Why is the axiom of choice controversial?

Question

In other words, what are the arguments for ZF over ZFC, and what philosophical issues have people raised against including it as a standard axiom of set theory?

Answer 1

This axiom is controversial because although it seems like a relatively intuitive idea, there are still some issues. One of the best ways of understanding it is in this way: Take the set of all pairs of shoes and find a way to pick one shoe from each pair in order to form a new set. Easy! Just let our choice function be take every right shoe from each pair and you get the set of all right shoes. Consider this now: Take the set of all pairs of socks. The axiom of choice tells us there is a way to pick one sock from each pair to form a new set, but how you make that choice is not easy to describe. So although the axiom says a choice function always exists, what that choice function can be or how it is defined is not always apparent or even impossible to define in many cases.

Some more things: The axiom of choice is indeed an extremely useful axiom in many areas of math. However, it gives rise to the Banach-Tarski Paradox and the existence of nonmeasurable subsets of the real numbers. Also, the axiom of choice is equivalent to the statement that any set can be well-ordered, i.e., every nonempty set can be endowed with a total order such that every nonempty subset has a least element. Therefore, this means that $\mathbb{R}$ can be well-ordered. However, no one has ever been able to explicitly state how one does this, and if I'm not mistaken, it may be impossible to do so with the current axioms we have available. I also want to add that one person in the comments section mentioned, we need AC in order to talk about cardinalities of infinite sets.