Bounds on provability in set theories Toshiyasu Arai(Chiba) 21/2/2012

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Lifting up the ordinal anayses to ZF

For any rec and Π11 -sound theories T, its proof-theoretic ordinal |T| (the supremum of the order types of the provably recursive well orderings) is rec, |T| < ω1CK . Ordinal analyses of T computes or describes the rec ordinal |T|. [Buchholz92] introduced operator controlled derivations, hereby he gave a convincing ordinal analysis for KPi of recursively inaccessible ordinals, which is a rec analogue of ZF. In an operator controlled derivation, ordinals occurring in the derivation are controlled by a Skolem hull H. Through this we see that these ordinals are in H. Suppose that a formal theory on sets proves a Σ1 -sentence ∃x < ω1CK θ. Then the technique tells us how many times do we iterate Skolem hulllings to bound a rec ordinal x, a witness for θ. The technique has been applied successfully to set theories of reflecting ordinals, rec analogues of the indescribable cardinals. We now ask: Can we lift up ordinal analyses through a non-effective ordinal analysis? Specifically does the technique work also for set theories of (small) large cardinals? Collapsing functions α 7→ Ψκ,n α < κ are introduced for each uncountable regular cardinal κ ≤ I, the least weakly inaccessible cardinal I, and n < ω, cf. Definition 3.1. Let ωn (I + 1) denote the tower of ω with the next epsilon number εI+1 = sup{ωn (I + 1) : n < ω} above I. It is easy to see that the predicate x = Ψκ,n α is a Σn+1 -predicate for α < εI+1 , and for each n, k < ω ZF + (V = L) proves ∀α < ωk (I + 1)∀κ ≤ I∃x < κ[x = Ψκ,n α]. Conversely we show

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Theorem 1.1 ([A∞].) |ZF + (V = L)|ω1 := inf{α ≤ ω1 : ∀ϕ[ZF + (V = L) ⊢ ∃x ∈ Lω1 ϕ ⇒ ∃x ∈ Lα ϕ]} = Ψω1 εI+1 The countable ordinal Ψω1 εI+1 := sup{Ψω1 ,n ωn (I + 1) : n < ω} is the limit of ZF + (V = L)-provable countable ordinals: Ψω1 εI+1 = sup{α < ω1 : α is a ZF + (V = L)-definable ordinal}. From Theorem 1.1 we see that if ZF + (V = L) proves the existence of a real a ∈ ω ω enjoying a first order condition ϕ(a), ZF + (V = L) ⊢ ∃a ∈ ω ωϕ(a), then such a real a is already in the level LΨω1 εI+1 of the constructible hierarchy.

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Σn -Skolem hulls

ZF + (V = L) is interpreted in KPω + (V = L) with the axioms saying that the universe is regular, and there are regular ordinals unboundedly. We see that a limit ordinal κ is regular iff the set of critical points α < κ of Σ1 -elementary embeddings π with π(α) = κ is club in κ. Definition 2.1 For X ⊂ Lσ , HullσΣn (X) denotes Σn -Skolem hull of X in Lσ . a ∈ HullσΣn (X) ⇔ {a} is Σn (X)-definable on Lσ (a ∈ Lσ ). Definition 2.2 (Mostowski collapsing function F ) F −1 : Lγ ∼ = HullσΣn (X) ≺Σn Lσ for an ordinal γ ≤ σ s.t. F ¹ Y = id ¹ Y for any transitive Y ⊂ HullσΣn (X). Let us denote, though σ ̸∈ dom(F ) = HullσΣn (X) F (σ) := γ. Σn We write FX (x) for F (x).

Theorem 2.3 (Cf. [A97].) Let σ be an ordinal such that Lσ |= KPω + ∆2 -Separation, and ω ≤ α < κ < σ with α a multiplicative principal number and κ a limit ordinal. Then the following conditions are mutually equivalent: 1. α

κ ∩ Lσ ⊂ Lκ 2

2. Lσ |= α < cf (κ) 3. There exists a critical point x such that α < x < κ and Σ1 HullσΣ1 (x ∪ {κ}) ∩ κ ⊂ x & Fx∪{κ} (σ) < κ Σ1 4. For the Mostowski collapse Fx∪{κ} (y) Σ1 Σ1 ∃x, y[α < x = Fx∪{κ} (κ) < y = Fx∪{κ} (σ) < κ &

∀Σ1 ϕ∀a < x(Lσ |= ϕ[κ, a] → Ly |= ϕ[x, a])] Definition 2.4 T(I) = KPω + (V = L) plus the axioms: ∀α∃x[α < x = FxΣn (I)]

(1)

In (1), x = FxΣn (I) means the reflection principle ∀Σn ϕ∀a ∈ Lx (ϕ[a] → Lx |= ϕ[a])

(2)

