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Locally finite theories

Published online by Cambridge University Press:  12 March 2014

Jan Mycielski
Jan Mycielski*
Affiliation:
Department of Mathematics, University of Colorado, Boulder, Colorado 80309
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Extract

We say that a first order theory T is locally finite if every finite part of T has a finite model. It is the purpose of this paper to construct in a uniform way for any consistent theory T a locally finite theory FIN(T) which is syntactically (in a sense) isomorphic to T.

Our construction draws upon the main idea of Paris and Harrington [6] (I have been influenced by some unpublished notes of Silver [7] on this subject) and generalizes the syntactic aspect of their result from arithmetic to arbitrary theories. (Our proof is syntactic, and it is simpler than the proofs of [5], [6] and [7]. This reminds me of the simple syntactic proofs of several variants of the Craig-Lyndon interpolation theorem, which seem more natural than the semantic proofs.)

The first mathematically strong locally finite theory, called FIN, was defined in [1] (see also [2]). Now we get much stronger ones, e.g. FIN(ZF).

From a physicalistic point of view the theorems of ZF and their FIN(ZF)-counterparts may have the same meaning. Therefore FIN(ZF) is a solution of Hilbert's second problem. It eliminates ideal (infinite) objects from the proofs of properties of concrete (finite) objects.

In [4] we will demonstrate that one can develop a direct finitistic intuition that FIN(ZF) is locally finite. We will also prove a variant of Gödel's second incompleteness theorem for the theory FIN and for all its primitively recursively axiomatizable consistent extensions.

The results of this paper were announced in [3].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 51 , Issue 1 , March 1986 , pp. 59 - 62
Copyright
Copyright © Association for Symbolic Logic 1986

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References

REFERENCE

[1]Mycielski, J., Analysis without actual infinity, this Journal, vol. 46 (1981), pp. 625–633.Google Scholar
[2]Mycielski, J., Finitistic real analysis, Real Analysis Exchange, vol. 6 (1981), pp. 127–130.Google Scholar
[3]Mycielski, J., Locally finite counterparts of consistent theories, Abstracts of Papers Presented to the American Mathematical Society, vol. 5 (1984), pp. 229.Google Scholar
[4]Mycielski, J., Finite intuitions supporting the consistency of ZF and ZF + AD (to appear).Google Scholar
[5]Paris, J., Some independence results for Peano's arithmetic, this Journal, vol. 43 (1978), pp. 725–731.Google Scholar
[6]Paris, J. and Harrington, L., A mathematical incompleteness in Peano's arithmetic, Handbook of mathematical logic (Barwise, J, editor), North-Holland, Amsterdam, 1977, pp. 1133–1142.CrossRefGoogle Scholar
[7]Silver, J., Harrington's version of Paris' result, mimeographed notes, 1977.Google Scholar