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Hypersets. (English) Zbl 0756.03026
Several elegant solutions of problems in theoretical computer science, e.g. the constructions of natural models of the $$\lambda$$-calculus by M. von Rimscha [Arch. Math. Logik Grundlagenforsch. 20, 65-73 (1980; Zbl 0428.03045)], have brought to the attention of a larger audience the usefulness of alternatives to the axiom of foundation. A systematic study of such principles (generalizations of Mostowski’s collapse combined with various forms of extensionality) has been initiated by P. Hajek [Z. Math. Logik Grundlagen Math. 11, 103-115 (1965; Zbl 0171.264)] and M. Boffa [ibid. 14, 329-334 (1968; Zbl 0169.306)]. In the paper under review the authors give an introduction into one of these alternatives, the antifoundation axiom AFA in P. Aczel’s terminology [Non-well-founded sets (1988; Zbl 0668.04001)]. The purpose of this paper is to convince the reader that AFA “is an interesting, mathematically and philosophically respectable alternative” to the axiom of foundation.
Reviewer: N.Brunner (Wien)

##### MSC:
 03E65 Other set-theoretic hypotheses and axioms 03E75 Applications of set theory 03-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations
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