Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-18T18:17:40.776Z Has data issue: false hasContentIssue false
  • English
  • Français

Positive abstraction and extensionality

Published online by Cambridge University Press:  12 March 2014

Roland Hinnion  and
Thierry Libert
Roland Hinnion
Affiliation:
Service de Logique Mathématique, Université Libre de Bruxelles, CP211, Boulevard du Triomphe, 1050 Brussels, Belgium, E-mail: rhinnion@nlb.ac.be
Thierry Libert
Affiliation:
Service de Logique Mathématique, Université Libre de Bruxelles, CP211, Boulevard du Triomphe, 1050 Brussels, Belgium, E-mail: tlibert@ulb.ac.be
Get access Rights & Permissions [Opens in a new window]

Abstract

It is proved in this paper that the positive abstraction scheme is consistent with extensionality only if one drops equality out of the language. The theory obtained is then compared with GPK, a well-known set theory based on an extended positive comprehension scheme.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 3 , September 2003 , pp. 828 - 836
Copyright
Copyright © Association for Symbolic Logic 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Brady, R. T., The consistency of the axioms of abstraction and extensionality in a three-valued logic, Notre Dame Journal of Formal Logic, vol. 12 (1971), pp. 447–453.Google Scholar
[2] Esser, O., Interprétations mutuelles entre une théorie positive des ensembles et une extension de la théorie de Kelley-Morse, Ph.D. thesis , Université Libre de Bruxelles, 1997, Unpublished. Available at http://homepages.ulb.ac.be/~oesser.Google Scholar
[3] Esser, O., Inconsistency of the axiom of choice with the positive theory GPK∞ + , this Journal, vol. 65 (2000), pp. 1911–1916.Google Scholar
[4] Forti, M. and Hinnion, R., The consistency problem for positive comprehension principles, this Journal, vol. 54 (1989), pp. 1401–1418.Google Scholar
[5] Gilmore, R C., The consistency of partial set theory without extensionality, Proceedings of Symposia in Pure Mathematics, vol. 13, 1974, pp. 147–153.Google Scholar
[6] Hinnion, R., Le paradoxe de Russell dans des versions positives de la théorie naïve des ensembles, Comptes Rendus de l‘ Académie des Sciences, vol. 12 (1987), pp. 307–310.Google Scholar
[7] Hinnion, R., Naive set theory with extensionality in partial logic and in paradoxical logic, Notre Dame Journal of Formai Logic, vol. 35 (1994), pp. 15–40.Google Scholar
[8] Skolem, T., Investigations on a comprehension axiom without negation in the defining propositional functions, Notre Dame Journal of Formal Logic, vol. 1 (1960), pp. 13–22.Google Scholar
[9] Skolem, T., Studies on the axiom of comprehension, Notre Dame Journal of Formal Logic, vol. 4 (1963), pp. 162–170.Google Scholar