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Classification of non-well-founded sets and an application

✍ Scribed by Nitta Takashi; Okada Tomoko; Athanassios Tzouvaras

Publisher
John Wiley and Sons
Year
2003
Tongue
English
Weight
233 KB
Volume
49
Category
Article
ISSN
0044-3050

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✦ Synopsis

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 (awf) 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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