Download The Axiom of Choice

The Axiom of Choice
If you read about set theory, or take more advanced math classes, one thing you will
almost certainly hear about is the axiom of choice. I thought you might like to hear a
little bit about the axiom of choice, and related subjects.
As you probably know, it is possible to make discussions involving sets precise via
“axiomatic set theory.” In this class, we have chosen a more informal approach to
sets, simply thinking of them as bins that we throw objects (numbers, other sets, or
whatever) into. In fact, this is how the vast majority of mathematicians think about
sets. It is rare for mathematicians to make any direct mention of set-theoretic axioms,
with one exception: the axiom of choice. Proofs which make use of the axiom of choice
have a very different flavor from those that do not, and there is a feeling that using
the axiom of choice is legitimate (if it is needed), but not very satisfying. Therefore,
authors using axiom of choice often say directly that they are using it.
So what is the axiom of choice? Basically, it is the following very believable idea: if
I have a set X whose elements are nonempty sets, then then I can choose one element
from each of the sets in X. You may be surprised that we need an axiom for this! It
seems downright obvious.
But it’s a little bit like how we needed an axiom to justify proof by induction. Recall
that in that case, if we had any finite number of cases to check, and we can prove that
the first is true and each one implies the next, then we will have that all of them are
true, without any “axiom of induction.” The trouble comes when we try to do this for
an infinite number of cases: a “proof” (by definition) is only allowed to have finitely
many steps. We have that case 1 is true, and since 1 =⇒ 2 case 2 is true, and similarly
case 3 is true, and case 4 is true; but to conclude that all cases n ≥ 1 are true requires
the induction axiom.
Similarly, if X were any finite set consisting of nonempty sets, then there would be
no problem in choosing one element from each of the members of X. The trouble is
that when X has infinitely many members, how do we know that we can make all of
the choices in a “finite amount of time”? So we need an axiom for this.
Interestingly, there are several statements that turn out to be logically equivalent
to the axiom of choice, although they do not appear to be the same at all. I think this
is partly why the axiom of choice shows up so often. Let me mention a couple of these
which have to do with order relations.
One statement that’s equivalent to the axiom of choice is the well-ordering theorem.
It says that for any set A, there exists a total order ≤ on A such that for any nonempty
subset K ⊆ A, K has a minimum element. Such a total order is called a well-order. For
example, the standard order on the natural numbers N is a well-order. The standard
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order on the integers Z is not a well-ordering: choose any subset which is not bounded
below, e.g. {0, −1, −2, . . .} (or for that matter all of Z), and such a subset will not
have a least element. However, you can come up with a different total ordering of the
integers which is a well-ordering; for example, define ≤ by 0 ≤ 1 ≤ −1 ≤ 2 ≤ −2 ≤ . . ..
That was easy, but note that it is not at all clear we can define a well-order on the real
numbers. To do that, it turns out that we will need something like the axiom of choice.
At this point, it may be possible to start to give a sense of why things that use
the axiom of choice feel different from things that do not. We did not need to use
any mumbo jumbo to define a well-ordering on the integers; we simply wrote one down
explicitly. For the real numbers, we were not able to do this. We can show, using the
axiom of choice, that there exists a well-ordering on the real numbers, but we will never
be able to give any explicit description of what it is (this can be proven).
Statements that “there exists ...”, without saying (or being able to say) what it is,
abound in modern mathematics, and often involve use of the axiom of choice. This
bothers “constructivists,” mathematicians who believe that to prove the existence of
something one must explicitly write down what it is. Most mathematicians today are
willing to put up with non-constructive existence theorems, but prefer constructive ones
when possible.
Another statement that is equivalent to the axiom of choice is called Zorn’s lemma.
This is the version of the axiom of choice I have seen by far the most often in my
studies. In order to understand it, let’s define the notion of a “chain.” Let (A, ≤) be
a partially ordered set. We define a chain in A to be a totally-ordered subset of A.
That is, we call K ⊆ A a chain if for every x, y ∈ K, we have either x ≤ y or y ≤ x.
Then Zorn’s lemma says this: if (A, ≤) is a partially ordered set such that every chain
has an upper bound, then A must contain a maximal element (possibly more than one
maximal element).
