Classification of non‐well‐founded sets and an application

Mathematical Logic Quarterly 49 (2):187-200 (2003)
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Abstract

A complete list of Finsler, Scott and Boffa sets whose transitive closures contain 1, 2 and 3 elements is given. An algorithm for deciding the identity of hereditarily finite Scott sets is presented. Anti-well-founded sets, i. e., non-well-founded sets whose all maximal ∈-paths are circular, are studied. For example they form transitive inner models of ZFC minus foundation and empty set, and they include uncountably many hereditarily finite awf sets. A complete list of Finsler and Boffa awf sets with 2 and 3 elements in their transitive closure is given. Next the existence of infinite descending ∈-sequences in Aczel universes is shown. Finally a theorem of Ballard and Hrbáček concerning nonstandard Boffa universes of sets is considerably extended

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Athanassios Tzouvaras
Aristotle University of Thessaloniki (PhD)

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References found in this work

Standard foundations for nonstandard analysis.David Ballard & Karel Hrbacek - 1992 - Journal of Symbolic Logic 57 (2):741-748.
Hereditarily finite finsler sets.David Booth - 1990 - Journal of Symbolic Logic 55 (2):700-706.
Non‐circular, non‐well‐founded set universes.Athanassios Tzouvaras - 1993 - Mathematical Logic Quarterly 39 (1):454-460.

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