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Theories incomparable with respect to relative interpretability

Published online by Cambridge University Press:  12 March 2014

Richard Montague
Richard Montague*
Affiliation:
University of California, Los Angeles
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Extract

The present paper concerns the relation of relative interpretability introduced in [8], and arises from a question posed by Tarski: are there two finitely axiomatizable subtheories of the arithmetic of natural numbers neither of which is relatively interpretable in the other? The question was answered affirmatively (without proof) in [3], and the answer was generalized in [4]: for any positive integer n, there exist n finitely axiomatizable subtheories of arithmetic such that no one of them is relatively interpretable in the union of the remainder. A further generalization was announced in [5] and is proved here: there is an infinite set of finitely axiomatizable subtheories of arithmetic such that no one of them is relatively interpretable in the union of the remainder. Several lemmas concerning the existence of self-referential and mutually referential formulas are given in Section 1, and will perhaps be of interest on their own account.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 27 , Issue 2 , June 1962 , pp. 195 - 211
Copyright
Copyright © Association for Symbolic Logic 1962

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References

[1]Feferman, S. and Montague, R., The method of arithmetization and some of its applications, Amsterdam, forthcoming.Google Scholar
[2]Kalish, D. and Montague, R., Relations between interpretability and relative interpretability, in preparation.Google Scholar
[3]Montague, R., Two theorems on relative interpretability. Summaries of talks presented at the Summer Institute for Symbolic Logic, Cornell University, 1957, second edition, Princeton (Institute for Defense Analysis), 1960, pp. 263–264.Google Scholar
[4]Montague, R., Incomparable theories, Notices of the American Mathematical Society, vol. 6 (1958), pp. 362–363.Google Scholar
[5]Montague, R., The continuum of relative interpretability types, this Journal, vol. 23 (1958), p. 460.Google Scholar
[6]Mostowski, A., On models of axiomatic systems, Fundamenta mathematicae, vol. 39 (1952), pp. 133–158.CrossRefGoogle Scholar
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[8]Tarski, A., Mostowski, A., and Robinson, R. M., Undecidable theories, Amsterdam (North Holland), 1953.Google Scholar