One of the standard statements of the axiom of choice is the following:
Let S be a given nonvoid `universal' set. To each i in a nonvoid set I associate a nonvoid subset X_i of S. (Other wordings are `` let (X_i:i\in I) be a family of subsets S)". Then there is a function f that maps each i to some element x_i in X_i (in other words the cartesian product \prod X_i is non void).
In my opinion this statement does not make sense, since in the hypothesis one already uses a special case of the conclusion. The statement "one associates X_i" is nothing but the assumption that there is a map iota from I into P(S), the power set of S. In other words, iota is an element of P(S)^I, the cartesian product of I copies of P(S). Thus a special case of the axiom of choice serves as the hypotheses for what is claimed....
How to go around in this version of the axiom of choice?