Slim models of zermelo set theory

Journal of Symbolic Logic 66 (2):487-496 (2001)
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 Φ(λ, 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 λ A λ = {α ∣Φ(λ, a)}
Keywords Zermelo Set Theory   Fruitful Class   Zermelo Tower   Supertransitive Model
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DOI 10.2307/2695026
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A. R. D. Mathias (2001). The Strength of Mac Lane Set Theory. Annals of Pure and Applied Logic 110 (1-3):107-234.
Penelope Maddy (2005). Mathematical Existence. Bulletin of Symbolic Logic 11 (3):351-376.
Akihiro Kanamori (2004). Zermelo and Set Theory. Bulletin of Symbolic Logic 10 (4):487-553.

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