Introduction

We recall from Chapter 3 the definition of the ω-inconsistency of a theory T (whose language contains 0 and s): T is ω-inconsistent iff for some formula A(x), T⊢∃xA(x), and for every natural number n, T⊢¬A(n). T is ω-consistent iff it is not ω-inconsistent. If T is ω-consistent, then T⊬∃xx ≠ x, and therefore T is consistent.

It is easy to show, however, that the converse does not hold: Let T be the theory that results when Bew(┌⊥┐) is added to PA. Since PA does not prove ¬Bew(┌⊥┐), T is consistent and for every n, the Δ sentence ¬Pf(n, ┌⊥┐) is true. Thus for every n, PA⊢¬Pf(n, ┌⊥┐), and so for every n, T⊢¬Pf(n, ┌⊥┐) (T extends PA). But T⊢Bew(┌⊥┐), that is, T⊢∃yPf(y, ┌⊥┐). So, despite its consistency, T is ω-inconsistent.

As a sentence S is said to be inconsistent with T if the theory whose axioms are those of T together with S itself is inconsistent, so S is ω-inconsistent (with T) if the theory whose axioms are those of T together with S is ω-inconsistent. S is ω-consistent iff not ω-inconsistent.

We call a sentence S ω-provable in T iff ¬S is ω-inconsistent with T. So if S is provable in T, S is ω-provable in T.