The Independence of a Strong Axiom of Choice

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.