Confused? You’re not alone. There’s a joke, popular among mathematicians,
that goes like this: “The axiom of choice is obviously true, the well-ordering theorem is obviously false, and who can tell about Zorn’s lemma?” The joke is that many
mathematicians find the axiom of choice to be intuitive, the well-ordering theorem
to be counterintuitive, and Zorn’s lemma to be too complex for any intuition. (See
http://en.wikipedia.org/wiki/Axiom_of_choice#Quotes.)
Here’s how Zorn’s lemma often comes up: let B be some set. Suppose there is some
property P which is satisfied by some, but not all, of the subsets of B. We sometimes
want to show that there is a maximal subset satisfying property P : a subset which
satisfies property P , and which is not contained in any larger subset that also satisfies
property P . Let’s see how one might use Zorn’s lemma to show that there is such a
maximal subset.
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Let A be all the subsets of B satisfying property P (so A is a subset of the power
set of B). We’ll define a partial order on A like this: for X, Y ∈ A (i.e. X and Y are
subsets of B satisfying property P ), X ≤ Y if and only if X ⊆ Y . This is the standard
partial order on the power set of a set (or on a subset of the power set); your book
mentions it.
Suppose we can show that the union of subsets satisfying P is a subset which also
satisfies P . Then we’re in business: suppose {Xi }i∈I (I some index set) is a chain of
subsets of B satisfying P (remember, chain means totally-ordered [
subset, so here that
means that for any i, j ∈ I, either Xi ⊆ Xj or Xj ⊆ Xi ). Then
Xi is a subset of
i∈I
B satisfying P , and containing all the Xi ’s, i.e. it is an upper bound of the elements
Xi ∈ A. Therefore, we can apply Zorn’s lemma – since an arbitrary chain in A had an
upper bound, A must have a maximal element! That is to say, a subset of B satisfying
P such that there is no subset containing it that satisfies P , which is exactly what we
were seeking.
Does it seem somehow unsatisfying that we magically showed that there has to be a
maximal subset satisfying P , without giving any indication of what it might be? Again,
this is a typical example of a proof using the axiom of choice (in Zorn’s lemma form).
Zorn’s lemma can also be used to prove statements like this: Every vector space has
a basis (specifically, a “Hamel” basis). See http://planetmath.org/encyclopedia/
EveryVectorSpaceHasABasis.html. You may have seen that this is an easy fact for
finite-dimensional vector spaces, but it’s not even true for infinite-dimensional vector
spaces, unless we assume the axiom of choice (or equivalently, Zorn’s lemma). Somehow,
we would like the statement that every vector space has a basis to be true. We know
it’s true for finite-dimensional vector spaces, and the whole thing feels neater if it’s true
in general. So by assuming the axiom of choice, we can prove that some things behave
as they “should” in general.
But, the axiom of choice can also be used to prove some very bizarre things. For
example, one can use it to prove that a solid ball in 3-dimensional space can be split
into a finite number of non-overlapping pieces, which can then be put back together
in a different way to yield two identical copies of the original ball. The reassembly
process involves only moving the pieces around and rotating them, without changing
their shape. See http://en.wikipedia.org/wiki/Banach-Tarski.
So, is the axiom of choice true, or untrue? Initially, it met with a lot of resistance
among mathematicians. Gradually, it gained acceptance, so that today its use is completely standard, and only a tiny minority object. But this doesn’t really answer the
question: is it true or not? My personal opinion is that it’s really meaningless to say
whether the axiom of choice is true or not. It is merely an axiom, an assumption. One
can look at the mathematics that one gets by assuming it, and the math one gets if
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one doesn’t assume it, and compare the two, and the differences will be interesting.
Neither is “right” or “wrong” in the absolute; they are simply implied by the initial
assumptions (axioms).
I think an analogy can be made with playing games. Different games have different
rules, just as one can explore the consequences of different sets of mathematical axioms.
A collection of rules for playing a game can’t be “wrong” (at least, unless the rules
contradict each other), but different rules will certainly yield different games. Still,
some rules are more basic than others. For instance, in most games it is an unwritten
rule that one may not kill opposing players. Similarly, most of math is based in some
way on sets, and all of math is based on the basic axioms of logic. Maybe even these
axioms are not “true” in any absolute sense. But altering these most basic axioms
might result in something too foreign, too distant from intuition for mathematicians
to find it interesting. Then again, if the axioms yielded a rich, surprising, beautiful
structure, who knows?
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