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On first-order theories with provability operator

Published online by Cambridge University Press:  12 March 2014

Sergei Artëmov  and
Franco Montagna
Sergei Artëmov
Affiliation:
Steklov Mathematical Institute, 42, Vavilov Str., Moscow, 117966, Russia, E-mail: [email protected]
Franco Montagna
Affiliation:
Dipartimento di Matematica, Universita di Siena, 53100-Siena, Italia, E-mail: [email protected]
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Abstract

In this paper the modal operator “x is provable in Peano Arithmetic” is incorporated into first-order theories. A provability extension of a theory is defined. Presburger Arithmetic of addition, Skolem Arithmetic of multiplication, and some first order theories of partial consistency statements are shown to remain decidable after natural provability extensions. It is also shown that natural provability extensions of a decidable theory may be undecidable.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 59 , Issue 4 , December 1994 , pp. 1139 - 1153
Copyright
Copyright © Association for Symbolic Logic 1994

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References

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