Antifoundation and Transitive Closure in the System of Zermelo

Notre Dame Journal of Formal Logic 40 (2):197-205 (1999)
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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



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Bemerkungen, das Fundierungsaxiom Betreffend.Kurt Hauschild - 1966 - Mathematical Logic Quarterly 12 (1):51-56.
Universality and strong extensionality.Michael Rimscha - 1981 - Archive for Mathematical Logic 21 (1):179-193.

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