Models as universes. (English) Zbl 1417.03236

Halimi, Brice

Notre Dame J. Formal Logic 58, No. 1, 47-78 (2017).

Summary: 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.

MSC:

03C62	Models of arithmetic and set theory
03A05	Philosophical and critical aspects of logic and foundations
03C55	Set-theoretic model theory
03C70	Logic on admissible sets
03B45	Modal logic (including the logic of norms)

Keywords:

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