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A simple model for a weak system of arithmetic

Published online by Cambridge University Press:  17 April 2009

D.W. Barnes  and
G.P. Monro
D.W. Barnes
Affiliation:
Department of Pure Mathematics, University of Sydney, Sydney, New South Wales;
G.P. Monro
Affiliation:
School of Mathematics, University of New South Wales, Kensington, New South Wales.
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Abstract

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The natural, first order version of Peano's axioms (the theory T with 0, the successor function and an induction schema) is shown to possess the following nonstandard model: the natural numbers together with a collection of ‘infinite’ elements isomorphic to the integers. In fact, a complete list of the models of this theory is obtained by showing that T is equivalent to the apparently weaker theory with the induction axiom replaced by axioms stating that there are no finite cycles under the successor function and that 0 is the only non-successor.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 11 , Issue 3 , December 1974 , pp. 321 - 323
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Bell, J.L. and Slomson, A.B., Models and ultraproducts: an introduction (North-Holland, Amsterdam, London, 1969).Google Scholar