Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-19T17:40:04.768Z Has data issue: false hasContentIssue false
  • English
  • Français

Typical ambiguity and the axiom of choice

Published online by Cambridge University Press:  12 March 2014

Marcel Crabbé
Marcel Crabbé*
Affiliation:
Université de Louvain, 1348 Louvain-la-Neuve, Belgique
Get access Rights & Permissions [Opens in a new window]

Extract

E. Specker has proved that the axiom of choice (AC) is false in NF [6]. Since AC is stratified, one can, according to another famous result of Specker [7], prove directly ¬AC in type theory (TT) plus some finite set of ambiguity axioms, i.e. sentences of the form φ ↔ φ+, where φ+ results from φ by adding one to its type indices.

We shall in §2 of this paper give a disproof of AC directly in TT plus some axioms of ambiguity. The argument will be split into two parts. The first one (contained in Proposition 2) concerns cardinal arithmetic and has nothing to do with typical ambiguity. Though carried out in TT, it could have been done in other set theories such as Zermelo's Z or ZF. The second part is an application of this to the cardinals of the universes at different types. This is made possible through the introduction of an appropriate definition of 2α in §1 enabling one to express shifting sentences as “typed properties” of the universe, in Boffa's sense. The disproof of AC is then completed in TT plus two extra ambiguity axioms. In §3, we show that this is in a sense the best possible result: that means that every single ambiguity axiom is consistent with TT plus AC, thus giving a positive solution to a conjecture of Specker [7, p. 119].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 49 , Issue 4 , December 1984 , pp. 1074 - 1078
Copyright
Copyright © Association for Symbolic Logic 1984

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]Boffa, M., Sets equipollent to their power set in NF this Journal, vol. 40 (1975), pp. 149–150.Google Scholar
[2]Crabbé, M., A propos de 2α, La théorie des ensembles de Quine, Cahiers du Centre de Logique, no. 4, Louvain-la-Neuve, 1982, pp. 17–22.Google Scholar
[3]Forster, Th., Quine's new foundations, Cahiers du Centre de Logique, no. 5, Louvain-la-Neuve, 1983.Google Scholar
[4]Pétry, A., On the typed properties in Quine's “new foundations”, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 25 (1979), pp. 99–102.CrossRefGoogle Scholar
[5]Shoenfield, J. R., Unramified forcing, Axiomatic set theory, Proceedings of Symposia in Pure Mathematics, vol. 13, part 1, American Mathematical Society, Providence, Rhode Island, 1971, pp. 357–381.CrossRefGoogle Scholar
[6]Specker, E., The axiom of choice in Quine's new foundations for mathematical logic, Proceedings of the National Academy of Science of the United States of America, vol. 39 (1953), pp. 972–975.CrossRefGoogle Scholar
[7]Specker, E., Typical ambiguity, Logic, Methodology, and Philosophy of Science: Proceedings of the 1960 International Congress, Stanford University Press, Stanford, California, 1962, pp. 116–124.Google Scholar