.

Sakaé Fuchino, Pre-Hilbert spaces without orthonormal bases,submitted.

Bernhard König, Generic compactness reformulated, Archivewof Mathematical Logic 43, (2004), 311 326.

Menachem Magidor, Large cardinals and sgrong logics, Lecturenotes of the Advanced Course on Large Cardinals and StrongLogics, IRP LARGE CARDINALS AND STRONG LOGICS,CRM, Bacelona September 19 to 23, (2016).

Stevo Todorčević, On a conjecture of Rado, Journal of LondonMathematical Society, Vol.S2-27, (1), (1983), 1–8.

https://arxiv.org/pdf/1606.03869v2

Thank you for your attention.

Diagonal Reflection Principle◮ For a regular cardinal θ > ℵ1:

DRP(θ, IC): There are stationarily many M ∈ [H((θℵ0)+)]ℵ1 s.t.

(1) M ∩H(θ) is✿✿✿✿✿✿✿✿✿✿✿✿

internally club ;

(2) for all R ∈ M s.t. R is a stationary subset of [θ]ℵ0 ,

R ∩ [θ ∩M]ℵ0 is stationary in [θ ∩M]ℵ0 .

◮ For a regular cardinal λ > ℵ1

(∗)λ: For any countable expansion A of 〈H(λ),∈〉, if〈Sa : a ∈ H(λ)〉, is a family of stationary subsets of [H(λ)]ℵ0 ,then there is an internally club M ∈ [H(λ)]ℵ1 s.t. A ↾ M ≺ A

and Sa ∩ [M]ℵ0 is stationary in [M]ℵ0 , for all a ∈ M.

Proposition 1. T.F.A.E.: (a) The global version of DiagonalReflection Principle of S.Cox for internal clubness (i.e. DRP(θ, IC)for all regular θ > ℵ1) holds.

(b) (∗)λ for all regular λ > ℵ1 holds.

(c) SDLS−(Lℵ0stat , <ℵ2) holds.

もどる

Reflection Principles RP??

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

RPIC For any uncountable cardinal λ, stationary S ⊆ [H(λ)]ℵ0 andany countable expansion A of the structure 〈H(λ),∈〉, there isan

✿✿✿✿✿✿✿✿✿✿✿✿✿

internally club M ∈ [H(λ)]ℵ1 s.t. (1) A ↾ M ≺ A; and (2)S ∩ [M]ℵ0 is stationary in [M]ℵ0 .

RPIU For any uncountable cardinal λ, stationary S ⊆ [H(λ)]ℵ0 andany countable expansion A of the structure 〈H(λ),∈〉, there isan

✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿

internally unbounded M ∈ [H(λ)]ℵ1 s.t. (1) A ↾ M ≺ A;and (2) S ∩ [M]ℵ0 is stationary in [M]ℵ0 .

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

Lemma 1. RPIC implies RPIU.

RPIU is also called Axiom R in Set-Theoretic Topology.

Theorem 2. ([F., Juhász et al.,2010]) RPIU implies FRP.

もどる

Stationary subsets of [X ]ℵ0

◮ C ⊆ [X ]ℵ0 is club in [X ]ℵ0 if (1) for every u ∈ [X ]ℵ0 , there is v ∈ C

with u ⊆ v ; and (2) for any countable increasing chain F in C wehave

⋃F ∈ C .

⊲ S ⊆ [X ]ℵ0 is stationary in [X ]ℵ0 if S ∩C 6= ∅ for all club C ⊆ [X ]ℵ0 .

◮ A set M is internally unbounded if M ∩ [M]ℵ0 is cofinal in [M]ℵ0

(w.r.t. ⊆)

⊲ A set M is internally stationary if M ∩ [M]ℵ0 is stationary in [M]ℵ0

⊲ A set M is internally club if M ∩ [M]ℵ0 contains a club in [M]ℵ0 .

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

Fodor-type Reflection Principle (FRP)(FRP) For any regular κ > ω1, any stationary E ⊆ Eκ

ω and anymapping g : E → [κ]ℵ0 with g(α) ⊆ α for all α ∈ E , there isγ ∈ Eκ

ω1s.t.

