Choices from finite sets and choices of finite subsets
HTML articles powered by AMS MathViewer
- by Martin M. Zuckerman PDF
- Proc. Amer. Math. Soc. 27 (1971), 133-138 Request permission
Abstract:
In set theory without the axiom of choice we prove a consistency result involving certain “finite versions” of the axiom of choice. Assume that it is possible to select a nonempty finite subset from each nonempty set. We determine sets $Z$, of integers, which have the property that $n \in Z$ is a necessary and sufficient condition for the possibility of choosing an element from every $n$-element set. Given any nonempty set $P$ of primes, the set ${Z_p}$, consisting of integers which are not “linear combinations” of primes of $P$, is such a set $Z$.References
- K. Gëdel′, The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory, Uspehi Matem. Nauk (N.S.) 3 (1948), no. 1(23), 96–149 (Russian). MR 0024870
- A. Lévy, Axioms of multiple choice, Fund. Math. 50 (1961/62), 475–483. MR 139528, DOI 10.4064/fm-50-5-475-483
- Elliott Mendelson, The independence of a weak axiom of choice, J. Symbolic Logic 21 (1956), 350–366 (1957). MR 84462, DOI 10.2307/2268356 A. Mostowski, Über die Unabhängigkeit des Wohlordnungssätzes vom Ordnungsprinzip, Fund. Math. 32 (1939), 201-252.
- Andrzej Mostowski, Axiom of choice for finite sets, Fund. Math. 33 (1945), 137–168. MR 16352, DOI 10.4064/fm-33-1-137-168
- Martin M. Zuckerman, Multiple choice axioms, Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971, pp. 447–466. MR 0280360
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 27 (1971), 133-138
- MSC: Primary 02.60
- DOI: https://doi.org/10.1090/S0002-9939-1971-0270905-5
- MathSciNet review: 0270905