of choice is simple to state; it says that if we have a collection of nonempty
sets, we can choose a member of each set. In fact, when I first saw this
axiom, my initial reaction was “Why do we need this axiom? Choosing
things from sets is like shopping with an inexhaustible budget. Just go
into a store [set], and say, ‘I want this.’ ” Nonetheless, the axiom of choice
is highly controversial—insofar as an axiom could ever be considered
controversial.
The controversy centers around the word choose. Just as there are activist
judges and strict constructionists, there are liberal mathematicians and
strict constructionists when it comes to the word choose. Is choice an ac-
tive process, in which one must specify the choices made (or a procedure
for making those choices), or is it merely a statement of existence, in that
choices can be made (this is somewhat reminiscent of Henry Kissinger’s
remark that “mistakes were made in Administrations of which I was a
part”)?^7 If you are a strict constructionist who wants a recipe for choice,
you won’t have any problem doing this with a collection of sets of positive
integers—you could just choose the smallest integer in any set. In fact,
there are many collections of sets in which constructing a choice function
(a function whose value for each set is the choice that is made for that set)
presents no problem. However, if one considers the collection of all non-
empty subsets of the real line, there is no obvious way to do this—nor is
there an unobvious way, as no one has yet done it and the betting of many
mathematical logicians is that it can’t be done.
There is a significant difference between “sets of positive integers” and
“sets of real numbers”—and that is the existence of a smallest positive
integer in any nonempty set of positive integers—but there is no obvious
smallest real number in any nonempty set of real numbers. If there were,
we could find a choice function in exactly the same manner that we did
for sets of positive integers—we’d simply choose t he smallest real number
in the nonempty set.
It may have occurred to you that there are sets of real numbers that
clearly have no smallest member, such as the set of all positive real num-
bers. If you think you have the smallest such number, half of it is still
positive, but smaller. However, there might conceivably be a way to ar-
range the real numbers in a different order than the usual one, but one
such that every nonempty set of real numbers has a smallest member. If
there were, then the choice function would be the one defined in the last
paragraph—the smallest number in each set. As a matter of fact, this
idea is known as the well-ordering principle, and is logically equivalent to
the axiom of choice.
If finding a choice function for the collection of all subsets of real num-
22 How Math Explains the World