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Original Paper
Generic cuts in models of arithmetic
Article first published online: 7 MAR 2008
DOI: 10.1002/malq.200710017
Copyright © 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1521-3870/asset/cover.gif?v=1&s=3df32dbed8d9153ac6eafe1eaa786854b9b48345 "Mathematical Logic Quarterly")
Additional Information
How to Cite
Kaye, R. (2008), Generic cuts in models of arithmetic. Mathematical Logic Quarterly, 54: 129–144. doi: 10.1002/malq.200710017
Publication History
- Issue published online: 7 MAR 2008
- Article first published online: 7 MAR 2008
- Manuscript Accepted: 8 JUN 2007
- Manuscript Revised: 6 JUN 2007
- Manuscript Received: 7 FEB 2007
- Abstract
- References
- Cited By
Keywords:
- Models of arithmetic;
- Peano arithmetic;
- initial segments
Abstract
We present some general results concerning the topological space of cuts of a countable model of arithmetic given by a particular indicator Y.
The notion of “indicator” is de.ned in a novel way, without initially specifying what property is indicated and is used to de.ne a topological space of cuts of the model. Various familiar properties of cuts (strength, regularity, saturation, coding properties) are investigated in this sense, and several results are given stating whether or not the set of cuts having the property is comeagre.
A new notion of “generic cut” is introduced and investigated and it is shown in the case of countable arithmetically saturated models M ⊧ PA that generic cuts exist, indeed the set of generic cuts is comeagre in the sense of Baire, and furthermore that two generic cuts within the same “small interval” of the model are conjugate by an automorphism of the model.The paper concludes by outlining some applications to constructions of cuts satisfying properties incompatible with genericity, and discussing in model-theoretic terms those properties for which there is an indicator Y. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)