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Non-classical Models of ZF

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Abstract

This paper contributes to the generalization of lattice-valued models of set theory to non-classical contexts. First, we show that there are infinitely many complete bounded distributive lattices, which are neither Boolean nor Heyting algebra, but are able to validate the negation-free fragment of \(\mathsf {ZF}\). Then, we build lattice-valued models of full \(\mathsf {ZF}\), whose internal logic is weaker than intuitionistic logic. We conclude by using these models to give an independence proof of the Foundation axiom from \(\mathsf {ZF}\).

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Acknowledgements

We thank Rodolfo Ertola-Biraben for the useful suggestions and comments on the content of this paper. His invaluable help made this work possible. We also thank Sourav Tarafder for the valuable comments and discussions. We finally thank the many useful suggestions and corrections of an anonymous referee, who helped to improve and to correct many aspects of our work. We also thank Studia Logica for the fine evaluation and publication process; so rare in this days. The second author acknowledges support from the FAPESP Jovem Pesquisador Grant No. 2016/25891-3. The first author acknowledges support from the FAPESP Grant No. 2017/23853-0.

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Correspondence to S. Jockwich Martinez.

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Presented by Daniele Mundici

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Jockwich Martinez, S., Venturi, G. Non-classical Models of ZF. Stud Logica 109, 509–537 (2021). https://doi.org/10.1007/s11225-020-09915-0

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  • DOI: https://doi.org/10.1007/s11225-020-09915-0

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