A Substitute for the Axiom of Choice
نویسنده
چکیده
This theorem is well known to be equivalent to the axiom of choice (though there does not seem to be a proof of this fact in the literature) and it has been suggested as an alternative for this axiom. The purpose of this note (which is purely methodological) is to propose a simpler but equivalent formulation of (A) as a substitute for the Zermelo axiom. The simplicity lies in the fact that we make no assumptions concerning the relation R which replaces partial order. Let Q be a set and R an arbitrary binary relation on Q. A subset of Q will be termed R-simple if for any pair of its elements, a and b, we have either aRb or bRa. The version of (A) we propose is:
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