Is the Axiom of Choice a Logical or Set-Theoretical Principle?

Abstract

A generalization of the axioms of choice says that all the Skolem functions of a true first-order sentence exist. This generalization can be implemented on the first-order level by generalizing the rule of existential instantiation into a rule of functional instantiation. If this generalization is carried out in first-order axiomatic set theory (FAST), it is seen that in any model of FAST, there are sentences S which are true but whose Skolem functions do not exist. Since this existence is what the truth of S means in a combinational (model-theoretical) sense, in any model of FAST there are sentences which are set-theoretical “true” but false in the normal sense of the word. This shows that the assumptions on which the axiom of choice rests cannot be fully implemented in FAST. The axiom choice is not a set-theoretical principle.