Generic cuts in models of arithmetic

This paper has been accepted for the Mathematical Logic Quaterly. A preliminary version only is available here in PDF format: Generic cuts in models of arithmetic.


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 defined in a novel way, without initially specifying what property is indicated and is used to define 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 of 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.

Additional resources

A note by Tin Lok Wong and Richard Kaye explaining that there are always countably infinitely many conjugacy classes of generic cuts in a model M of PA. (In fact, most of the results generalise to counting the number of conjugacy classes of a dense subset of the space ZY.)

Richard Kaye's Desktop PC, Home page.