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Continuum hypothesis and diamond principle Continuum hypothesis and diamond principle Kousuke Ogawa( ) Department of Mathematics and Information Sciences Tokyo Metroporitan University (

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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

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Continuum hypothesis and diamond principle 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

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Continuum hypothesis and diamond principle Continuum hypothesis

Continuum hypothesis

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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 < 2λ.

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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 < 2λ. = ⇒ For an infinite cardinal λ, does there exist a cardinal µ such that λ < µ < 2λ ?

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Continuum hypothesis and diamond principle Continuum hypothesis

Continuum hypothesis

Definition The Continuum Hypothesis(CH) is the statement 2ω = ω1. For an infinite cardinal λ, CHλ is the statement 2λ = λ+. The statement ∀λ CHλ is called the Generalized Continuum Hypothesis(GCH).

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Continuum hypothesis and diamond principle Continuum hypothesis

Continuum hypothesis

Definition The Continuum Hypothesis(CH) is the statement 2ω = ω1. For an infinite cardinal λ, CHλ is the statement 2λ = λ+. The statement ∀λ CHλ is called the Generalized Continuum Hypothesis(GCH). Theorem (G¨

del, Cohen)

GCH is independent of the axioms of ZFC.

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Continuum hypothesis and diamond principle Diamond principle

Diamond principle

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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)

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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 ̸= ∅.

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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

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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 ⟨Sδ : δ ∈ S⟩ such that for every A ⊂ κ, {δ ∈ S : A ∩ δ = Sδ} is stationary.

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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 ⟨Sδ : δ ∈ S⟩ such that for every A ⊂ κ, {δ ∈ S : A ∩ δ = Sδ} is stationary. Remark If S ⊂ T then ♢S → ♢T.

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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 → 2λ = λ+.

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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 → 2λ = λ+. Proof. Assume ⟨Sδ : 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 = Sδ. Then 2λ = |P(λ)| ≤ |S ∩ C| = λ+.

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Continuum hypothesis and diamond principle Diamond principle

Diamond principle

Fact (Jensen) (2ω = ω1) + ¬♢ω1 is consistent.

？

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Continuum hypothesis and diamond principle Diamond principle

Diamond principle

Fact (Jensen) (2ω = ω1) + ¬♢ω1 is consistent. Question 1 Assume λ is an uncountable cardinal. What kind of S ⊂ λ+ entail 2λ = λ+ → ♢S ?

？

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Continuum hypothesis and diamond principle Diamond principle

Diamond principle

Fact (Jensen) (2ω = ω1) + ¬♢ω1 is consistent. Question 1 Assume λ is an uncountable cardinal. What kind of S ⊂ λ+ entail 2λ = λ+ → ♢S ? Question 2 For every uncountable cardinal λ, 2λ = λ+ → ♢λ+ holds？

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Continuum hypothesis and diamond principle Shelah’s theorem

Shelah’s theorem

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Continuum hypothesis and diamond principle Shelah’s theorem

Shelah’s theorem

Assume µ < κ are regular cardinals. Sκ

µ := {α < κ : cf(α) = µ},

Sκ

̸=µ := {α < κ : cf(α) ̸= µ}.

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Continuum hypothesis and diamond principle Shelah’s theorem

Shelah’s theorem

Assume µ < κ are regular cardinals. Sκ

µ := {α < κ : cf(α) = µ},

Sκ

̸=µ := {α < κ : cf(α) ̸= µ}.

Lemma Assume µ < κ are regular cardinals. Then Sκ

µ is stationary.

For uncountable cardinal λ, Sλ+

̸=cf(λ) is stationary.

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Continuum hypothesis and diamond principle Shelah’s theorem

Shelah’s theorem

Theorem (Shelah) Assume λ is an uncountable cardinal and S ⊂ Sλ+

̸=cf(λ) is a stationary set.

Then, 2λ = λ+ → ♢S holds.

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Continuum hypothesis and diamond principle Shelah’s theorem

Shelah’s theorem

Theorem (Shelah) Assume λ is an uncountable cardinal and S ⊂ Sλ+

̸=cf(λ) is a stationary set.

Then, 2λ = λ+ → ♢S holds. Corollary For every uncountable cardinal λ, 2λ = λ+ → ♢λ+ holds.

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Continuum hypothesis and diamond principle Shelah’s theorem

Shelah’s theorem

Theorem (Shelah) Assume λ is an uncountable cardinal and S ⊂ Sλ+

̸=cf(λ) is a stationary set.

Then, 2λ = λ+ → ♢S holds. Corollary For every uncountable cardinal λ, 2λ = λ+ → ♢λ+ holds. Question 2 is solved !!

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

A problem about Shelah’s theorem

？

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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 (2λ = λ+) + ¬♢Sλ+

cf(λ) is consistent.

？

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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 (2λ = λ+) + ¬♢Sλ+

cf(λ) is consistent.

