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MINIMUM MODELS OF SECOND-ORDER SET THEORIES

Published online by Cambridge University Press:  08 April 2019

KAMERYN J. WILLIAMS
KAMERYN J. WILLIAMS*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF HAWAI‘I AT MĀNOA 2565 MCCARTHY MALL, KELLER 401A HONOLULU, HI 96822, USAE-mail: [email protected]: http://kamerynjw.net
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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.

Keywords

Primary 03E70Secondary 03C62Kelley–MorseGödel–Bernayselementary transfinite recursionminimum modelsecond-order set theory
Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 2 , June 2019 , pp. 589 - 620
Copyright
Copyright © The Association for Symbolic Logic 2019 

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