Page "Axiom of choice" Paragraph 28

The difficulty appears when there is no natural choice of elements from each set.

If we cannot make explicit choices, how do we know that our set exists?

For example, suppose that X is the set of all non-empty subsets of the real numbers.

First we might try to proceed as if X were finite.

If we try to choose an element from each set, then, because X is infinite, our choice procedure will never come to an end, and consequently, we will never be able to produce a choice function for all of X.

Next we might try specifying the least element from each set.

But some subsets of the real numbers do not have least elements.

For example, the open interval ( 0, 1 ) does not have a least element: if x is in ( 0, 1 ), then so is x / 2, and x / 2 is always strictly smaller than x.

So this attempt also fails.