About the coexistence of “classical sets” with “non-classical” ones: A survey
DOI:
https://doi.org/10.12775/LLP.2003.004Keywords
set theory, consistency, paraconsistency, extensionality, choiceAbstract
This is a survey of some possible extensions of ZF to a larger universe, closer to the “naive set theory” (the universes discussed here concern, roughly speaking : stratified sets, partial sets, positive sets, paradoxical sets and double sets).References
Batens, D., and K. De Clercq, ‘A rich paraconsistent extension of full positive logic’, Logique et Analyse (2000).
Chang, C.C., and H. J. Keisler, ‘Model Theory’, North-Holland, 1973.
Crabb´e, M., ‘Soyons positifs: la complétude de la théorie naïve des ensembles’, Cahiers du Centre de Logique 7 (Université Catholique de Louvain) (1992), 51–68.
Esser, O., ‘Interprétations mutuelles entre une théorie positive des ensembles et une extension de la th´eorie de Kelley-Morse’, Thèse de Doctorat en Sciences, Université Libre de Bruxelles, 1997.
Esser, O., ‘An interpretation of the Zermelo-Fraenkel set theory in a positive theory’, Mathematical Logic Quarterly 43 (1997), 369–377.
Esser, O., ‘Inconsistency of the axiom of choice with the positive theory GPK+ ’, to appear in The Journal of Symbolic Logic.
Forster, T. E., ‘Quine’s New Foundations’, Cahiers du Centre de Logique 5 (Universit´e Catholique de Louvain) (1983).
Forti, M., and R. Hinnion, ‘The consistency problem for positive comprehension principles’, The Journal of Symbolic Logic 54 (1989), 1401–1418.
Forti, M., and F. Honsell, ‘Choice Principles in Hyperuniverses’, Annals of Pure and Applied Logic 77 (1996), 35–53; this should be completed with (and corrected by) : ‘Addendum and Corrigendum’, to appear in the same journal.
Gilmore, P.C., ‘The consistency of partial set theory without extensionality’, Axiomatic Set Theory, Proceedings of Symposia in Pure Mathematics, American Mathematical Society, 13 (1974), 147–153.
Hinnion, R., ‘Sur la théorie des ensembles de Quine’, Thèse de Doctorat en Sciences, Université Libre de Bruxelles, 1975.
Hinnion, R., ‘Le paradoxe de Russell dans des versions positives de la théorie naïve des ensembles’, Comptes Rendus de l’Académie des Sciences de Paris, Série I, 304 (1987), 307–310.
Hinnion, R., ‘Stratified and positive comprehension seen as superclass rules over ordinary set theory’, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 36 (1990), 519–534.
Hinnion, R., ‘Naive set theory with extensionality in partial logic and in paradoxical logic’, Notre Dame Journal of Formal Logic 35 (1994), 15–40.
Holmes, R., ‘Elementary set theory with a universal set’, Cahiers du Centre de Logique 10 (Université Catholique de Louvain) (1998).
Jensen, R. B., ‘On the consistency of a slight (?) modification of Quine’s New Foundations’, Synthese 19 (1969), 250–263.
Kisielewicz, A., ‘Double extension set theory’, Reports on Mathematical Logic 23 (1989), 81–89.
Kisielewicz, A., ‘A very strong set theory?’, Studia Logica 61 (1998), 171–178.
Malitz, R. J., ‘Set theory in which the axiom of foundation fails’, Ph.D. Thesis, UCLA, Los Angeles (1976), umpublished.
Quine, W.V.O., ‘New Foundations for Mathematical Logic’, The American Mathematical Monthly 44 (1937), 70–80.
Specker, E., ‘The Axiom of Choice in Quine’s New Foundations for Mathematical Logic’, Proceedings of the National Academy of Sciences, USA, 39 (1953), 972–975.
Weydert, E., ‘How to approximate the naive comprehension scheme inside of classical logic’, Bonner Mathematische Schriften 194 (1989), 1–40.
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