Question1-2 Assume λ is a singular cardinal. Then, For every stationary S ⊂ Sλ+

cf(λ),

2λ = λ+ → ♢S holds？

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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 ∩ Sλ+

>cf(λ) ∃Sγ ⊂ γ ∩ S stationary

( sup d [ [Sγ]2] < cf(λ) ) .

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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 ∩ Sλ+

>cf(λ) ∃Sγ ⊂ γ ∩ S stationary

( sup d [ [Sγ]2] < cf(λ) ) . Tr(S) = {α < λ+ : cf(α) > ω, S ∩ α is stationary in α},

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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 ∩ Sλ+

>cf(λ) ∃Sγ ⊂ γ ∩ S stationary

( sup d [ [Sγ]2] < 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 ) .

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

A problem about Shelah’s theorem

Theorem (Rinot) Assume λ is a singular cardinal and S ⊂ λ+ is a stationary set. Then, if I[S; λ] contains a stationary set, 2λ = λ+ → ♢S holds.

？？

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

A problem about Shelah’s theorem

Theorem (Rinot) Assume λ is a singular cardinal and S ⊂ λ+ is a stationary set. Then, if I[S; λ] contains a stationary set, 2λ = λ+ → ♢S holds. Question1-3 Assume λ is a singular cardinal. Then for every stationary S ⊂ Sλ+

cf(λ),

does I[S; λ] contain stationary sets？ Does I[Sλ+

cf(λ); λ] contain stationary sets？

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Continuum hypothesis and diamond principle Sketch of the proof of Shelah’s theorem

Sketch of the proof of Shelah’s theorem

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Continuum hypothesis and diamond principle Sketch of the proof of Shelah’s theorem

Sketch of the proof of Shelah’s theorem

Theorem (Shelah) Assume λ is an uncountable cardinal and S ⊂ Sλ+

̸=cf(λ) is a stationary set.

Then, 2λ = λ+ → ♢S holds.

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Continuum hypothesis and diamond principle Sketch of the proof of Shelah’s theorem

Sketch of the proof of Shelah’s theorem

Theorem (Shelah) Assume λ is an uncountable cardinal and S ⊂ Sλ+

̸=cf(λ) is a stationary set.

Then, 2λ = λ+ → ♢S holds. Sketch of proof (Komj´ ath).

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Continuum hypothesis and diamond principle Sketch of the proof of Shelah’s theorem

Sketch of the proof of Shelah’s theorem

Theorem (Shelah) Assume λ is an uncountable cardinal and S ⊂ Sλ+

̸=cf(λ) is a stationary set.

Then, 2λ = λ+ → ♢S holds. Sketch of proof (Komj´ ath). Claim There exist a sequence ⟨ Aδ ∈ [δ]<λ : δ ∈ S ⟩ and an enumeration of [λ × λ+]≤λ, ⟨Xβ : β < λ+⟩ , such that for every Z ⊂ λ × λ+, SZ := { δ ∈ S : sup { α ∈ Aδ : ∃β ∈ Aδ(Z ∩ (λ × α) = Xβ) } = δ } is stationary.

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Continuum hypothesis and diamond principle Sketch of the proof of Shelah’s theorem

Sketch of the proof of Shelah’s theorem

Now, we define a sequence of subsets of λ+, ⟨Yγ : γ < λ⟩, and a decreasing sequence of clubs, ⟨Cγ : γ < λ⟩.

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Continuum hypothesis and diamond principle Sketch of the proof of Shelah’s theorem

Sketch of the proof of Shelah’s theorem

Now, we define a sequence of subsets of λ+, ⟨Yγ : γ < λ⟩, and a decreasing sequence of clubs, ⟨Cγ : γ < λ⟩. Put Y0 := C0 := λ+.

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Continuum hypothesis and diamond principle Sketch of the proof of Shelah’s theorem

Sketch of the proof of Shelah’s theorem

Now, we define a sequence of subsets of λ+, ⟨Yγ : γ < λ⟩, and a decreasing sequence of clubs, ⟨Cγ : γ < λ⟩. Put Y0 := C0 := λ+. Assume ⟨Yτ : τ < γ⟩ and ⟨Cτ : τ < γ⟩ are defined for some γ < λ. If there exist Y ⊂ λ+ and club C ⊂ ∩

τ<γ Cτ such that for every

δ ∈ S ∩ C, we have δ = ∪ { α ∈ Aδ : ∃β ∈ Aδ∀τ < γ ( Yτ ∩ α = (Xβ)τ )} then ∃ ⟨α, β⟩ ∈ Aδ × Aδ [ ∀τ < γ ( Yτ ∩ α = (Xβ)τ ) ∧ Y ∩ α ̸= (Xβ)γ ] , then, put Yγ = Y , Cγ = C. Otherwise, terminate the recursion. (Xβ)γ := {ξ : ⟨γ, ξ⟩ ∈ Xβ} ⊂ λ+.

