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On recursively enumerable and arithmetic models of set theory

Published online by Cambridge University Press:  12 March 2014

Michael O. Rabin
Michael O. Rabin*
Affiliation:
Institute for Advanced Study and the Hebrew University, Jerusalem
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Extract

In this note we shall prove a certain relative recursiveness lemma concerning countable models of set theory (Lemma 5). From this lemma will follow two results about special types of such models.

Kreisel [5] and Mostowski [6] have shown that certain (finitely axiomatized) systems of set theory, formulated by means of the ϵ relation and certain additional non-logical constants, do not possess recursive models. Their purpose in doing this was to construct consistent sentences without recursive models. As a first corollary of Lemma 5, we obtain a very simple proof, not involving any formal constructions within the system of the notions of truth and satisfiability, of an extension of the Kreisel-Mostowski theorems. Namely, set theory with the single non-logical constant ϵ does not possess any recursively enumerable model. Thus we get, as a side product, an easy example of a consistent sentence containing a single binary relation which does not possess any recursively enumerable model; this sentence being the conjunction of the (finitely many) axioms of set theory.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 23 , Issue 4 , December 1958 , pp. 408 - 416
Copyright
Copyright © Association for Symbolic Logic 1958

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References

REFERENCES

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