GraduateSchoolofSystemInformatics,KobeUniversity,Japan SakaéFuchino ( ) Rado’sConjectureandHamburger’sHypothesis

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Rado’s Conjecture and Hamburger’s Hypothesis

Sakaé Fuchino (

渕野昌

)

Graduate School of System Informatics, Kobe University, Japan

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

(2020

年

11

月

20

日

(16:19 JST) version)

2020

年

11

月

20

日

(JST,

於

RIMS set theory workshop) This presentation is typeset by upL

^A

TEX with beamer class.

The most up-to-date version of these slides is downloadable as https://fuchino.ddo.jp/slides/RIMS2020-pf.pdf

This research is supported by

Kakenhi Grant-in-Aid for Scientiﬁc Research (C) 20K03717

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Rado Conjecture (RC) RC and HH (2/17)

(RC): For any tree T , if T is not special, then there is a non-special subtree T

0

of T of size < ℵ

2

.

▶ A tree T is special if T is the union T = ∪

n∈ω

T

n

where each T

n

is pairwise incomparable (or antichain in tree terminology).

▷ Note that T

₀

as above must be of size = ℵ

1

since any countable tree is special.

▷ Note also that the assertion of RC holds, if T has height > ω

1

. Proposition 1. If ω

2

is generically supercompact by σ-closed p.o.s,

then RC holds. In particular, the consistency of RC follows from the existence of a supercompact cardinal (actually a strongly compact cardinal is enough to prove the consistency of RC).

▷ We shall see a proof of a more general assertion later.

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Rado Conjecture (RC) (2/3) RC and HH (3/17)

▶ The relation of RC to other principles:

Theorem 1. (S. Todorčević, 1993) RC implies

Chang’s Conjecture (CC) and Singular Cardinal Hypothesis (SCH).

Theorem 2. (Ph. Doebler, 2013) RC implies Semi-Stationary Reﬂection Principle (SSR).

Theorem 3. (S.F, H. Sakai, V.Torres-Perez, T.Usuba) RC implies Fodor-type Reﬂection Principle (FRP).

Theorem 4. (B.König, 2004) The condition “ω

₂

is a generically supercompact cardinal by σ-closed p.o.s” can be characterized as a reﬂection statement on non-existence of winning strategy for the second player of certain game (Game Reﬂection Principle (GRP)).

▶ By the Theorem on the previous slide, GRP implies RC.

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Rado Conjecture (RC) (3/3) RC and HH (4/17)

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Semi-stationary Reflection (SSR)

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Fodor-type

Reflection Principle (FRP)

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Game Reflection Principle (GRP)

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CH

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There is a Laver-generically supercompact cardinal for -closed pos

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!2is generically supercompact by -closed forcing

<latexit sha1_base64="v1mmLF5YWhr9+qlsgup+m9dXmjw=">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</latexit>

Rado Conjecture (RC)

<latexit sha1_base64="xRoaZBWoxXwvPTpJo1W5sbNBLUw=">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</latexit>

Chang’s Conjecture <latexit sha1_base64="il3V76Sy35f+MeoXdSo9x61ewA0=">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</latexit>

Total failure of square principle

<latexit sha1_base64="cecTtRvykN5MDOJfoQzrYmcOtBQ=">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</latexit>

Singular Cardinal Hypthesis

Proof

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Hamburger’s Hypothesis (HH) RC and HH (5/17)

▶ The consistency of the following statement is still open:

(HH): For any topological space X with χ(x, X ) ≤ ℵ

0

for all x ∈ X , if X is non-metrizable, then there is a non-metrizable subspace Y of X of cardinality < ℵ

2

.

▶ The following is a theorem in ZFC:

Theorem 5. (A. Dow) If X is a non-metrizable compact space then there is a non-metrizable subspace Y of X of cardinality < ℵ

2

.

▶ The following statement is shown to be equivalent (over ZFC) to the Fodor-type Reﬂection Principle (FRP):

▷ If X is a non-metrizable locally compact space then there is a subspace Y of X of cardinality < ℵ

2

s.t. Y is also non-metrizable.

▷ ^{Z. Balogh} proved that the statement follows form Axiom R.

[S.F., Juhász, Soukup, Szentmiklóssy, Usuba] and [S.F., Sakai, Torres-Perez,

Usuba] show the equivalence.

