Journal of Symbolic Logic

Slim Models of Zermelo Set Theory

A. R. D. Mathias

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Working in Z + KP, we give a new proof that the class of hereditarily finite sets cannot be proved to be a set in Zermelo set theory, extend the method to establish other failures of replacement, and exhibit a formula $\Phi(\lambda, a)$ such that for any sequence $\langle A_{\lambda} \mid \lambda \text{a limit ordinal} \rangle$ where for each $\lambda, A_{\lambda} \subseteq ^{\lambda}2$, there is a supertransitive inner model of Zermelo containing all ordinals in which for every $\lambda A_{\lambda} = \{\alpha \mid\Phi(\lambda, a)\}$.

Article information

J. Symbolic Logic, Volume 66, Issue 2 (2001), 487-496.

First available in Project Euclid: 6 July 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 03C30: Other model constructions
Secondary: 03E30: Axiomatics of classical set theory and its fragments 03E45: Inner models, including constructibility, ordinal definability, and core models

Zermelo Set Theory Fruitful Class Zermelo Tower Supertransitive Model


Mathias, A. R. D. Slim Models of Zermelo Set Theory. J. Symbolic Logic 66 (2001), no. 2, 487--496.

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