Forcing Arguments in Second-Order Arithmetic Kenji Omoto Tohoku University

2012/2/20

Reverse Mathematics I

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Setting: Second order arithmetic.

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Main Question: What axioms are necessary to prove the theorems of Mathematics?

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Axiom systems(Big 5): RCA0 ≡ ∆11 -Comprehension + Σ01 -Induction + Semiring ax WKL0 ≡ RCA0 + Weak K¨onig’s lemma ACA0 ≡ RCA0 + Arithmetic Comprehension ATR0 1 Π1 -CA0

≡ ACA0 + Arithmetic Transfinite recursion ≡ ACA0 + Π11 -Comprehension

Theories of Second-Order Arithmetic I ▶

∪ A set is called arithmetical ( n<ω Σ0n (X )) if it is defined by a formula consisting of only number quantifiers(no set quantifiers).

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ACA0 is a theory consists of RCA0 and arithmetical comprehension.

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ACA0 is a Π11 -conservative extension of PA.

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ACA0 satisfies good theorems about arithmetical properties.

Fact (examples of properties of ACA0 ) 1. ARITH(X ) ⊨ ACA0 for any X ⊆ ω. Where ARITH(X ) is a class of all sets which are arithmetical in X. 2. Every ω model M of ACA0 is arithmetically closed. That is, for any X ∈ M, ARITH(X ) ⊆ M.

Theories of Second-Order Arithmetic II RCA0 WKL0

∆01

Comprehension Weak K¨onig’s Lemma

∆01 Recursive, Computable ∪

Σ0n Arithmetical ∪ ∆11 , µ<ωCK Σ0µ n<ω

ACA0

Arithmetical Comprehension

? ATR0 Π11 -CA0

What axiom is placed here? Arithmetical Transfinite Recursion Π11 Comprehension

1

Hyperarithmetical Π11

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A set defined by transfinite induction of arithmetical definition. is called hyperarithmetical set.

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A ∪ class of0 sets which hyperarithmetical in1 X is denoted by µ<ω X Σµ . In fact, It is equivalent to ∆1 (X )

Theories of Second-Order Arithmetic III

∆01 Comprehension Weak K¨onig’s lemma

RCA0 WKL0

∆01 Recursive, Computable ∪

Σ0n Arithmetical ∪ ∆11 , µ<ωCK Σ0µ n<ω

ACA0

Arithmetical Comprehension

? ATR0 Π11 -CA0

What axiom is placed here? Arithmetical Transfinite Recursion Π11 Comprehension

1

Hyperarithmetical Π11

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Question: What theory of second-order arithmetic corresponds to Hyperarithmetical?

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∆11 -CA0 ?

Theory of Hyperarithemetic Analysis I

Definition A theory T is called theory of hyperarithmetic analysis if it satisfies following: 1. HYP(X ) ⊨ T for any X ⊆ ω. Where HYP(X ) is a class of all sets which are hyperarithmetical in X. 2. Every ω model of T is hyperarithmetically closed. ▶

∆11 -CA0 is a theory of hyperarithmetic analysis.

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However, beside ∆11 -CA0 , there exists other various theories of hyperarithmetic.

Theory of Hyperarithemetic Analysis II

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the following are theories of hyperarithmetic analysis. Σ11 -AC0 ≡ ACA0 + Σ11 -Axiom of Choice Π11 -SEP0 ≡ ACA0 + Π11 -Separation ∆11 -CA0 ≡ ACA0 + ∆11 -Comprehension

weak-Σ11 -AC0 ≡ ACA0 + weak-Σ11 -Axiom of Choice INDEC ≡ ACA0 + Jullien’s Indecomposability Theorem ABW0 ≡ ACA0 + Arithmetical Bolzano-Weierstraß’s Theorem JI ≡ ACA0 + (∀X )(∀α){[(∀β < α)X (β) exists] → X (α) exists}

