On certain elementary extensions of models of set theory

Enayat, Ali

doi:10.1090/S0002-9947-1984-0737894-1

On certain elementary extensions of models of set theory
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by Ali Enayat

Trans. Amer. Math. Soc. 283 (1984), 705-715

DOI: https://doi.org/10.1090/S0002-9947-1984-0737894-1

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Abstract:

In $\S 1$ we study two canonical methods of producing models of $\operatorname {ZFC}$ with no elementary end extensions. $\S 2$ 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.

References

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