Reflection principles and iterated consistency assertions

Extract

This paper compares the strength of two sorts of sentences of PA (classical first-order arithmetic with induction): reflection principles and sentences that may be called iterated consistency assertions.

Let Bew(x) be the standard provability predicate for PA, and for any sentence S of PA, let ⌈S⌉ be the numeral for the Gödel number of S. The reflection principle for S is the sentence Bew(⌈S⌉) → S, and a reflection principle is simply the reflection principle for some sentence. Nothing false (in the standard model for PA) is provable in PA, and therefore every reflection principle is true. Löb's theorem asserts that S is provable (in PA) if the reflection principle for S is provable.

We shall suppose that the 0-ary propositional connectives ⊤ and ⊥ are taken as primitives in the formulation of PA. We define the iterated consistency assertions Con^m by: Con⁰ = ⊤; Con^_m−1 = − Bew(⌈ − Con^m⌉). Con¹ may be taken to be the sentence of PA that expresses the consistency of PA; Conⁿ⁻¹, the sentence that expresses the consistency of PA ⋃ {Conⁿ}.

Our starting point is the observation that Con¹ is equivalent (in PA) to the reflection principle for ⊥. (The second incompleteness theorem thus follows in a well-known way from Löb's theorem: if PA is consistent, then ⊥ is not provable, the reflection principle for ⊥ is not provable, and the consistency of PA is not provable either.)