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.
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.