(*) for any I ∈ [γ]ℵ1 closed w.r.t. g and club in γ, if〈Iα : α < ω1〉 is a filtration of I then sup(Iα) ∈ E andg(sup(Iα)) ⊆ Iα hold for stationarily many α < ω1.

⊲ F = 〈Iα : α < λ〉 is a filtration of I if F is a continuously increasing⊆-sequence of subsets of I of cardinality < | I | s.t. I =

⋃

α<λ Iα.

◮ FRP follows from Martin’s Maximum or Rado’s Conjecture.MA+(σ-closed) already implies FRP but PFA does not imply FRPsince PFA does not imply stationary reflection of subsets of Eω2

ω

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

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

⊲ FRP is known to be equivalent to the reflection of uncountablecoloring number of graphs down to cardinality < ℵ2. もどる


Axiom R = RPIU


RPIC



MM+ω1



もどる

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 subspacesY of X of cardinality < κ are metrizable.

Proof.◮ Let κ′ ≥ κ be of cofinality ≥ κ, ω1.

⊲ 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′) > ℵ0.⊲ 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 [F., Juhász et al.,2010]) For a regularcardinal κ ≥ ℵ2 if, there is a

✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿

non-reflectingly stationary S ⊆ Eκω ,

then there is a non✿✿✿✿✿✿✿✿✿✿✿✿

meta-lindelöf (and hence non metrizable) lo-cally compact and locally countable topological space X of cardi-nality 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 〈aα : α ∈ S〉 be s.t. aα ∈ [I ∩ α]ℵ0 , aα 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) {aα ∪ {α} \ β : β < α} a neighborhood base of each α ∈ S .

◮ Then (X ,O) is not meta-lindelöf (by Fodor’s Lemma) but eachα < κ as subspace of X is metrizable (by induction on α).� もどる

Coloring number and chromatic number of a graph

◮ For a cardinal κ ∈ Card, a graph G = 〈G ,K 〉 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. もどる

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

◮ The chromatic number chr(G ) of a graph G = 〈G ,K 〉 is theminimal cardinal κ s.t. G can be partitioned into κ piecesG =

⋃

α<κ Gα s.t. each Gα 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 Tα, α < κ s.t. each Tα is anantichain (i.e. pairwise incomparable set).

もどる

Stationary subset of Eκω

◮ For a cardinal κ,

Eκω = {γ < κ : cf(γ) = ω}.

◮ A subset C ⊆ ξ of an ordinal ξ of uncountable cofinality, C is closedunbounded (club) in ξ if (1): C is cofinal in ξ (w.r.t. the canonicalordering 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 ofuncountable cofinality s.t.S ∩ η is stationary in η. Thus:

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

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Proof of Theorem 1.CH ⇒ SDLS(Lℵ0,II , < ℵ2): For a structure A with a countablesignature L and underlying set A, let θ be large enough and

A = 〈H(θ),A,A,∈〉. where A = A A for a unary predicate symbol

A and A = AA for a constant symbol A. Let B ≺ A be

s.t.|B | = ℵ1 for the underlying set B of B and [B]ℵ0 ⊆ B .

B = A ↾ AB is then as desired.

SDLS(Lℵ0 , < ℵ2) ⇒ CH: Suppose A = {ω2 ∪ [ω2]ℵ0 ,∈}. Consider

the Lℵ0-formula ϕ(X ) = ∃x∀y (y ∈ x ↔ y ε X ).If B = 〈B , ...〉 is s.t. |B | ≤ ℵ1 and B ≺Lℵ0 , then for C ∈ [B]ℵ0 ,since A |= ϕ(C ), we have B |= ϕ(C ). It dollows that [B]ℵ0 ⊆ B

and 2ℵ0 ≤ (|B |)ℵ0 ≤ |B | = ℵ1.

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reﬂection principles formulated as löwenheim-skolem ...reﬂection principles formulated as...

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