Theory of Hyperarithemetic Analysis III CDG-CA ≡ ACA0 + Given a sequence {Tn | n ∈ N} of completely determined trees, there exists a set X s.t. n ∈ X ⇔ I has a winning strategy in Tn CDG-AC ≡ ACA0 + Given a sequence {Tn | n ∈ N} of completely determined trees, Σn∈N Tn is also completely determined. DG-CA ≡ ACA0 + Given a sequence {Tn | n ∈ N} of determined trees, there exists a set X s.t. n ∈ X ⇔ I has a winning strategy in Tn DG-AC ≡ ACA0 + Given a sequence {Tn | n ∈ N} of determined trees, Σn∈N Tn is also completely determined. etc...

Theory of Hyperarithemetic Analysis IV → is proved over RCA0

Σ11 -AC0

⇒ is proved over RCA0 + Σ11 -IND ?

DG-AC H



×

̸→ is proved by tagged tree forcing



Π11 -SEP0 J




ABW0 o

× × ×



∆11 -CA0 o J

'

×

/ DG-CA



INDEC H ×

"



weak-Σ11 -AC0 K



o × CDG-CA H ×



JI

/ CDG-AC

Definition of Tagged Tree I We shall see how Π11 -SEP0 ̸→ Σ11 -AC0 is proved by Montalb´an.

Definition Σ11 -AC0 is ACA0 plus following scheme: (∀n)(∃X )ψ(n, X ) ⇒ (∃Z )(∀n)ψ(n, Zn ), where ψ(n, X ) is Σ11 .

Definition Π11 -SEP0 is ACA0 plus following scheme: ¬(∃n)[¬ψ(n) ∧ ¬ϕ(n)] ⇒ (∃Z )(∀n)[(¬ψ(n) → n ∈ Z ) ∧ (¬ϕ(n) → n ∈ / Z )], where ψ(n) and ϕ(n) are Σ11 formula.

Definition of Tagged Tree II Definition p = (T p , f p , hp ) ∈ P if and only if ▶ T p ⊆ ω <ω is a finite tree. ▶

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f p is a non-empty finite partial function such that dom(f p ) ⊆ ω, ran(f p ) ⊆ T p . ▶

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T p grows to a infinite tree T G .

Every f p (i) grows to a path f G (i) of T G .

hp : T p → ω1CK ∪ {∞} s.t. ▶

(∀s, t ∈ T p )[s ⊊ t → hp (s) > hp (t)] (∀s ∈ T p )[(∃i)[s ⊆ f p (i)] → hp (s) = ∞]

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Every hp grows to a rank function hG of T G

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Definition of Tagged Tree III

Definition For any P-generic G, we define objects as follows: ▶

T G := ∪p∈G T p

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αiG = f G (i) := ∪p∈G ,i∈dom(f p ) f p (i)

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hG := ∪p∈G hp

▶ ▶

⊕ MFG := {X | (∃µ < ω1CK )[X is Σ0µ (T G ⊕ i∈F αG (i))]} ∪ G := G G G M∞ F ⊆fin ω MF = HYP({T } ∪ {α (i) | i ∈ F })

where, MFG ∩ [T G ] = {αiG | i ∈ F } holds.

G M∞ ⊭ Σ11 -AC0

Theorem G ⊭ Σ1 -AC M∞ 0 1 ▶

If there finitely arbitrary many sets satisfy a Σ11 formula, Σ11 -AC0 certifies that it is possible to take infinitely many sets satisfying the formula.

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”X is an infinite path of T” is also Σ11 formula with argument X.

