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

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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DOI 10.1016/j.apal.2008.07.005
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References found in this work BETA

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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Citations of this work BETA

A Note on Standard Systems and Ultrafilters.Fredrik Engström - 2008 - Journal of Symbolic Logic 73 (3):824-830.
Models of Expansions of Equation Image with No End Extensions.Saharon Shelah - 2011 - Mathematical Logic Quarterly 57 (4):341-365.

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