Minimum models of second-order set theories

Journal of Symbolic Logic 84 (2):589-620 (2019)
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Abstract

In this article I investigate the phenomenon of minimum and minimal models of second-order set theories, focusing on Kelley–Morse set theory KM, Gödel–Bernays set theory GB, and GB augmented with the principle of Elementary Transfinite Recursion. The main results are the following. (1) A countable model of ZFC has a minimum GBC-realization if and only if it admits a parametrically definable global well order. (2) Countable models of GBC admit minimal extensions with the same sets. (3) There is no minimum transitive model of KM. (4) There is a minimum β-model of GB+ETR. The main question left unanswered by this article is whether there is a minimum transitive model of GB+ETR.

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10.1017/jsl.2019.27

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Citations of this work

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References found in this work

Classes and truths in set theory.Kentaro Fujimoto - 2012 - Annals of Pure and Applied Logic 163 (11):1484-1523.
Models of set theory with definable ordinals.Ali Enayat - 2005 - Archive for Mathematical Logic 44 (3):363-385.
Powers of regular cardinals.William B. Easton - 1970 - Annals of Mathematical Logic 1 (2):139.

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