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Completeness properties of heyting's predicate calculus with respect to re models

Published online by Cambridge University Press:  12 March 2014

Dov M. Gabbay
Dov M. Gabbay*
Affiliation:
Bar-Ilan University, Ramat-Gan, Israel
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Extract

Validity in recursive structures was investigated by several authors. Kreisel [10] has shown that there exists a consistent sentence of classical predicate calculus (CPC) that does not possess a recursive model. The sentence is a conjunction of the axioms of a variant of Bernays set theory, including the axiom of infinity. The language contains additional constants besides ϵ. Later Kreisel [2] and Mostowski [3] presented a sentence (not possessing recursive models) which was a conjunction of axioms of a variant of Bernays set theory without the axiom of infinity but still with additional constants besides ϵ. Later Mostowski [4] improved the result by giving a sentence which can be demonstrated in Heyting arithmetic to be consistent and to have no recursive models. Rabin [6] obtained a simple proof that some sentence of set theory with the single nonlogical constant ϵ does not have any recursively enumerable models.

More generally, Mostowski [5] has shown that the set of all sentences valid in all RE models is not arithmetical and Vaught [1] improved this result by showing that it holds for a language of one binary relation. In fact, Vaught gives· a way of translating n-place relations to 2-place ones that preserves the RE characteristic of the model. For further results pertaining to recursive models see Vaught [1].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 41 , Issue 1 , March 1976 , pp. 81 - 94
Copyright
Copyright © Association for Symbolic Logic 1976

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Footnotes

1

This research has been supported in part by the National Science Foundation under NSF grant GJ-443X. I am grateful to Professor G. Kreisel for most valuable suggestions and criticisms.

References

REFERENCES

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