It is nonconstructive - it conjures up a set without providing any sort of procedure for its construction. For The acceptance of AC is based on the belief that our intuition about finite sets can be extended to infinite sets. The main argument for accepting it is that it is useful. Many important, intuitively plausible theorems are equivalent to it or depend on it. Every nonempty partially ordered set P in which each chain has an upper bound in P has a maximal element.
Every infinite set has a denumerable subset.
Every Boolean algebra has an ultrafilter on it. If set theory is done in such a logical formal system the Axiom of Choice will be a theorem. Accept a contradictory axiom such as the Axiom of Determinacy. Use a completely different framework for mathematics such as Category Theory. Cambridge University Press, 1983.