Abstract
The role of foundation with respect to transitive closure in the Zermelo system Z has been investigated by Boffa; our aim is to explore the role of antifoundation. We start by showing the consistency of "Z $+$ antifoundation $+$ transitive closure" relative to Z (by a technique well known for ZF). Further, we introduce a "weak replacement principle" (deductible from antifoundation and transitive closure) and study the relations among these three statements in Z via interpretations. Finally, we give some adaptations for ZF without infinity.
Citation
Olivier Esser. Roland Hinnion. "Antifoundation and Transitive Closure in the System of Zermelo." Notre Dame J. Formal Logic 40 (2) 197 - 205, Spring 1999. https://doi.org/10.1305/ndjfl/1038949536
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