Continuum hypothesis and diamond principle Kousuke Ogawa( ) - PowerPoint PPT Presentation

Continuum hypothesis and diamond principle Continuum hypothesis and diamond principle Kousuke Ogawa( 小川孝典 ) Department of Mathematics and Information Sciences Tokyo Metroporitan University ( 首都大学東京 理工学研究科 数理情報科学専攻 ) 「数学基礎論若手の会」 2011

Continuum hypothesis and diamond principle Contents Contents 1 Continuum hypothesis 2 Diamond principle 3 Shelah’s theorem 4 A problem about Shelah’s theorem 5 Sketch of the proof of Shelah’s theorem

Continuum hypothesis and diamond principle Continuum hypothesis Continuum hypothesis

Continuum hypothesis and diamond principle Continuum hypothesis Continuum hypothesis Theorem (Cantor’s theorem) For every set X, we have | X | < |P ( X ) | . For a cardinal λ , we have λ < 2 λ .

Continuum hypothesis and diamond principle Continuum hypothesis Continuum hypothesis Theorem (Cantor’s theorem) For every set X, we have | X | < |P ( X ) | . For a cardinal λ , we have λ < 2 λ . = ⇒ For an infinite cardinal λ , does there exist a cardinal µ such that λ < µ < 2 λ ?

Continuum hypothesis and diamond principle Continuum hypothesis Continuum hypothesis Definition The Continuum Hypothesis(CH) is the statement 2 ω = ω 1 . For an infinite cardinal λ , CH λ is the statement 2 λ = λ + . The statement ∀ λ CH λ is called the Generalized Continuum Hypothesis(GCH).

Continuum hypothesis and diamond principle Continuum hypothesis Continuum hypothesis Definition The Continuum Hypothesis(CH) is the statement 2 ω = ω 1 . For an infinite cardinal λ , CH λ is the statement 2 λ = λ + . The statement ∀ λ CH λ is called the Generalized Continuum Hypothesis(GCH). Theorem (G¨ odel, Cohen) GCH is independent of the axioms of ZFC.

Continuum hypothesis and diamond principle Diamond principle Diamond principle

Continuum hypothesis and diamond principle Diamond principle Diamond principle C ⊂ κ is a club set iff ( ) 0 < ∀ δ < κ sup( C ∩ δ ) = δ → δ ∈ C (closed) sup C = κ (unbounded)

Continuum hypothesis and diamond principle Diamond principle Diamond principle C ⊂ κ is a club set iff ( ) 0 < ∀ δ < κ sup( C ∩ δ ) = δ → δ ∈ C (closed) sup C = κ (unbounded) S ⊂ κ is a stationary set iff for every club set C, S ∩ C ̸ = ∅ .

Continuum hypothesis and diamond principle Diamond principle Diamond principle C ⊂ κ is a club set iff ( ) 0 < ∀ δ < κ sup( C ∩ δ ) = δ → δ ∈ C (closed) sup C = κ (unbounded) S ⊂ κ is a stationary set iff for every club set C, S ∩ C ̸ = ∅ . club set : measure 1 stationary set : measure positive

Continuum hypothesis and diamond principle Diamond principle Diamond principle C ⊂ κ is a club set iff ( ) 0 < ∀ δ < κ sup( C ∩ δ ) = δ → δ ∈ C (closed) sup C = κ (unbounded) S ⊂ κ is a stationary set iff for every club set C, S ∩ C ̸ = ∅ . club set : measure 1 stationary set : measure positive Definition Assume κ is a regular cardinal and S ⊂ κ is a stationary set. Then, ♢ S is the following statement: there exists ⟨ S δ : δ ∈ S ⟩ such that for every A ⊂ κ , { δ ∈ S : A ∩ δ = S δ } is stationary.

Continuum hypothesis and diamond principle Diamond principle Diamond principle C ⊂ κ is a club set iff ( ) 0 < ∀ δ < κ sup( C ∩ δ ) = δ → δ ∈ C (closed) sup C = κ (unbounded) S ⊂ κ is a stationary set iff for every club set C, S ∩ C ̸ = ∅ . club set : measure 1 stationary set : measure positive Definition Assume κ is a regular cardinal and S ⊂ κ is a stationary set. Then, ♢ S is the following statement: there exists ⟨ S δ : δ ∈ S ⟩ such that for every A ⊂ κ , { δ ∈ S : A ∩ δ = S δ } is stationary. Remark If S ⊂ T then ♢ S → ♢ T .

Continuum hypothesis and diamond principle Diamond principle Diamond principle Proposition Assume λ is an infinite cardinal and S ⊂ λ + is a stationary set. Then, ♢ S → 2 λ = λ + .

Continuum hypothesis and diamond principle Diamond principle Diamond principle Proposition Assume λ is an infinite cardinal and S ⊂ λ + is a stationary set. Then, ♢ S → 2 λ = λ + . Proof. Assume ⟨ S δ : δ ∈ S ⟩ is a ♢ S -sequence. Since C := { δ < λ + : δ > λ } = ( λ, λ + ) is club, for every A ⊂ λ , there exists δ ∈ S ∩ C such that A = A ∩ δ = S δ . Then 2 λ = |P ( λ ) | ≤ | S ∩ C | = λ + .

？ Continuum hypothesis and diamond principle Diamond principle Diamond principle Fact (Jensen) (2 ω = ω 1 ) + ¬ ♢ ω 1 is consistent.

？ Continuum hypothesis and diamond principle Diamond principle Diamond principle Fact (Jensen) (2 ω = ω 1 ) + ¬ ♢ ω 1 is consistent. Question 1 Assume λ is an uncountable cardinal. What kind of S ⊂ λ + entail 2 λ = λ + → ♢ S ?

