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Theories of arithmetics in finite models

Published online by Cambridge University Press:  12 March 2014

Michał Krynicki  and
Konrad Zdanowski
Michał Krynicki
Affiliation:
Cardinal Stefan Wyszyński University, Warszawa, Poland, E-mail: krynicki@uksw.edu.pl
Konrad Zdanowski
Affiliation:
Institute of Mathematics, Polish Academy of Science Śniadeckich 8, 00–956 Warszawa, Poland, E-mail: konrad.zdanowski@wp.pl
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Abstract

We investigate theories of initial segments of the standard models for arithmetics. It is easy to see that if the ordering relation is definable in the standard model then the decidability results can be transferred from the infinite model into the finite models. On the contrary we show that the Σ2–theory of multiplication is undecidable in finite models. We show that this result is optimal by proving that the Σ1–theory of multiplication and order is decidable in finite models as well as in the standard model. We show also that the exponentiation function is definable in finite models by a formula of arithmetic with multiplication and that one can define in finite models the arithmetic of addition and multiplication with the concatenation operation.

We consider also the spectrum problem. We show that the spectrum of arithmetic with multiplication and arithmetic with exponentiation is strictly contained in the spectrum of arithmetic with addition and multiplication.

Keywords

Finite modelsarithmeticdefinabilityspectrumcomplexity
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 1 , March 2005 , pp. 1 - 28
Copyright
Copyright © Association for Symbolic Logic 2005

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