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Continuum hypothesis and diamond principle Sketch of the proof of Shelah’s theorem

Sketch of the proof of Shelah’s theorem

Claim There exists γ∗ < λ such that the recursion terminates in γ∗. Then, for every Y ⊂ λ+ and club C ⊂ ∩

γ<γ∗ Cγ, there exists δ ∈ S ∩ C,

such that δ = ∪ { α ∈ Aδ : ∃β ∈ Aδ∀γ < γ∗( Yγ ∩ α = (Xβ)γ )} and ∀ ⟨α, β⟩ ∈ Aδ × Aδ [ ∀γ < γ∗( Yγ ∩ α = (Xβ)γ ) → Y ∩ α = (Xβ)γ∗] .

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Continuum hypothesis and diamond principle Sketch of the proof of Shelah’s theorem

Sketch of the proof of Shelah’s theorem

Claim There exists γ∗ < λ such that the recursion terminates in γ∗. Then, for every Y ⊂ λ+ and club C ⊂ ∩

γ<γ∗ Cγ, there exists δ ∈ S ∩ C,

such that δ = ∪ { α ∈ Aδ : ∃β ∈ Aδ∀γ < γ∗( Yγ ∩ α = (Xβ)γ )} and ∀ ⟨α, β⟩ ∈ Aδ × Aδ [ ∀γ < γ∗( Yγ ∩ α = (Xβ)γ ) → Y ∩ α = (Xβ)γ∗] . Put C ∗ := ∩

γ<γ∗ Cγ.

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Continuum hypothesis and diamond principle Sketch of the proof of Shelah’s theorem

Sketch of the proof of Shelah’s theorem

Claim There exists γ∗ < λ such that the recursion terminates in γ∗. Then, for every Y ⊂ λ+ and club C ⊂ ∩

γ<γ∗ Cγ, there exists δ ∈ S ∩ C,

such that δ = ∪ { α ∈ Aδ : ∃β ∈ Aδ∀γ < γ∗( Yγ ∩ α = (Xβ)γ )} and ∀ ⟨α, β⟩ ∈ Aδ × Aδ [ ∀γ < γ∗( Yγ ∩ α = (Xβ)γ ) → Y ∩ α = (Xβ)γ∗] . Put C ∗ := ∩

γ<γ∗ Cγ.

For δ ∈ S ∩ C ∗, define Sδ := ∪

⟨α,β⟩∈Aδ×Aδ

{ (Xβ)γ∗ : ∀γ < γ∗( Yγ ∩ α = (Xβ)γ )} .

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Continuum hypothesis and diamond principle Sketch of the proof of Shelah’s theorem

Sketch of the proof of Shelah’s theorem

Then, for every Y ⊂ λ+ and club C ⊂ λ+, there exists δ ∈ S ∩ C ∗ ∩ C, we have Y ∩ δ = Y ∩ ∪ { α ∈ Aδ : ∃β ∈ Aδ∀γ < γ∗( Yγ ∩ α = (Xβ)γ )} = ∪

⟨α,β⟩∈Aδ×Aδ

{ Y ∩ α : ∀γ < γ∗( Yγ ∩ α = (Xβ)γ )} = ∪

⟨α,β⟩∈Aδ×Aδ

{ (Xβ)γ∗ : ∀γ < γ∗( Yγ ∩ α = (Xβ)γ )} = Sδ.

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Continuum hypothesis and diamond principle Sketch of the proof of Shelah’s theorem

Sketch of the proof of Shelah’s theorem

Then, for every Y ⊂ λ+ and club C ⊂ λ+, there exists δ ∈ S ∩ C ∗ ∩ C, we have Y ∩ δ = Y ∩ ∪ { α ∈ Aδ : ∃β ∈ Aδ∀γ < γ∗( Yγ ∩ α = (Xβ)γ )} = ∪

⟨α,β⟩∈Aδ×Aδ

{ Y ∩ α : ∀γ < γ∗( Yγ ∩ α = (Xβ)γ )} = ∪

⟨α,β⟩∈Aδ×Aδ

{ (Xβ)γ∗ : ∀γ < γ∗( Yγ ∩ α = (Xβ)γ )} = Sδ. ⟨Sδ : δ ∈ S ∩ C ∗⟩ is ♢S∩C ∗-sequence. Then, ♢S holds.

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Continuum hypothesis and diamond principle References

References

S.Shelah, Diamonds, Proceedings of the American Mathematical Society, vol.138 (2010). P.Komj´ ath, Shelah’s proof of diamond, Annales Universitatis Scientiarum Budapestinensis de Rolando E¨

tv¨
s Nominatae.

Sectio Mathematica, vol.51 (2008). A.Rinot, A relative of the approachability ideal, diamond and non-saturation, Journal of Symbolic Logic, vol.75 (2010).