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Inconsistency of the Axiom of Choice with the positive theory

Published online by Cambridge University Press:  12 March 2014

Olivier Esser
Olivier Esser*
Affiliation:
Université Libre de Bruxelles, Service de Logique C.P.211, Boulevard du Triomphe, B1050 Bruxelles, Belgium, E-mail:[email protected]
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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 .

Keywords

GPK theoryPositive TheoryAxiom of Choice
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 4 , December 2000 , pp. 1911 - 1916
Copyright
Copyright © Association for Symbolic Logic 2000

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References

REFERENCES

[1]Esser, O., Inconsistency of GPK + AFA, Mathematical Logic Quarterly, vol. 42 (1996), pp. 104–108.CrossRefGoogle Scholar
[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. 369–377.CrossRefGoogle Scholar
[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des ensembles et une extension de la theorie de Kelley-Morse., Ph.D. thesis, UniversitÉ Libre de Bruxelles, 1997, unpublished, available at http://homepages.ulb.ac.be/~oesser.Google Scholar
[4]Esser, O., On the consistency of a positive theory, MLQ. Mathematical Logic Quarterly, vol. 45 (1999), pp. 105–116.CrossRefGoogle Scholar
[5]Forti, M. and Hinnion, R., The consistency problem for positive comprehension principles, this Journal, vol. 54 (1989), pp. 1401–1418.Google Scholar
[6]Forti, M. and Honsell, R., Models of self-descriptive set theories, Partial differential equations and the calculus of variations, essays in honor of Ennio De Giorgi vol. i, BirkhÄuser Boston, Boston, MA, 1989, pp. 473–518.Google Scholar
[7]Forti, M. and Honsell, R., Weak foundation and antifoundation properties of positively comprehensive hyperuniverses, L'anti-fondation en logique et en theorie des ensembles (Hinnion, R., editor), Cahiers du Centre de Logique, UnivercitÉ Catholique Louvain, Louvain-la-Neuve, 1992, pp. 31–43.Google Scholar
[8]Forti, M. and Honsell, R., Choice principles in hyperuniverses, Annals of Pure and Applied Logic, vol. 77 (1996), pp. 35–52.CrossRefGoogle Scholar
[9]Forti, M. and Honsell, R., A general construction of hyperuniverses, Theoretical Computer Science, vol. 156 (1996), pp. 203–215.CrossRefGoogle Scholar
[10]Forti, M. and Honsell, R., Addendum and corrigendum to “Choice principles in hyperuniverses” [Annals of Pure and Applied Logic 11 (1996), 35–52], Annals of Pure and Applied Logic, vol. 92 (1998), pp. 211–214.CrossRefGoogle Scholar
[11]Malitz, R.J., Set theory in which the axiom of foundation fails., Ph.D. thesis, UCLA, Los Angeles, 1976, unpublished.Google Scholar
[12]Weydert, E., How to approximate the naive comprehension scheme inside of classical logic, Ph.D. thesis, Bonner Mathematische Schriften 194, Bonn, 1989.Google Scholar