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The Independence of a Strong Axiom of Choice

Published online by Cambridge University Press:  03 November 2016

Shaligram Singh
Shaligram Singh*
Affiliation:
Mathematics Department, University of Bihar, India
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Extract

There is a growing realization among mathematicians and logicians of the many-sided role played by the axiom of choice in various branches of mathematics. Many of them tend to accept the axiom of choice as a legitimate principle provided, of course, it is proved to be independent in a suitable axiom system. This tendency has been accelerated by Gödel’s proof of the compatibility of this axiom in a reasonably broad system of axioms [2]. Such a view seems to have been shared by Fraenkel and Bar-Hillel [1; pp. 44-80] in their excellent exposition of the function of the axiom of choice in the modern mathematics in general and the axiomatic set theory in particular.

Type
Research Article
Information
The Mathematical Gazette , Volume 46 , Issue 356 , May 1962 , pp. 126 - 129
Copyright
Copyright © Mathematical Association 1962

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References

1. Fraenkel, Abraham A. and Bar-Hillel, Yehoshua, “Foundations of Set Theory”, Studies in Logic and Foundations of Mathematics, Amsterdam (North-Holland Publ. Co.), 1958 Google Scholar
2. Gödel, Kurt, “The Consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory”, Annals of Mathematics Studies, No. 3, Princeton (Princeton University Press), 1940.Google Scholar
3. Mendelson, Elliott, “ The Independence of a Weak Axiom of Choice”, The Journal of Symbolic Logic, Vol. 21 (1956), pp. 350–66.CrossRefGoogle Scholar
4. Robinson, A, “ On the Metamathematics of Algebra”, Studies in Logic and the Foundations of Mathematics, Amsterdam (North-Holland Publ. Co.), 1951.Google Scholar
5. Shoenfield, J. R., “ The Independence of the Axiom of Choice”, The Journal of Symbolic Logic, Vol. 20 (1955), p. 200 (Abstract).Google Scholar
6. Shukla, R., “ A System for General Set Theory”, The Journal of Indian Mathematical Society, n.s., Vol. 12 (1948), pp. 121–4.Google Scholar