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Generic cuts in models of arithmetic

✍ Scribed by Richard Kaye

Publisher
John Wiley and Sons
Year
2008
Tongue
English
Weight
188 KB
Volume
54
Category
Article
ISSN
0044-3050

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✦ Synopsis

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)

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