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If F is large enough, MFG contains many paths, therefore it is G possible to take they from M∞

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However, every MFG contains at most finitely many paths. Therefore it is impossible to take infinitely many paths from G. M∞

Forcing relation I

Definition For p ∈ P and a formula ψ, p ⊩ ψ is defined as follows: ▶

p ⊩ σ ∈ T ⇔def σ ∈ T p ∨ (1 ≤ hp (σ − ) ∧ σ − ∈ T p )

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p ⊩ (¯ n, m) ¯ ∈ αi ⇔def i ∈ dom(f p ) ∧ f p (i)(n) = m

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p ⊩ n ∈ Sν,F ,e ⇔def p ⊩ (∃x)θ(e, x, n, Hν,F ) (where (∃x)θ(e, x, y , X ) is a universal Σ01 formula for some θ Σ00 formula.)

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p ⊩ (e, n, ν) ∈ Hµ,F ⇔def ν < µ ∧ p ⊩ n ∈ Sν,F ,e

Forcing relation II Definition ▶

p ⊩ ∀XFµ ϕ(XFµ ) ⇔def (∀ν < µ)(∀e ∈ ω)p ⊩ ϕ(Sν,F ,e )

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p ⊩ ∀X µ ϕ(X µ ) ⇔def (∀ν < µ)(∀e ∈ ω)(∀F ⊆fin ω)p ⊩ ϕ(Sν,F ,e )

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p ⊩ ∀XF ϕ(XF ) ⇔def (∀ν < ω1CK )(∀e ∈ ω)p ⊩ ϕ(Sν,F ,e )

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p ⊩ ∀X ϕ(X ) ⇔def (∀ν < ω1CK )(∀e ∈ ω)(∀F ⊆fin ω)p ⊩ ϕ(Sν,F ,e )

where a subscript F of the variable X means that values of X are in MFG , a superscript µ of the variable X means that values of X are Σ0µ set.

Retagging I

Definition

Ret(µ, F , p, p ∗ ) (p is a µ-F -absolute-retagging of p ∗ ) if and only if ∗

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Tp = Tp

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(∀i ∈ F )[f p (i) = f p (i)]

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(∀s ∈ T p )[hp (s) ≥ µ → hp (s) ≥ µ]

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(∀s ∈ T p )[hp (s) < µ → hp (s) = hp (s)]

∗

∗

Lemma

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Ret(ω · µ, F , p, p ∗ ) ∧ q ≤ p ∧ ν < µ ⇒ (∃q ∗ ≤ p ∗ )Ret(ω · ν, F , q, q ∗ )

Retagging II Definition A formula ψ is F -restricted if every subscript of variable of ψ is a subset of F .

Lemma Let a formula ψ be such that F-restricted and rk(ψ) < ω1CK . Let p, p ∗ ∈ P. Then Ret(ω · rk(ψ), F , p, p ∗ ) ⇒ (p ⊩ ψ ⇒ p ∗ ⊩ ψ) p⊩ψ

Ret





p∗

q⊩ψ

Ret

q∗

Retagging III

Lemma For a formula ψ with rank µ, 0(µ) is able to decide whether p ⊩ ψ uniformly in p, ψ, and µ.

Σ-over-L∞ Definition ▶

A formula ψ is F -restricted if every subscript of variable of ψ is a subset of F .

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A formula ψ is ranked if every variable of ψ is ranked.

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A formula is Σ-over-LF if it is built up from ranked and F -restricted formulas using ∧, (∀n), (∃X ).

Definition For any formula ψ and µ < ω1CK , ψ µ is a result of replacing of (∃X ) by (∃X µ ).

Lemma Let F ⊆fin ω, ψ be Σ-over-LF sentence, and µ < ω1CK . Then, Ret(ω · rk(ψ) + ω 2 + ω, F , p, p ∗ ) ∧ dom(f p ) ⊆ F

⇒ (p ⊩ ¬ψ µ ⇒ p ∗ ⊩ ¬ψ µ )

G M∞ ⊨ Π11 -SEP0 I

Theorem G ⊨ Π1 -SEP M∞ 0 1

Proof ▶

Let ϕ(n) and ψ(n) be a Σ-over-LF formulas(Σ11 formulas) with a free variable n such that G M∞ ⊨ (∀n)[ψ(n) ∨ ϕ(n)]

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G will be constructed such that D ∈ M∞ G M∞ ⊨ (∀n)[(¬ϕ(n) → n ∈ D) ∧ (n ∈ D → ψ(n))]

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Let p ∈ G be such that p ⊩ (∀n)[ψ(n) ∨ ϕ(n)].

