Journal of Symbolic Logic

Blunt and Topless End Extensions of Models of Set Theory

Matt Kaufmann

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Abstract

Let $\mathscr{U}$ be a well-founded model of ZFC whose class of ordinals has uncountable cofinality, such that $\mathscr{U}$ has a $\Sigma_n$ end extension for each $n \in \omega$. It is shown in Theorem 1.1 that there is such a model which has no elementary end extension. In the process some interesting facts about topless end extensions (those with no least new ordinal) are uncovered, for example Theorem 2.1: If $\mathscr{U}$ is a well-founded model of ZFC, such that $\mathscr{U}$ has uncountable cofinality and $\mathscr{U}$ has a topless $\Sigma_3$ end extension, then $\mathscr{U}$ has a topless elementary end extension and also a well-founded elementary end extension, and contains ordinals which are (in $\mathscr{U}$) highly hyperinaccessible. In $\S 3$ related results are proved for $\kappa$-like models ($\kappa$ any regular cardinal) which need not be well founded. As an application a soft proof is given of a theorem of Schmerl on the model-theoretic relation $\kappa \rightarrow \lambda$. (The author has been informed that Silver had earlier, independently, found a similar unpublished proof of that theorem.) Also, a simpler proof is given of (a generalization of) a characterization by Keisler and Silver of the class of well-founded models which have a $\Sigma_n$ end extension for each $n \in \omega$. The case $\kappa = \omega_1$ is investigated more deeply in $\S 4$, where the problem solved by Theorem 1.1 is considered for non-well-founded models. In Theorems 4.1 and 4.4, $\omega_1$-like models of ZFC are constructed which have a $\Sigma_n$ end extension for all $n \in \omega$ but have no elementary end extension. $\omega_1$-like models of ZFC which have no $\Sigma_3$ end extension are produced in Theorem 4.2. The proof uses a notion of satisfaction class, which is also applied in the proof of Theorem 4.6: No model of ZFC has a definable end extension which satisfies ZFC. Finally, Theorem 5.1 generalizes results of Keisler and Morley, and Hutchinson, by asserting that every model of ZFC of countable cofinality has a topless elementary end extension. This contrasts with the rest of the paper, which shows that for well-founded models of uncountable cofinality and for $\kappa$-like models with $\kappa$ regular, topless end extensions are much rarer than blunt end extensions.

Article information

Source
J. Symbolic Logic, Volume 48, Issue 4 (1983), 1053-1073.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183741414

Mathematical Reviews number (MathSciNet)
MR727794

Zentralblatt MATH identifier
0537.03024

JSTOR
links.jstor.org

Citation

Kaufmann, Matt. Blunt and Topless End Extensions of Models of Set Theory. J. Symbolic Logic 48 (1983), no. 4, 1053--1073. https://projecteuclid.org/euclid.jsl/1183741414


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