Complete topoi representing models of set theory

Annals of Pure and Applied Logic 57 (1):1-26 (1992)
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Abstract

By a model of set theory we mean a Boolean-valued model of Zermelo-Fraenkel set theory allowing atoms (ZFA), which contains a copy of the ordinary universe of (two-valued,pure) sets as a transitive subclass; examples include Scott-Solovay Boolean-valued models and their symmetric submodels, as well as Fraenkel-Mostowski permutation models. Any such model M can be regarded as a topos. A logical subtopos E of M is said to represent M if it is complete and its cumulative hierarchy, as defined by Fourman and Hayashi, coincides with the usual cumulative hierarchy of M. We show that, although M need not be a complete topos, it has a smallest complete representing subtopos, and we describe this subtopos in terms of definability in M. We characterize, again in terms of definability, those models M whose smallest representing topos is a Grothendieck topos. Finally, we discuss the extent to which a model can be reconstructed when its smallest representing topos is given.

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Citations of this work

Category Theory.Jean-Pierre Marquis - 2008 - Stanford Encyclopedia of Philosophy.
Forcing Under Anti‐Foundation Axiom: An Expression of the Stalks.Sato Kentaro - 2006 - Mathematical Logic Quarterly 52 (3):295-314.

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References found in this work

The Axiom of Choice.Thomas J. Jech - 1973 - Amsterdam, Netherlands: North-Holland.
Topos Theory.P. T. Johnstone - 1982 - Journal of Symbolic Logic 47 (2):448-450.

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