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Hamburger’s Hypothesis (HH) (2/2) RC and HH (6/17)

Semi-stationary Reflection (SSR)

Fodor-type

Reflection Principle (FRP)

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Hamburger’s Hypothesis for locally compact spaces

Game Reflection Principle (GRP)

<latexit sha1_base64="/rf5OwF/TUW9+Icg2BOV0/d9q3w=">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</latexit>

CH

<latexit sha1_base64="Y51wG9bc43ujvvblWk1qImoJ4DU=">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</latexit>

consistency unknown

<latexit sha1_base64="n33Vsub+VNpJC/f1456eJ1Scnys=">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</latexit>

Hamburger’s Hypothesis (HH)

Rado Conjecture (RC)

<latexit sha1_base64="hhId2UU8CmOaC4vTOVnxO05nQcY=">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</latexit>

Singular Cardinal Hypothesis

(7)

‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌

In a larger picture RC and HH (7/17)

Semi-stationary Reflection (SSR)

Fodor-type

Reflection Principle (FRP)

<latexit sha1_base64="+MDm1Bm4S5J8Zdkwk8svMP7rYK4=">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</latexit>

Hamburger’s Hypothesis for locally compact spaces

Game Reflection Principle (GRP)

<latexit sha1_base64="Y51wG9bc43ujvvblWk1qImoJ4DU=">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</latexit>

consistency unknown

<latexit sha1_base64="1ftX2m1QilclsgtP6Iv8qv5KUX4=">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</latexit>

MA^+!¹( -closed)

<latexit sha1_base64="n33Vsub+VNpJC/f1456eJ1Scnys=">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</latexit>

Hamburger’s Hypothesis (HH)

Rado Conjecture (RC)

<latexit sha1_base64="cecTtRvykN5MDOJfoQzrYmcOtBQ=">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</latexit>

Singular Cardinal Hypthesis

<latexit sha1_base64="zLyt2XKlOy8yXGZGqIH5weLwUng=">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</latexit>

SDLS ( L

^@stat⁰

, < @

2

)

<latexit sha1_base64="XUK2M8uL6zFYiWswvPKWCwOAUbY=">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</latexit>

SDLS( L

^@stat⁰^,II

, < @

2

)

<latexit sha1_base64="lTK5638dyblK2ObX6C3olpNdzBI=">AAAC9XicdVLLbhMxFHWGVymvFJZsLFKkIpVkJhI03YXHgkUqFSVpI+I08jh3Eiue8ci+AwmjWfEd7BBbfoHf4AfYwifgSQOiEVzJ1vG594x975kwVdKi73+reJcuX7l6bev69o2bt27fqe7cPbE6MwL6QittBiG3oGQCfZSoYJAa4HGo4DScvyjzp2/BWKmTHi5TGMV8mshICo6OGlffMCu4glAvcr/eKoZuG+Us0hoTjWDle6A5sxHtvux0i928OHu8x5ygc5YzJ0tnY78Y5xY5Fvus02NCJyiTLIsf7Rbjas2v+74fBAEtQXDw1Hfg8LDVDFo0KFMuamQdx+Odygc20SKLIUGhuLXDwE9xP0sVR1iMcm5QCgXFNssspFzM+RSGDiY8BjvKV8Mo6EPHTGikjVsJ0hX7tyLnsbXLOHSVMceZ3cyV5D9zU8PTmRQLd3/M58DdrBHBfXsCEXPCMMzL8napnBqdpZQdtfuUucueHXXDPzKNM7j4ojDUalJsUPFGnxg5Z2SSZgiJOG8zyhRFTUtf6UQaEKiWDnBhpJsUFTNuuHBPdH2shHmjb92p0eVzDo0eDBqryUpsuL8iM7ruDqVtv72h/wcnzXrwpO6/btbaz9cGbpH75AHZIwE5IG3yihyTPhHkK/lOfpCf3jvvo/fJ+3xe6lXWmnvkQnhffgFUc/Qc</latexit>

SDLS ( L

^@_stat⁰

, < 2

^@⁰

)

<latexit sha1_base64="FQGKB5p62Jh9ZP9umQl8+THGA5E=">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</latexit>

