On ω-consistency and related properties

Abstract

1. This paper grew out of an attempt to answer a question raised in [4]. Let a logic L containing “numerals” z1, z2, … and a certain statement N(x) (intended to express the proposition that x is a natural number) be called ω-inconsistent if there is a statement such that ⊦ F(zk) for k = 1, 2, …, and ⊦ ∼(x)·N(x) ⊃ F(x); then it is evident that L cannot have a model in which N(x) is satisfied by the images of the numerals and nothing else if L is ω-inconsistent.Question: If L is ω-consistent, i.e. not ω-inconsistent, must there be such a model? Calling a model of the kind just described a special model, we ask for necessary and sufficient conditions on L to insure the existence of a special model. We give several sets of such conditions, applicable to a certain very inclusive class of logics, in Theorem 1 and Theorems 3 and 4. Theorem 2 shows that a logic may be ω-consistent but still not have a special model.This paper was close to completion when [3] appeared. For systems with only denumerably many symbols our results include Henkin's, for, by adjoining a new predicate N(x) to each of the systems considered in [3] which have only a denumerable number of constant symbols and then adding as an axiom (x)N(x), these systems become special cases of the systems we consider. It is easily seen that Henkin's Theorem 7 essentially proves the equivalence of conditions (2) and (3) in our Theorem 1, and Theorem 3 of [3] corresponds to our Theorem 2. Incidentally, our argument of Theorem 2 could also serve to prove Henkin's Theorem 6.