Continuum hypothesis and diamond principle Diamond principle Diamond principle Fact (Jensen) (2 ω = ω 1 ) + ¬ ♢ ω 1 is consistent. Question 1 Assume λ is an uncountable cardinal. What kind of S ⊂ λ + entail 2 λ = λ + → ♢ S ? Question 2 For every uncountable cardinal λ , 2 λ = λ + → ♢ λ + holds ？

Continuum hypothesis and diamond principle Shelah’s theorem Shelah’s theorem

Continuum hypothesis and diamond principle Shelah’s theorem Shelah’s theorem Assume µ < κ are regular cardinals. S κ µ := { α < κ : cf ( α ) = µ } , S κ ̸ = µ := { α < κ : cf ( α ) ̸ = µ } .

Continuum hypothesis and diamond principle Shelah’s theorem Shelah’s theorem Assume µ < κ are regular cardinals. S κ µ := { α < κ : cf ( α ) = µ } , S κ ̸ = µ := { α < κ : cf ( α ) ̸ = µ } . Lemma Assume µ < κ are regular cardinals. Then S κ µ is stationary. For uncountable cardinal λ , S λ + ̸ = cf ( λ ) is stationary.

Continuum hypothesis and diamond principle Shelah’s theorem Shelah’s theorem Theorem (Shelah) Assume λ is an uncountable cardinal and S ⊂ S λ + ̸ = cf ( λ ) is a stationary set. Then, 2 λ = λ + → ♢ S holds.

Continuum hypothesis and diamond principle Shelah’s theorem Shelah’s theorem Theorem (Shelah) Assume λ is an uncountable cardinal and S ⊂ S λ + ̸ = cf ( λ ) is a stationary set. Then, 2 λ = λ + → ♢ S holds. Corollary For every uncountable cardinal λ , 2 λ = λ + → ♢ λ + holds.

Continuum hypothesis and diamond principle Shelah’s theorem Shelah’s theorem Theorem (Shelah) Assume λ is an uncountable cardinal and S ⊂ S λ + ̸ = cf ( λ ) is a stationary set. Then, 2 λ = λ + → ♢ S holds. Corollary For every uncountable cardinal λ , 2 λ = λ + → ♢ λ + holds. Question 2 is solved !!

？ Continuum hypothesis and diamond principle A problem about Shelah’s theorem A problem about Shelah’s theorem

？ Continuum hypothesis and diamond principle A problem about Shelah’s theorem A problem about Shelah’s theorem Fact If λ is a regular cardinal, then (2 λ = λ + ) + ¬ ♢ S λ + cf ( λ ) is consistent.

Continuum hypothesis and diamond principle A problem about Shelah’s theorem A problem about Shelah’s theorem Fact If λ is a regular cardinal, then (2 λ = λ + ) + ¬ ♢ S λ + cf ( λ ) is consistent. Question1-2 Assume λ is a singular cardinal. Then, For every stationary S ⊂ S λ + cf ( λ ) , 2 λ = λ + → ♢ S holds ？

Continuum hypothesis and diamond principle A problem about Shelah’s theorem A problem about Shelah’s theorem Definition Assume λ is a singular cardinal and S ⊂ λ + is a stationary set. I [ S ; λ ] is a set such that T ∈ I [ S ; λ ] ↔ T ⊂ Tr ( S ) and ∃ d : [ λ + ] 2 → cf ( λ ) normal, subadditive ∃ C ⊂ λ + club ∀ γ ∈ T ∩ C ∩ S λ + > cf ( λ ) ∃ S γ ⊂ γ ∩ S stationary ( ) [ S γ ] 2 ] [ sup d < cf ( λ ) .

Continuum hypothesis and diamond principle A problem about Shelah’s theorem A problem about Shelah’s theorem Definition Assume λ is a singular cardinal and S ⊂ λ + is a stationary set. I [ S ; λ ] is a set such that T ∈ I [ S ; λ ] ↔ T ⊂ Tr ( S ) and ∃ d : [ λ + ] 2 → cf ( λ ) normal, subadditive ∃ C ⊂ λ + club ∀ γ ∈ T ∩ C ∩ S λ + > cf ( λ ) ∃ S γ ⊂ γ ∩ S stationary ( ) [ S γ ] 2 ] [ sup d < cf ( λ ) . Tr ( S ) = { α < λ + : cf ( α ) > ω, S ∩ α is stationary in α } ,

Continuum hypothesis and diamond principle A problem about Shelah’s theorem A problem about Shelah’s theorem Definition Assume λ is a singular cardinal and S ⊂ λ + is a stationary set. I [ S ; λ ] is a set such that T ∈ I [ S ; λ ] ↔ T ⊂ Tr ( S ) and ∃ d : [ λ + ] 2 → cf ( λ ) normal, subadditive ∃ C ⊂ λ + club ∀ γ ∈ T ∩ C ∩ S λ + > cf ( λ ) ∃ S γ ⊂ γ ∩ S stationary ( ) [ S γ ] 2 ] [ sup d < cf ( λ ) . Tr ( S ) = { α < λ + : cf ( α ) > ω, S ∩ α is stationary in α } , For d : [ λ + ] 2 → cf ( λ ), d is subadditive ↔ ∀ α ≤ ∀ β ≤ ∀ γ < λ + ( ) d ( α, γ ) ≤ max { d ( α, β ) , d ( β, γ ) } , d is normal ↔ ∀ β < λ + ∀ i < cf ( λ ) ( ) |{ α < β : d ( α, β ) ≤ i }| < λ .