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Inconsistency of the Axiom of Choice with the positive theory

  • Olivier Esser (a1)
Abstract

The idea of the positive theory is to avoid the Russell's paradox by postulating an axiom scheme of comprehension for formulas without “too much” negations. In this paper, we show that the axiom of choice is inconsistent with the positive theory .

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[1]Esser, O., Inconsistency of GPK + AFA, Mathematical Logic Quarterly, vol. 42 (1996), pp. 104108.
[2]Esser, O., An interpretation of the Zermelo-Fraenkel set theory and the Kelley-Morse set theory in a positive theory, Mathematical Logic Quarterly, vol. 43 (1997), pp. 369377.
[3]Esser, O., InterprÉtations mutuelles entre une theorie positive des ensembles et une extension de la theorie de Kelley-Morse., Ph.D. thesis, UniversitÉ Libre de Bruxelles, 1997, unpublished, available at http://homepages.ulb.ac.be/~oesser.
[4]Esser, O., On the consistency of a positive theory, MLQ. Mathematical Logic Quarterly, vol. 45 (1999), pp. 105116.
[5]Forti, M. and Hinnion, R., The consistency problem for positive comprehension principles, this Journal, vol. 54 (1989), pp. 14011418.
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[8]Forti, M. and Honsell, R., Choice principles in hyperuniverses, Annals of Pure and Applied Logic, vol. 77 (1996), pp. 3552.
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[11]Malitz, R.J., Set theory in which the axiom of foundation fails., Ph.D. thesis, UCLA, Los Angeles, 1976, unpublished.
[12]Weydert, E., How to approximate the naive comprehension scheme inside of classical logic, Ph.D. thesis, Bonner Mathematische Schriften 194, Bonn, 1989.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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