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Non-standard models for formal logics

Published online by Cambridge University Press:  12 March 2014

J. Barkley Rosser  and
Hao Wang
J. Barkley Rosser
Affiliation:
Cornell University
Hao Wang
Affiliation:
Harvard University
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Extract

In his doctor's thesis [1], Henkin has shown that if a formal logic is consistent, and sufficiently complex (for instance, if it is adequate for number theory), then it must admit a non-standard model. In particular, he showed that there must be a model in which that portion of the model which is supposed to represent the positive integers of the formal logic is not in fact isomorphic to the positive integers; indeed it is not even well ordered by what is supposed to be the relation of ≦.

For the purposes of the present paper, we do not need a precise definition of what is meant by a standard model of a formal logic. The non-standard models which we shall discuss will be flagrantly non-standard, as for instance a model of the sort whose existence is proved by Henkin. It will suffice if we and our readers are in agreement that a model of a formal logic is not a standard model if either:

(a) The relation in the model which represents the equality relation in the formal logic is not the equality relation for objects of the model.

(b) That portion of the model which is supposed to represent the positive integers of the formal logic is not well ordered by the relation ≦.

(c) That portion of the model which is supposed to represent the ordinal numbers of the formal logic is not well ordered by the relation ≦.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 15 , Issue 2 , June 1950 , pp. 113 - 129
Copyright
Copyright © Association for Symbolic Logic 1950

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