Reflection principles and iterated consistency assertions

This paper compares the strength of two sorts of sentences of PA (classical firstorder 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 FS] be the numeral for the Gddel number of S. The reflection principle for S is the sentence Bew(QS]) -* 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. Lob'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 T and I are taken as primitives in the formulation of PA. We define the iterated consistency assertions Conm by: Con0 = T; Conm+1 -Bew(Q Conmi). Con' may be taken to be the sentence of PA that expresses the consistency of PA; Conn+l, the sentence that expresses the consistency of PA U {Conn}. Our starting point is the observation that Con1 is equivalent (in PA) to the reflection principle for I. (The second incompleteness theorem thus follows in a well-known way from Lob's theorem: if PA is consistent, then I is not provable, the reflection principle for I is not provable, and the consistency of PA is not provable either.) Con0 is trivially equivalent to the conjunction of 0 reflection principles and Con2 is equivalent to the conjunction of the two reflection principles Bew(FBew(FI])]) -* Bew(r 11) and Bew (F l]) -* I. In like manner, for any n > 2, Conn is equivalent to a conjunction of n reflection principles. We shall show in this note that every conjunction of n reflection principles that implies Conn is in fact equivalent to Con". It will follow that no conjunction of fewer than n reflection principles implies Con", and thus that the bound n on the number of conjuncts in a conjunction of reflection principles that is equivalent to Conn is as low as possible. We shall also provide an example of an unprovable reflection principle that is strictly weaker than Con1. Our proof will make use of certain techniques and notions of propositional modal logic. The system G of modal logic has as its axioms all tautologies and all sentences of the forms LI(A -* B) -* (EM A -* L1B), WA -* DEJA, and LI(LIA -* A) -* ClA; its two rules of inference are modus ponens and necessitation.2 We use 'O' to range over functions from the sentences of modal logic to sentences of PA that commute with propositional connectives (including T and I) and are such that