2

^@⁰

= @

2

<latexit sha1_base64="Ext0IgTgz4jM1NtINlBGuFDm8ZU=">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</latexit>

CH

<latexit sha1_base64="XH+S0i6JQfFEDd+y9HL5dHAilZU=">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</latexit>

DRP(IC

_@₀

)

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Axiom R RP_IU_@0

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PFA

^+!¹

<latexit sha1_base64="Du/5mhTA0hZY2JyETaln17X1ffo=">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</latexit>

There is a Laver-generically supercompact cardinal for proper pos

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Reﬂection principles with higher reﬂection points RC and HH (8/17)

▶ We also want to consider reﬂection principles with the reﬂection point < 2

^ℵ⁰

or ≤ 2

^ℵ⁰

(i.e. < (2

^ℵ⁰

)

⁺

). More generally, for a cardinal κ, let

(RC(< κ)): For any tree T , if T is not special, then there is a non- special subtree T

0

of T of size < κ.

(HH(< κ)): For any topological space X with χ(x, X ) < κ for all x ∈ X , if X is non-metrizable, then there is a non-metrizable subspace Y of X of cardinality < κ.

Lemma 6. (Hajnal-Juhász, 1976) HH(< κ) is equivalent to the follo- wing seemingly weaker reﬂection principle. In particular, HH(< ℵ

2

) is equivalent to HH:

・ For any topological space X with χ(x, X ) ≤ ℵ

0

for all

x ∈ X , if X is non-metrizable, then there is a non-metrizable

subspace Y of X of cardinality < κ.

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Reﬂection principles with higher reﬂection points (2/2) RC and HH (9/17)

▶ A p.o. P preserves the non-metrizability, if, for any non-metrizable topological space X = ⟨ X , τ ⟩ , we have ∥ –

_P

“ X is non-metrizable ” . A property P of p.o.s preserves the non-metrizability, if P preserves the non-metrizability for all P | = P .

▶ A p.o. P preserves non-specialty, if, for any non-special tree T , we have ∥ –

_P

“ T is non-special ” . A property P of p.o.s preserves the non-specialty, if P preserves the non-specialty for all P | = P .

Proposition 7. Suppose that κ is generically supercompact by P . (1) If P preserves non-metrizability, then HH(< κ) holds.

(2) If P preserves non-specialty, then RC(< κ) holds.

Proof.

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Preservation and non-preservation RC and HH (10/17)

Proposition 8. (1) (Todorčević) σ-closed p.o.s preserve non- specialty.

(2) σ-centered p.o.s preserve non-specialty.

(3) FS-iterations of σ-centered p.o.s preserve non-specialty.

(4) (Todorčević) There is a ccc p.o. which does not preserve non-

specialty. □

▶ The example ccc p.o. for the proof (4) above can be used to show the following:

Proposition 9. (Todorčević) RC(< κ) implies ma < κ.

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Preservation and non-preservation (2/2) RC and HH (11/17)

Theorem 10. (1) (Dow, Tall, and Weiss) Generalized Cohen forcing (for adding multiple Cohen reals) preserves non-metrizability.

(2) (van Douwen) Hechler forcing does not preserve non-

metrizability. □

▶ The topological space constructed in the proof of Theorem 2, (2) also shows the following:

Lemma 11. (van Douwen) HH(< κ) implies b < κ. □

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Consistency results RC and HH (12/17)

Theorem 12. RC(< 2

^ℵ⁰

) + HH(< 2

^ℵ⁰

) is consistent (modulo a su- percompact cardinal).

Proof. Let κ be a supercompact cardinal, P = Fn(κ, 2), and G a (V, P )-generic set. Then, in V[ G ], we have κ = 2

^ℵ⁰

and κ is generically supercompact for { Fn(λ, 2) : λ ∈ On } .

▶ By Proposition 8, (3) and Theorem 10, (1), it follows that

V[ G ] | = RC(< 2

^ℵ⁰

) + HH(< 2

^ℵ⁰

). □ Theorem 13. RC(< 2

^ℵ⁰

) + ¬ HH(< 2

^ℵ⁰

) is consistent (modulo a

supercompact cardinal).

