Barwise, Jon; Moss, Lawrence Vicious circles. On the mathematics of non-wellfounded phenomena. (English) Zbl 0865.03002 CSLI Lecture Notes. 60. Stanford, CA: CSLI, Center for the Study of Language and Information. x, 390 p. £16.95/pbk; $ 24.95/pbk; £50.00/ hbk; $ 64.95/hbk (1996). This is a book on theory and applications of non-wellfounded sets. Starting from explanations where circularity phenomena are important, e.g. in computer science or philosophy, first the basic theory of hypersets (comprising the wellfounded and the non-wellfounded sets) is given. Then a first chapter on elementary applications (to graphs, games, semantic paradoxes, streams etc.) follows. On this background, an extension of the theory is given discussing greatest fixed points, uniform operators, and corecursion. And these topics are applied in a last chapter to further applications. The book is well written and suitable for the classroom as well as for individual study. Each chapter ends with a section of exercises and some historical remarks. Reviewer: S.Gottwald (Leipzig) Cited in 5 ReviewsCited in 70 Documents MSC: 03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations 00A30 Philosophy of mathematics 03-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations 68R99 Discrete mathematics in relation to computer science 03E70 Nonclassical and second-order set theories 03A05 Philosophical and critical aspects of logic and foundations Keywords:antifoundation axiom; self-reference; nonstandard set theory; non-wellfounded sets; circularity; hypersets; graphs; games; semantic paradoxes; streams; greatest fixed points; uniform operators; corecursion PDFBibTeX XMLCite \textit{J. Barwise} and \textit{L. Moss}, Vicious circles. On the mathematics of non-wellfounded phenomena. Stanford, CA: CSLI, Center for the Study of Language and Information (1996; Zbl 0865.03002)