Notre Dame Journal of Formal Logic

Antifoundation and Transitive Closure in the System of Zermelo

Olivier Esser and Roland Hinnion


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.

Article information

Notre Dame J. Formal Logic, Volume 40, Number 2 (1999), 197-205.

First available in Project Euclid: 3 December 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03E35: Consistency and independence results
Secondary: 03E65: Other hypotheses and axioms 03E70: Nonclassical and second-order set theories


Esser, Olivier; Hinnion, Roland. Antifoundation and Transitive Closure in the System of Zermelo. Notre Dame J. Formal Logic 40 (1999), no. 2, 197--205. doi:10.1305/ndjfl/1038949536.

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