Proof. Superompact long FS-iteration of Heckler forcing will do

(see Proposition 8, (3) and Lemma 11). □

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Consistency results (2/2) RC and HH (13/17) Theorem 14. ¬ RC(< 2

^ℵ⁰

) + ¬ HH(< 2

^ℵ⁰

) is consistent with the

continuum being Laver-generically supercompact for ccc p.o.s.

Proof. If 2

^ℵ⁰

is Laver-generically supercompact for ccc p.o.s, then

MA holds. □

▶ Existence of a Laver-generically supercompact for ccc p.o.s implies

that the continuum is fairly large.

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Mixed support iteration RC and HH (14/17)

▶ The consistency results in the previous slides can be still

strengthened by adding the consistency of the following principles.

The proof is done by Mixed support iteration of supercompact lenth along with a Laver function with a preparatory iteration.

・ SDLS

^int₊

( L

^ℵstat⁰

, < 2

^ℵ⁰

), GRP

^<²^ℵ0

( ≤ 2

^ℵ⁰

),

・ SDLS

^int₊

(L

^PKL_stat

, < 2

^ℵ⁰

);

・ A certain fragment of MA

⁺⁺

(ccc).

Some Open Prblems:

・ Hamburger’s Problem.

・ Consistency of HH(< 2

^ℵ⁰

) + ¬ RC(< 2

^ℵ⁰

).

・ Cardinal invariants under HH(< 2

^ℵ⁰

) or RC(< 2

^ℵ⁰

).

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References RC and HH (15/17)

[1] Sakaé Fuchino, André Ottenbereit Maschio Rodriques and Hiroshi Sakai, Strong downward Löwenheim-Skolem theorems for stationary logics, I, Archive for Mathematical Logic (2020). 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, to appear in Archive for Mathematical Logic.

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 — mixed support iteration, to appear in the Proceedings of the Asian Logic Conference 2019. https://fuchino.ddo.jp/papers/SDLS-III-xx.pdf

[4] Sakaé Fuchino, and André Ottenbereit Maschio Rodriques, Reﬂection principles, generic large cardinals, and the Continuum Problem, in: Advances in

Mathematical Logic / Dedicated to the Memory of Professor Gaisi Takeuti, SAML 2018, Kobe, Japan, September 2018, Springer Proceedings in Mathematics and Statistics, to appear.

https://fuchino.ddo.jp/papers/reﬂ_principles_gen_large_cardinals_continuum_problem-x.pdf

[5] Sakaé Fuchino, Rado’s Conjecture and Hamburger’s Hypothesis, in preparation.

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Thank you for your attention!

ご清聴ありがとうございました．

Muchas gracias por su atención.

(17)

‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌‌ ‌

asante sana kwa umakini wako

1

^日本語

すべての人間は、生まれながらにして自由であり、かつ、尊厳と権利とについて平等である。人間は、理性と良心とを授けられており、互いに同胞の精神をもって行動しなければならない。

2

^{中国語・簡体字} ^简体中文

人人生而自由，在尊严和权利上一律平等。他们赋有理性和良心，并应以兄弟关系的精神相对待。

3

^{中国語・繁体字} ^繁體中文

人人生而自由，在尊嚴和權利上一律平等。他們賦有理性和良心，並應以兄弟關係的精神相對待。

4

^韓国語 한국어

끝까지 들어 주셔서 감사합니다．

1

^日本語

すべての人間は、生まれながらにして自由であり、かつ、尊厳と権利とについて平等である。人間は、理性と良心とを授けられており、互いに同胞の精神をもって行動しなければならない。

2

^{中国語・簡体字} ^简体中文谢谢您的倾听。

3

^{中国語・繁体字} ^繁體中文

人人生而自由，在尊嚴和權利上一律平等。他們賦有理性和良心，並應以兄弟關係的精神相對待。

4

^韓国語 한국어

끝까지 들어 주셔서 감사합니다．

1

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Laver generically large cardinals

▶ A cardinal κ is said to be a Laver-generically supercompact for P for a property P of p.o.s, if, for any λ ≥ κ and any P | = P , there are a p.o. Q | = P with P ≤

^◦

Q and (V, Q)-generic ﬁlter H s.t. there are classes M , j ⊆ V[ H ] with

・ M is an inner model of V[G] and j : V →

^≼

M ,

・ crit (j ) = κ,

・ j (κ) > λ,

・ P , H ∈ M and

・ j

^′′

λ ∈ M .

