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 1 / 6

GL GLT Fast provability Slow provability G¨ odel-L¨ ob provability logic GL • K together with L¨ ob’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. 2 / 6

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¨ om-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¨ om) GLT is sound and complete w.r.t. Lindstr¨ om-frames. 3 / 6

GL GLT Fast provability Slow provability Fast provability PA ∗ is Peano Arithmetic ( PA ) together with Parikh’s rule: if � PA ϕ , then ϕ . Theorem (Parikh) PA ∗ has speed-up over PA . Theorem (Lindstr¨ om) GLT is the joint provability logic of � PA and △ p . Lemma (Lindstr¨ om) PA ⊢ ▽ p ϕ ↔ � ω PA ϕ 4 / 6

GL GLT Fast provability Slow provability Slow provability Friedman, Rathjen, and Weiermann: � PA ↾ F := { IΣ n | F ( n ) ↓} , n ∈ ω 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 ⊢ � PA ϕ ↔ ▽ ω 1 ϕ ii. PA ⊢ � PA ϕ ↔ ▽ ε 0 2 ϕ 5 / 6