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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

On certain elementary extensions of models of set theory


Author: Ali Enayat
Journal: Trans. Amer. Math. Soc. 283 (1984), 705-715
MSC: Primary 03C62
MathSciNet review: 737894
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Abstract: In $ \S1$ we study two canonical methods of producing models of $ \operatorname{ZFC} $ with no elementary end extensions. $ \S2$ is devoted to certain "completeness" theorems dealing with elementary extensions, e.g., using $ {\diamondsuit _{{\omega _1}}}$ we show that for a consistent $ T \supseteq \operatorname{ZFC} $ the property "Every model $ \mathfrak{A}$ of $ T$ has an elementary extension fixing $ {\omega ^\mathfrak{A}}$" is equivalent to $ T\vdash$ "There exists an uncountable measurable cardinal". We also give characterizations of $ T\vdash$ "$ \kappa $ is weakly compact" and $ T\vdash$ "$ \kappa $ is measurable" in terms of elementary extensions.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1984-0737894-1
PII: S 0002-9947(1984)0737894-1
Article copyright: © Copyright 1984 American Mathematical Society