Back to the 1.diag.

Back to the 2.diag.

Back to the main diag.

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Proof of Proposition 7.

▶ We prove (1). (2) is shown similarly. We have to show the following:

・ Suppose that κ is generically supercompact by P where P preserves non-metrizability. Then HH(< κ) holds.

Proof. Let ⟨ X , τ ⟩ be a non-metrizable space with ( † ) χ(x, X ) < κ for all x ∈ X . W.l.o.g., we may assume that X = λ for a cardinal λ and ( ‡ ) | τ | ≤ | X | .

▶ Let P | = P and (V , P )-generic G be s.t. there are classes j , M ∈ V[ G ] s.t. (1) M is an inner model of V[ G ] and j : V

^≼

→ M, (2) crit(j ) = κ, (3) j(κ) > λ, and (4) j

^′′

λ ∈ M .

▶ Let X

^∗

= j

^′′

X and τ

^∗

= { j

^′′

O : O ∈ τ } . By (4) and ( ‡ ),

⟨ X

^∗

, τ

^∗

⟩ ∈ M. By ( † ), ⟨ X

^∗

, τ

^∗

⟩ is a subspace of ⟨ j (X ), j (τ ) ⟩ .

▶ Since ⟨ X

^∗

, τ

^∗

⟩ ∼ = ⟨ X , τ ⟩ in V[ G ],

V[ G ] | =“ ⟨ X

^∗

, τ

^∗

⟩ is non-metrizable”

by P | = P .

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Proof of Proposition 1. (2/2)

▶ It follows that

M | = “ there is a non-metrizable subspace of j(X ) of size < j (κ)” .

▶ By elementarity, it follows that

V | =“ there is a non-metrizable subspace of X of size < κ”.

□

Back

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Fodor-type Reﬂection Principle (FRP)

(FRP) For any regular κ > ω

1

, any stationary E ⊆ E

_ω^κ

and any mapping g : E → [κ]

^ℵ⁰

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

_ω^κ₁

s.t.

(*) for any I ∈ [γ ]

^ℵ¹

closed w.r.t. g and club in γ , if

⟨ I

α

: α < ω

1

⟩ is a ﬁltration of I then sup(I

α

) ∈ E and g (sup(I

_α

)) ⊆ I

_α

hold for stationarily many α < ω

₁

.

▷ F = ⟨ I

_α

: 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

I

_α

.

▶ FRP is also equivalent to the reﬂection of uncountable coloring number of graphs down to cardinality < ℵ

2

.

Back to RC frame Back to HH frame

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GRP implies CH

Proposition 1. Suppose that ω

₂

is generically supercompact by σ- closed forcing. Then CH holds.

Proof. Suppose that ω

2

is generically supercompact by σ-closed forcing but ¬ CH holds. Then there is a 1-1 i : ω

2

→ P (ω).

▶ Let λ = 2

^ℵ⁰

. Let P be a σ-closed p.o., and G be a (V, P )-generic set with classes j , M ⊆ V[ G ] s.t. (1) j : V →

^≼

M , (2) crit (j ) = ω

2

, (3) j (ω

₂

) > λ, and (4) j

^′′

λ ∈ M .

▶ By elementarity (1), M | =“ j (i) : j (ω

₂

) → P (ω) is 1-1”.

▷ This is a contradiction as P (ω)

^V

= P (ω)

^V[^G^]

. □

▶ Actually, the proposition above and its proof does not help us very much, since, to establish the proof of the equivalence of GRP and this generic supercompactness, we have to prove that GRP implies CH. This is done (in the proof of Bernhard König) by using a game deﬁned from a Bernstein set.

Back

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Generically supercompact κ by P

▶ A cardinal κ is said to be a generically supercompact cardinal by P for a property (of p.o.s) P if, for any cardinal λ ≥ κ, there is a p.o. P with P | = P and a (V, P )-generic G s.t. there are classes j , M ⊆ V[ G ] with

・ M is an inner model of V[G] and j : V →

^≼

M ;

・ crit (j ) = κ;

・ j (κ) > λ and

・ j

^′′

λ ∈ M .

