A standard model of Peano Arithmetic with no conservative elementary extension

Annals of Pure and Applied Logic 156 (2):308-318 (2008)
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Abstract

The principal result of this paper answers a long-standing question in the model theory of arithmetic [R. Kossak, J. Schmerl, The Structure of Models of Peano Arithmetic, Oxford University Press, 2006, Question 7] by showing that there exists an uncountable arithmetically closed family of subsets of the set ω of natural numbers such that the expansion of the standard model of Peano arithmetic has no conservative elementary extension, i.e., for any elementary extension of , there is a subset of ω* that is parametrically definable in but whose intersection with ω is not a member of . We also establish other results that highlight the role of countability in the model theory of arithmetic. Inspired by a recent question of Gitman and Hamkins, we furthermore show that the aforementioned family can be arranged to further satisfy the curious property that forcing with the quotient Boolean algebra collapses 1 when viewed as a notion of forcing

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Citations of this work

Destructibility and axiomatizability of Kaufmann models.Corey Bacal Switzer - 2022 - Archive for Mathematical Logic 61 (7):1091-1111.
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References found in this work

Models and types of Peano's arithmetic.Haim Gaifman - 1976 - Annals of Mathematical Logic 9 (3):223-306.
Definability of measures and ultrafilters.David Pincus & Robert M. Solovay - 1977 - Journal of Symbolic Logic 42 (2):179-190.
Models of arithmetic and closed ideals.Julia Knight & Mark Nadel - 1982 - Journal of Symbolic Logic 47 (4):833-840.

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