## Notre Dame Journal of Formal Logic

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

### Models as Universes

#### Abstract

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

**Source**

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

**Dates**

Received: 6 June 2011

Accepted: 9 March 2014

First available in Project Euclid: 14 December 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ndjfl/1481684566

**Digital Object Identifier**

doi:10.1215/00294527-3716058

**Mathematical Reviews number (MathSciNet)**

MR3595341

**Zentralblatt MATH identifier**

06686417

**Subjects**

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}

**Keywords**

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

#### Citation

Halimi, Brice. Models as Universes. Notre Dame J. Formal Logic 58 (2017), no. 1, 47--78. doi:10.1215/00294527-3716058. https://projecteuclid.org/euclid.ndjfl/1481684566