Can someone please explain the 'Axiom of Choice' to me?

Question

turingtest · Answer

viodara is quoting wikipedia http://en.wikipedia.org/wiki/Axiom_of_choice which you can read for yourself if you want

kinggeorge · Answer

Basically, let's suppose you have some number of sets. The axiom of choice says that you can "choose" exactly one element from each of those sets, and with those elements you chose, create a new set.

kinggeorge · Answer

For example. Suppose you have the sets {1} and {2}. Using the axiom of choice, you can create the set {1, 2}.

mathsolver · Answer

Yeah, I have read the wiki article already. Thanks KingGeorge, so what are the consequences of not taking the axiom of choice?

turingtest · Answer

lots of things can't be proven lol

anonymous · Answer

The important consequences come when you are dealing with infinite sets. The wiki mentions some examples, like every vector space having a basis. AC basically allows you to generalize and abstract more than you would be able to without it.

kinggeorge · Answer

The axiom of choice allows us to create new sets. Without it, it would be much more difficult to create new sets starting with single-element sets.

anonymous · Answer

KG that's not entirely accurate. You can make selections from other sets without AC if the sets are finite. AC comes into play mostly with infinite sets.

kinggeorge · Answer

Not entirely, but it does allow less freedom when constructing sets with multiple elements.

anonymous · Answer

Yes, but if he's trying to understand the implications of AC, it's best not to oversell them. You can make selections from finite sets pretty freely without AC, and you can even make selections from infinite sets without it in some cases too. For example, if you have the set of subsets of the natural numbers, you can always make selections from them using the well-ordering principle without having to resort to the axiom of choice.

kinggeorge · Answer

Fair enough.

turingtest · Answer

nice discussion :)

mathsolver · Answer

brilliant discussion :D

mathsolver · Answer

Since we have so many people literate in mathematics in one place, could someone please explain the implications of AC on the Banach–Tarski paradox?

anonymous · Answer

Banach-Tarski's original proof relied on the axiom of choice. Since then the same result has been proven without it, but the new proofs still rely on weaker versions of the axiom of choice. It has been shown that it cannot be proven with ZF alone. (Sorry for the multiple edits)