Journal of Symbolic Logic

Theories of arithmetics in finite models

Michał Krynicki and Konrad Zdanowski

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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.

Article information

Source
J. Symbolic Logic Volume 70, Issue 1 (2005), 1-28.

Dates
First available in Project Euclid: 1 February 2005

Permanent link to this document
http://projecteuclid.org/euclid.jsl/1107298508

Digital Object Identifier
doi:10.2178/jsl/1107298508

Mathematical Reviews number (MathSciNet)
MR2119121

Zentralblatt MATH identifier
05004786

Subjects
Primary: 03C13: Finite structures [See also 68Q15, 68Q19]
Secondary: 03C68: Other classical first-order model theory 68Q17: Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.) [See also 68Q15]

Keywords
Finite models arithmetic definability spectrum complexity

Citation

Krynicki, Michał; Zdanowski, Konrad. Theories of arithmetics in finite models. J. Symbolic Logic 70 (2005), no. 1, 1--28. doi:10.2178/jsl/1107298508. http://projecteuclid.org/euclid.jsl/1107298508.


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