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2012 Elementary Cuts in Saturated Models of Peano Arithmetic
James H. Schmerl
Notre Dame J. Formal Logic 53(1): 1-13 (2012). DOI: 10.1215/00294527-1626491

Abstract

A model $\mathscr{M} = (M,+,\times, 0,1,<)$ of Peano Arithmetic $({\sf PA})$ is boundedly saturated if and only if it has a saturated elementary end extension $\mathscr{N}$. The ordertypes of boundedly saturated models of $({\sf PA})$ are characterized and the number of models having these ordertypes is determined. Pairs $(\mathscr{N},M)$, where $\mathscr{M} \prec_{\sf end} \mathscr{N} \models({\sf PA})$ for saturated $\mathscr{N}$, and their theories are investigated. One result is: If $\mathscr{N}$ is a $\kappa$-saturated model of $({\sf PA})$ and $\mathscr{M}_0, \mathscr{M}_1 \prec_{\sf end} \mathscr{N}$ are such that $\aleph_1 \leq \mathrm{min}(\mathrm{cf}(M_0),\mathrm{dcf}(M_0)) \leq \mathrm{min}(\mathrm{cf}(M_1), \mathrm{dcf}(M_1)) < \kappa$, then $(\mathscr{N},M_0) \equiv (\mathscr{N},M_1)$.

Citation

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James H. Schmerl. "Elementary Cuts in Saturated Models of Peano Arithmetic." Notre Dame J. Formal Logic 53 (1) 1 - 13, 2012. https://doi.org/10.1215/00294527-1626491

Information

Published: 2012
First available in Project Euclid: 9 May 2012

zbMATH: 1244.03114
MathSciNet: MR2925265
Digital Object Identifier: 10.1215/00294527-1626491

Subjects:
Primary: 03C62

Rights: Copyright © 2012 University of Notre Dame

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Vol.53 • No. 1 • 2012
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