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Two forms of the axiom of choice for an elementary topos

Published online by Cambridge University Press:  12 March 2014

Anna Michaelides Penk
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Extract

The concept of a topos as a system, or world in which mathematics could be defined and interpreted, was developed by F. W. Lawvere and M. Tierney. Much of their work is embodied in the lecture notes Elementary toposes by A. Kock and G. C. Wraith [6].

In an early paper Lawvere set forth a set of axioms for approximately such a system [8]. The topos constructed there is a set-like category that includes among its axioms an axiom of infinity and an axiom of choice.

In its final form an elementary topos is freed from any such axioms.

The most prominent example of an elementary topos is a set theory with the usual Zermelo-Fraenkel or Godel-Bernays set of axioms.

In this paper I have tried to determine what, if any, is the effect of an axiom of choice introduced in a topos, and how are some of the set-theoretic equivalents of such an axiom related in topos theory.

Since the set-theoretic membership relation ∈, the notions of an “empty” set and a “power” set are definable in topos theory, it makes sense to talk about a “choice map” that picks a single element out of every nonempty object, provided that these objects can be somehow collected into a single object or “family.” In other words, an analogue of the usual choice axiom can be formulated in elementary topos language; this is axiom AC2 of the text.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 40 , Issue 2 , June 1975 , pp. 197 - 212
Copyright
Copyright © Association for Symbolic Logic 1975

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References

BIBLIOGRAPHY

[1]Barr, M., Catégories exactes, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, vol. 272 (1971), pp. 1501–1503.Google Scholar
[2]Freyd, Peter, Abelian categories, Harper and Row, New York, 1964.Google Scholar
[3]Freyd, Peter, Aspects of topoi, Bulletin of the Australian Mathematical Society, vol. 7 (1972), pp. 1–76.CrossRefGoogle Scholar
[4]Kock, Anders, Introduction to functorial semantics, Bertrand Russell Memorial Logic Conference, Uldum, Denmark, 1971, mimeographed notes.Google Scholar
[5]Kock, Anders and Mikkelsen, C. J., Topos theoretic factorization of nonstandard extensions, Aarhus Universität, 1972.Google Scholar
[6]Kock, Anders and Wraith, G. C., Elementary toposes, Aarhus Universität, Lecture Notes Series, no. 30, 1971.Google Scholar
[7]Lawvere, F. W., Adjointness in foundations, Dialectica, vol. 23 (1969), pp. 281–286.CrossRefGoogle Scholar
[8]Lawvere, F. W., An elementary theory of the category of sets, Proceedings of the National Academy of Sciences, vol. 52 (1964), pp. 1506–1511.CrossRefGoogle ScholarPubMed
[9]Lawvere, F. W., Functorial semantics of algebraic theories, Dissertation, Columbia University, 1963.CrossRefGoogle Scholar
[10]Mikkelsen, C. J., Colimits in toposes (in preparation).Google Scholar
[11]Mitchell, Barry, Theory of categories, Academic Press, New York, 1965.Google Scholar