Journal of Symbolic Logic

Blunt and Topless End Extensions of Models of Set Theory

Matt Kaufmann

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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 (The author has been informed that Silver had earlier, independently, found a similar unpublished proof of that theorem.)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.

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

First available in Project Euclid: 6 July 2007

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Kaufmann, Matt. Blunt and Topless End Extensions of Models of Set Theory. J. Symbolic Logic 48 (1983), no. 4, 1053--1073.

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