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2017 Models as Universes
Brice Halimi
Notre Dame J. Formal Logic 58(1): 47-78 (2017). DOI: 10.1215/00294527-3716058


Kreisel’s set-theoretic problem is the problem as to whether any logical consequence of ZFC is ensured to be true. Kreisel and Boolos both proposed an answer, taking truth to mean truth in the background set-theoretic universe. This article advocates another answer, which lies at the level of models of set theory, so that truth remains the usual semantic notion. The article is divided into three parts. It first analyzes Kreisel’s set-theoretic problem and proposes one way in which any model of set theory can be compared to a background universe and shown to contain internal models. It then defines logical consequence with respect to a model of ZFC, solves the model-scaled version of Kreisel’s set-theoretic problem, and presents various further results bearing on internal models. Finally, internal models are presented as accessible worlds, leading to an internal modal logic in which internal reflection corresponds to modal reflexivity, and resplendency corresponds to modal axiom 4.


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Brice Halimi. "Models as Universes." Notre Dame J. Formal Logic 58 (1) 47 - 78, 2017.


Received: 6 June 2011; Accepted: 9 March 2014; Published: 2017
First available in Project Euclid: 14 December 2016

zbMATH: 06686417
MathSciNet: MR3595341
Digital Object Identifier: 10.1215/00294527-3716058

Primary: 03A05, 03C62
Secondary: 03B45, 03C55, 03C70

Rights: Copyright © 2017 University of Notre Dame


Vol.58 • No. 1 • 2017
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