Σ1 Σ1 ∀β∃κ > β∀α < κ∃x[α < x = Fx∪{κ} (κ) < Fx∪{κ} (I) < κ]

(3)

And Σ1 Σ1 In (3), x = Fx∪{κ} (κ) < y = Fx∪{κ} (I) < κ means the reflection principle

∀Σ1 ϕ∀a ∈ Lx (ϕ[κ, a] → Ly |= ϕ[x, a])

(4)

Lemma 2.5 T(I) is equivalent to the Zermelo-Fraenkel’s set theory ZF + (V = L) with the axiom of constructibility. Proof. Using (2), ∀Σn ϕ∀a ∈ Lx (ϕ[a] → Lx |= ϕ[a]), we see that T(I) proves Separation and Collection. Power set axiom in T(I) follows from Theorem 2.3 and (4), ∀Σ1 ϕ∀a ∈ Lx (ϕ[κ, a] → Ly |= ϕ[x, a]). 2

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Bounding provably definable countable ordinals in ZF

Finite derivations in T(I) are transformed to infinitary ones with ordinal constants. k(A) denotes the set of ordinal constants occurring in sentences A. ⊢a A designates that there exists an inifinitary derivation of A such that the depth of the derivation (wellfounded trees of formulas) is bounded by a. 3

The idea of the controlled derivations is to consider a relation H ⊢a A where ordinal constants k(A) occurring in A together with the depth a of derivations are controlled by Skolem hulls H in such a way that {a} ∪ k(A) ⊂ H

(5)

and simultaneously the witnessing ordinal is to be bounded by the depths. ∨ ( ) Let δ := µz < β θ[z] := min{δ : (δ < β ∧ θ[δ]) ∨ (¬∃z < βθ[z] ∧ δ = 0)} (6) For an a(δ) < a if δ
(7)

∨ H ⊢a(δ) θ[δ] ( ) a H ⊢ ∃z < β θ[z]

(7) is to bound existential quantifiers in provable ∃z < β θ[z] in terms of the depths a ∈ H of derivations. k(∃z < β θ[z]) ⊂ H ⇒ δ ∈ H if H is closed under definability. Σ1 1 (FΣ x∪{κ} ) Let κ < I be regular, and x = Fx∪{κ} (κ) and y = F Σ1 (I) for F Σ1 : HullI (x ∪ {κ}) ∼ = Ly . x∪{κ}

x∪{κ}

Σ1

Assume a0 < a, A ∈ Σ1 and k(A) ⊂ HullIΣ1 (x ∪ {κ}). Then H ⊢a0 A 1 (FΣ x∪{κ} ) Σ1 H ⊢a Fx∪{κ} ”A Σ1 where Fx∪{κ} ”A denotes the result of replacing all ordinals δ in k(A) ∪ {I} Σ1 by Fx∪{κ} (δ). Σ1 Fx∪{κ} ”(ϕ[κ, a]) ≡ ϕLy [x, a] for a ∈ Lx and ϕ ∈ Σ1 .

∀Σ1 ϕ∀a ∈ Lx (ϕ[κ, a] → Ly |= ϕ[x, a]) Σn Σn n (FΣ : HullIΣn (x) ∼ = Lx . x ) (Cf. (2).) Let x = Fx (I) for Fx

If a0 < a, A ∈ Σn and then

k(A) ⊂ HullIΣn (x) H ⊢a0 A n (FΣ x ) H ⊢a FxΣn ”A 4

(4)

∀α∃x[α < x = FxΣn (I)]

(1)

Σ1 Σ1 (I) < κ] (κ) < Fx∪{κ} ∀β∃κ > β∀α < κ∃x[α < x = Fx∪{κ}

(3)

Ordinals occurring in derivations are essentially the followings: 1. regular ordinals κ in (3), Σ1 Σ1 2. critical points x = Fx∪{κ} (κ), and images Fx∪{κ} (δ),

3. and critical points x = FxΣn (I), and images FxΣn (δ). Therefore Skolem hulls H should be closed under 1. I > α 7→ ωα (, which is a Σ2 -map), 2. first order (Σn -)definability for the ordinal δ ∈ H in (6), Σ1 3. Mostowski collapsing maps Fx∪{κ} for regular κ and FxΣn with critical points x,

4. and critical points α (κ > α)

7→ x = FxΣn (I) > α Σ1 7→ x = Fx∪{κ} (κ) > α

Skolem hulls with these closure properties are defined through critical points Ψκ,n α and ΨI,n α. Definition 3.1 Hα,n (X) is a Skolem hull of {0, I} ∪ X under the functions +, α 7→ ω α , ΨI,n ¹ α, Ψκ,n ¹ α (regular κ < I), the Σn Skolem hulling: X 7→ HullIΣn (X ∩ I) and the Mostowski collapsing functions Σ1 (x = Ψκ,n γ, δ) 7→ Fx∪{κ} (δ) (κ ∈ R)

and (x = ΨI,n γ, δ) 7→ FxΣn (δ). For κ ≤ I Ψκ,n α := min{β ≤ κ : κ ∈ Hα,n (β) & Hα,n (β) ∩ κ ⊂ β}.

