Page "Axiom of choice" Paragraph 30

The reason that we are able to choose least elements from subsets of the natural numbers is the fact that the natural numbers are well-ordered: every nonempty subset of the natural numbers has a unique least element under the natural ordering.

One might say, " Even though the usual ordering of the real numbers does not work, it may be possible to find a different ordering of the real numbers which is a well-ordering.

Then our choice function can choose the least element of every set under our unusual ordering.

" The problem then becomes that of constructing a well-ordering, which turns out to require the axiom of choice for its existence ; every set can be well-ordered if and only if the axiom of choice holds.