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Arithmetic of divisibility in finite models

A. E. Wasilewska & M. Mostowski

Mathematical Logic Quarterly 50 (2):169 (2004)

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(1 other version)Definability and decision problems in arithmetic.Julia Robinson - 1949 - Journal of Symbolic Logic 14 (2):98-114.details In this paper, we are concerned with the arithmetical definability of certain notions of integers and rationals in terms of other notions. The results derived will be applied to obtain a negative solution of corresponding decision problems.In Section 1, we show that addition of positive integers can be defined arithmetically in terms of multiplication and the unary operation of successorS(whereSa=a+ 1). Also, it is shown that both addition and multiplication can be defined arithmetically in terms of successor and the relation (...) Download Export citation Bookmark 45 citations
Arithmetical definability over finite structures.Troy Lee - 2003 - Mathematical Logic Quarterly 49 (4):385.details Arithmetical definability has been extensively studied over the natural numbers. In this paper, we take up the study of arithmetical definability over finite structures, motivated by the correspondence between uniform AC0 and FO. We prove finite analogs of three classic results in arithmetical definability, namely that < and TIMES can first-order define PLUS, that < and DIVIDES can first-order define TIMES, and that < and COPRIME can first-order define TIMES. The first result sharpens the equivalence FO =FO to FO = (...) Download Export citation Bookmark 3 citations

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