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Original Paper
Arithmetic of divisibility in finite models
Article first published online: 3 FEB 2004
DOI: 10.1002/malq.200310086
Copyright © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1521-3870/asset/cover.gif?v=1&s=3df32dbed8d9153ac6eafe1eaa786854b9b48345 "Mathematical Logic Quarterly")
Additional Information
How to Cite
Mostowski, M. and Wasilewska, A. E. (2004), Arithmetic of divisibility in finite models. Mathematical Logic Quarterly, 50: 169–174. doi: 10.1002/malq.200310086
Publication History
- Issue published online: 3 FEB 2004
- Article first published online: 3 FEB 2004
- Manuscript Accepted: 6 NOV 2003
- Manuscript Revised: 4 NOV 2003
- Manuscript Received: 12 AUG 2003
- Abstract
- References
- Cited By
Keywords:
- Finite model;
- divisibility relation;
- arithmetical definability;
- FM-representability;
- IS-interpretability;
- FM-domain;
- undecidability
Abstract
We prove that the finite-model version of arithmetic with the divisibility relation is undecidable (more precisely, it has Π01-complete set of theorems). Additionally we prove FM-representability theorem for this class of finite models. This means that a relation R on natural numbers can be described correctly on each input on almost all finite divisibility models if and only if R is of degree ≤0′. We obtain these results by interpreting addition and multiplication on initial segments of finite models with divisibility only. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)