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Remarks on weak notions of saturation in models of Peano arithmetic

Published online by Cambridge University Press:  12 March 2014

Matt Kaufmann  and
James H. Schmerl
Matt Kaufmann
Affiliation:
Austin Research Center, Burroughs Corporation, Austin, Texas 78727
James H. Schmerl
Affiliation:
Department of Mathematics, University of Connecticut, Storrs, Connecticut 06268
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Extract

This paper is a sequel to our earlier paper [2] entitled Saturation and simple extensions of models of Peano arithmetic. Among other things, we will answer some of the questions that were left open there. In §1 we consider the question of whether there are lofty models of PA which have no recursively saturated, simple extensions. We are still unable to answer this question; but we do show in that section that these models are precisely the lofty models which are not recursively saturated and which are κ-like for some regular κ. In §2 we use diagonal methods to produce minimal models of PA in which the standard cut is recursively definable, and other minimal models in which the standard cut is not recursively definable. In §3 we answer two questions from [2] by exhibiting countable models of PA which, in the terminology of this paper, are uniformly ω-lofty but not continuously ω-lofty and others which are continuously ω-lofty but not recursively saturated. We also construct a model (assuming ◇) which is not recursively saturated but every proper, simple cofinal extension of which is ℵ1-saturated. Finally, in §4 we answer another question from [2] by proving that for regular κ ≥ ℵ1; every κ-saturated model of PA has a κ-saturated proper, simple extension which is not κ+-saturated.

Our notation and terminology are quite standard. Anything unfamiliar to the reader and not adequately denned here is probably defined in §1 of [2]. All models considered are models of Peano arithmetic.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 52 , Issue 1 , March 1987 , pp. 129 - 148
Copyright
Copyright © Association for Symbolic Logic 1987

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Footnotes

1

Partially supported by NSF grant no. MCS-8301603.

References

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