My question is essentially this: Why does the principle of transfinite induction not suffice to show the axiom of choice when the sets to be chosen from are indexed by a well ordered set?
I have read that one can prove the axiom of finite choice from simple induction. You induct on the size of the system of sets you are choosing from and pick an element from each set. I understand this. However, my grasp of the details is sketchy.
1) Why does standard induction alone not suffice to show the axiom of choice for systems of countable sets? Doesn't induction show the truth of the statement for all natural numbers, and therefore for any system of sets that can be indexed by the natural numbers (countable sets)? I know this to be false, but I do not know why.
2) Why can't the above "proof" that induction implies the AoC for countable sets not be repaired by using transfinite induction? Isn't this the purpose of transfinite induction, to allow one to induct on sets of infinite size? Shouldn't transfinite induction suffice to prove the axiom of choice for any system of sets indexed by a well-ordered set?
I am reading Jech right now, but my knowledge of ordinals and transfinite induction is very, very poor, so I would greatly prefer answers with a great amount of explanation and hand-holding.