Big and Small Steps for Fast and Slow Provability Paula Henk illc , - - PowerPoint PPT Presentation

GL GLT Fast provability Slow provability

Big and Small Steps for Fast and Slow Provability

Paula Henk

illc, University of Amsterdam

September 1, 2016

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GL GLT Fast provability Slow provability

G¨

• del-L¨
• b provability logic GL
• K together with L¨
• b’s axiom: (L) (A → A) → A
• Complete w.r.t. transitive converse well-founded trees

Theorem (Solovay)

GL is the provability logic of any reasonable theory T.

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GL GLT Fast provability Slow provability

The bimodal system GLT

Contains GL for both △ and , together with: (T1) △A → A (T1) A → ▽A (T2) A → △A (T2) ▽A → A (T3) A → △A (T3) ▽A → A (T4) △A → A (T4) A → ▽A Lindstr¨

• m-frame: W, ≺, ≺∞, with W, ≺ a GL-frame, and

x ≺∞ y :⇔|{z | x ≺ z ≺ y}|= ∞. x ▽A :⇔ y A for some y with x ≺ y. x A :⇔ y A for some y with x ≺∞ y.

Theorem (Lindstr¨

• m)

GLT is sound and complete w.r.t. Lindstr¨

• m-frames.

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GL GLT Fast provability Slow provability

Fast provability

PA∗ is Peano Arithmetic (PA) together with Parikh’s rule: if PAϕ, then ϕ.

Theorem (Parikh)

PA∗ has speed-up over PA.

Theorem (Lindstr¨

• m)

GLT is the joint provability logic of PA and △p.

Lemma (Lindstr¨

• m)

PA ⊢ ▽

pϕ ↔ ω

PAϕ 4 / 6

GL GLT Fast provability Slow provability

Slow provability

Friedman, Rathjen, and Weiermann: PA↾F :=

• n∈ω

{IΣn | F(n)↓}, where F is a certain recursive function with PA F↓. △s is the provability predicate of PA↾F.

Theorem (H. & Shavrukov)

GLT is the joint provability logic of △s and PA.

Theorem (Pakhomov, Freund)

There are slow provability predicates △1, △2, for which

• i. PA ⊢ PAϕ ↔ ▽ω

1 ϕ

• ii. PA ⊢ PAϕ ↔ ▽ε0

2 ϕ

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