Notre Dame Journal of Formal Logic

Models as Universes

Brice Halimi

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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.

Article information

Notre Dame J. Formal Logic, Volume 58, Number 1 (2017), 47-78.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03C62: Models of arithmetic and set theory [See also 03Hxx] 03A05: Philosophical and critical {For philosophy of mathematics, see also 00A30}
Secondary: 03C55: Set-theoretic model theory 03C70: Logic on admissible sets 03B45: Modal logic (including the logic of norms) {For knowledge and belief, see 03B42; for temporal logic, see 03B44; for provability logic, see also 03F45}

logical validity truth informal rigour Kreisel Boolos logical consequence of ZFC models of set theory modal logic recursively saturated structures


Halimi, Brice. Models as Universes. Notre Dame J. Formal Logic 58 (2017), no. 1, 47--78. doi:10.1215/00294527-3716058.

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