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Let µ < ω1CK be such that p ⊩ (∀n)[ψ µ (n) ∨ ϕµ (n)].

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Let ν < ω1CK be large enough for p ∈ Pν ∧ (∀n ∈ ω)[rk(ψ µ (¯ n) ∨ ϕµ (¯ n)) < ν]

G M∞ ⊨ Π11 -SEP0 II

▶ ▶

D is defined as follows. d ∈ D if and only if there exists q ∈ P such that 1. 2. 3. 4. 5. 6.

q ⊩ ¬ϕν (d) Tq ⊆ TG (∀i ∈ F )[f q (i) = αiG ∩ T q ] ran(hq (s)) ⊆ (ων + ω 2 + 2ω) ∪ {∞} hG (s) < ων + ω 2 + 2ω → g (s) = hG (s) hG (s) ≥ ων + ω 2 + 2ω → g (s) ≥ ν

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D is hyperarithmetical in T ⊕

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G. Therefore D ∈ M∞

⊕

i∈F

αiG .

(¬ϕ(n) → n ∈ D) and (n ∈ D → ψ(n)) will be proved in next page.

G M∞ ⊨ Π11 -SEP0 III

¬ϕ(n) → n ∈ D. ▶

G ⊨ ¬ϕµ (n) holds. Assume that M∞

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Let q ∗ ∈ G be such that q ∗ ≤ p ∧ q ∗ ⊩ ¬ϕµ (n).

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q ∗ satisfies almost every condition of D.

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However, it is not necessarily true that ∗ ran(hq (s)) ⊆ (ων + ω 2 + 2ω) ∪ {∞}. Let retagging q of q ∗ as follows. Then q is a witness of d ∈ D.

▶

∗

1. T q = T q ∗ 2. f q = f q { 3. hq (s) =

∗

hq (s) ∞

∗

(if hq (s) < ων + ω 2 + 2ω) (otherwise)

G M∞ ⊨ Π11 -SEP0 IV

¬ψ(n) → n ∈ /D ▶

Assume that ¬ψ(n) ∧ n ∈ D and we prove it by contradiction.

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G ⊨ ¬ψ µ (n), there exists r ≤ p s.t. r ⊩ ¬ψ µ (n). Since M∞

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G ⊨ n ∈ D, there exists q ≤ p such that q ⊩ ¬ϕµ (n). Since M∞

p ⊩ ψ µ (n) ∨ ϕµ (n) u

)

q ⊩ ¬ϕµ (n)

r ⊩ ¬ψ µ (n)



q ∗ ⊩ ¬ϕµ (n)



Ret

s∗

Ret



r ∗ ⊩ ¬ψ µ (n)

Let q ∗ ≤ q, r ∗ ≤ r , s ∗ ≤ p be such that Ret(ων + ω 2 + ω, F , q ∗ , s ∗ ) ∧ Ret(ων + ω 2 + ω, F , r ∗ , s ∗ ). Then s ∗ ⊩ ¬ϕµ (n) ∧ ¬ψ µ (n) holds. It is contradiction. Q.E.D.

Bibliography ▶

Antonio Montalb´an “Indecomposable linear ordering and hyperarithmetic analysis” Journal of Mathematical Logic, Vol.6, No.1, 89-120 2006.

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Antonio Montalb´an “On the Π11 -separation principle” Mathematical Logic Quarterly 2008.

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Itay Neeman “The strength of Jullien’s indecomposability theorem” Journal of Mathematical Logic 2008.

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Chris J. Conidis “Comparing theorems of hyperarithmetic analysis with the arithmetical Bolzano-Weierstrass theorem” to appear.

Thank you for your attention.