Proposition 1. If κ is a supercompact and µ < κ is an uncountable regular cardinal then for P = Col(µ, κ) and (V, P )-generic ﬁlter G , we have V[ G ] | = κ = µ

⁺

and κ is generically supercompact by

< µ-closed forcing.

Back

GraduateSchoolofSystemInformatics,KobeUniversity,Japan SakaéFuchino ( ) Rado’sConjectureandHamburger’sHypothesis

Rado’s Conjecture and Hamburger’s Hypothesis

Sakaé Fuchino (

)

Graduate School of System Informatics, Kobe University, Japan

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

(2020

11

20

(16:19 JST) version)

2020

11

20

(JST,

RIMS set theory workshop) This presentation is typeset by upL

TEX with beamer class.

The most up-to-date version of these slides is downloadable as https://fuchino.ddo.jp/slides/RIMS2020-pf.pdf

This research is supported by

Kakenhi Grant-in-Aid for Scientiﬁc Research (C) 20K03717

Rado Conjecture (RC) RC and HH (2/17)

(RC): For any tree T , if T is not special, then there is a non-special subtree T

of T of size < ℵ

.

▶ A tree T is special if T is the union T = ∪

T

where each T

is pairwise incomparable (or antichain in tree terminology).

▷ Note that T

as above must be of size = ℵ

since any countable tree is special.

▷ Note also that the assertion of RC holds, if T has height > ω

. Proposition 1. If ω

is generically supercompact by σ-closed p.o.s,

then RC holds. In particular, the consistency of RC follows from the existence of a supercompact cardinal (actually a strongly compact cardinal is enough to prove the consistency of RC).

▷ We shall see a proof of a more general assertion later.

Rado Conjecture (RC) (2/3) RC and HH (3/17)

▶ The relation of RC to other principles:

Theorem 1. (S. Todorčević, 1993) RC implies

Chang’s Conjecture (CC) and Singular Cardinal Hypothesis (SCH).

Theorem 2. (Ph. Doebler, 2013) RC implies Semi-Stationary Reﬂection Principle (SSR).

Theorem 3. (S.F, H. Sakai, V.Torres-Perez, T.Usuba) RC implies Fodor-type Reﬂection Principle (FRP).

Theorem 4. (B.König, 2004) The condition “ω

is a generically supercompact cardinal by σ-closed p.o.s” can be characterized as a reﬂection statement on non-existence of winning strategy for the second player of certain game (Game Reﬂection Principle (GRP)).

▶ By the Theorem on the previous slide, GRP implies RC.

Rado Conjecture (RC) (3/3) RC and HH (4/17)

Semi-stationary Reflection (SSR)

Fodor-type

Reflection Principle (FRP)

Game Reflection Principle (GRP)

Rado Conjecture (RC)

Hamburger’s Hypothesis (HH) RC and HH (5/17)

▶ The consistency of the following statement is still open:

(HH): For any topological space X with χ(x, X ) ≤ ℵ

for all x ∈ X , if X is non-metrizable, then there is a non-metrizable subspace Y of X of cardinality < ℵ

.

▶ The following is a theorem in ZFC:

Theorem 5. (A. Dow) If X is a non-metrizable compact space then there is a non-metrizable subspace Y of X of cardinality < ℵ

.

▶ The following statement is shown to be equivalent (over ZFC) to the Fodor-type Reﬂection Principle (FRP):

▷ If X is a non-metrizable locally compact space then there is a subspace Y of X of cardinality < ℵ

s.t. Y is also non-metrizable.

▷ Z. Balogh proved that the statement follows form Axiom R.

[S.F., Juhász, Soukup, Szentmiklóssy, Usuba] and [S.F., Sakai, Torres-Perez,

Usuba] show the equivalence.

Hamburger’s Hypothesis (HH) (2/2) RC and HH (6/17)

Semi-stationary Reflection (SSR)

Fodor-type

Reflection Principle (FRP)

Game Reflection Principle (GRP)

Hamburger’s Hypothesis (HH)

Rado Conjecture (RC)

In a larger picture RC and HH (7/17)

Semi-stationary Reflection (SSR)

Fodor-type

Reflection Principle (FRP)

Game Reflection Principle (GRP)

Hamburger’s Hypothesis (HH)

Rado Conjecture (RC)

SDLS ( L

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▷ ^{Z. Balogh} proved that the statement follows form Axiom R.