Notre Dame Journal of Formal Logic

Antifoundation and Transitive Closure in the System of Zermelo

Olivier Esser and Roland Hinnion

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.

Article information

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

Dates
First available in Project Euclid: 3 December 2002

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1038949536

Digital Object Identifier
doi:10.1305/ndjfl/1038949536

Mathematical Reviews number (MathSciNet)
MR1816888

Zentralblatt MATH identifier
0967.03044

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

Citation

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. https://projecteuclid.org/euclid.ndjfl/1038949536


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References

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