Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-09T08:48:23.790Z Has data issue: false hasContentIssue false

On ω-consistency and related properties

Published online by Cambridge University Press:  12 March 2014

Steven Orey*
Affiliation:
The University of Minnesota

Extract

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 ⊦ ∼(xN(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.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1956

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

[1]Henkin, L., The completeness of the first-order functional calculus, this Journal, vol. 14 (1949), pp. 159166.Google Scholar
[2]Henkin, L., Completeness in the theory of types, this Journal, vol. 15 (1950), pp. 8191.Google Scholar
[3]Henkin, L., A generalization of the concept of ω-consistency, this Journal, vol. 19 (1954), pp. 183196.Google Scholar
[4]Quine, W. V., On ω-consistency and a so-called axiom of infinity, this Journal, vol. 18 (1953), pp. 119124.Google Scholar
[5]Rasiowa, H. and Sikorski, R., A proof of the completeness theorem of Gödel, Fundamenta Mathematicae, vol. 37 (1950), pp. 193200.CrossRefGoogle Scholar
[6]Rasiowa, H. and Sikorski, R., A proof of the Skolem-Ldwenheim theorem, Fundamenta Mathematicae, vol. 38 (1951), pp. 230232.CrossRefGoogle Scholar
[7]Rosser, J. B., Gödel theorems for non-constructive logics, this Journal, vol. 2 (1937), pp. 129137.Google Scholar