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Ordinals Ψκ,n α and ΨI,n α are critical points: HullIΣ1 (Ψκ,n α ∪ {κ}) ∩ κ = Ψκ,n α (κ < I) HullIΣn (ΨI,n α) = ΨI,n α Hα,n (Ψκ,n α) ∩ κ = Ψκ,n α (κ ≤ I) Hence {δ, α, κ} ⊂ Hα,n (∅) & δ < κ ⇒ δ < Ψκ,n (α) ∈ Hα+1,n (∅) In other words for regular κ, cf. (1) and (3) α ∈ Hα,n ⊢a δ < κ ⇒ Hα+1,n ⊢a+1 ∃x[δ < x = Fx (κ)] Σ1 for x = Ψκ,n (α), and Fx = if κ = I then FxΣn else Fx∪{κ} . Let us bound existential quantifiers in provable formulas

Hα,n ⊢a ∃z < κ θ[z]. In view of (7) it suffices to bound the depths a of derivations to a < κ. If α ∈ Hα,n , then ∃z < Ψκ,n α θ[z] by a ∈ Hα,n ∩ κ. Roughly α ∈ Hα,n ⊢a ∃z < κ θ[z] ⇒ Hα+1,n ⊢Ψκ,n α ∃z < κ θ[z] is proved by induction on a. This ends a sketch of a proof of Theorem 1.1.

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Work in progress

Let X be a class of transitive sets. The Πi -recursively Mahlo operation RMi (X ) is defined through a universal Πi -formula Πi (a): P ∈ RMi (X )

:⇔ ∀b ∈ P [P |= Πi (b) → ∃Q ∈ X ∩ P (Q |= Πi (b))] (read:P is Πi -reflecting on X .)

A transitive set P is said to be Πi -reflecting if P ∈ RMi (L). For a definable relation ≺ ∩ P ∈ RM2 (a; ≺) :⇔ P ∈ {RM2 (RM2 (b; ≺)) : b ≺P a}, where b ≺P a :⇔ P |= b ≺ a. Π3 -reflecting ordinals are known to be recursive analogues to weakly compact cardinals. Ordinal analysis of Π3 -reflection has been done in [Rathjen94] and [A04]. An ordinal analysis yields a proof-theoretic reduction of Π3 -reflection in terms of iterations of recursively Mahlo operations as follows. KPℓ denotes a set theory for limits of admissibles. 6

Theorem 4.1 ([A09].) There exists a Σ1 -relation ¢ on ω such that KPΠ3 is Π11 (on ω)-conservative over the theory KPℓ + {L ∈ RM2 (a; ¢i ) : a ∈ ω}. [Jensen72] showed under V = L that for regular cardinals κ and the Mahlo operation M , κ is weakly compact iff ∀X ⊂ κ[κ ∈ M (X) ⇒ M (X) ∩ κ ̸= ∅] iff ∀X ⊂ κ[κ ∈ M (X) ⇒ κ ∈ M (M (X))]. The existence of a weakly compact cardinal over ZF seems to be prooftheoretically reducible to iterations of Mostowski collapsings and Mahlo operations.

References [A97] T. Arai, A sneak preview of proof theory of ordinals, an invited talk at Kobe seminar on Logic and Computer Science, 5-6 Dec. 1997. arXiv:1102.0596, to appear in Ann. Japan Assoc. Philos. Sci. [A04] T. Arai, Proof theory for theories of ordinals II:Π3 -Reflection, Ann. Pure Appl. Logic vol. 129 (2004), 39-92. [A09] T. Arai, Iterating the recursively Mahlo operations, in Proceedings of the thirteenth International Congress of Logic Methodology, Philosophy of Science, Ed. by C. Glymour, W. Wei and D. Westerstahl, College Publications, King’s College London (2009), pp. 21-35. arXiv:1005.1987 [A∞] T. Arai, Lifting up the proof theory to the countables: Zermelo-Fraenkel’s set theory, submitted. arXiv:1101.5660 [Buchholz92] W. Buchholz, A simplified version of local predicativity, P. H. G. Aczel, H. Simmons and S. S. Wainer(eds.), Proof Theory, Cambridge UP, 1992, pp. 115-147. [Jensen72] R. Jensen, The fine structure of the constructible hierarchy, Ann. Math. Logic 4(1972), 229-308. [Rathjen94] M. Rathjen, Proof theory of reflection, Ann. Pure Appl. Logic 68 (1994), 181-224.

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