bers gives you a headache, you might prefer the following version of the
dilemma, due to Bertrand Russell—if you have an infinite number of
pairs of shoes, it is easy to choose one shoe from each pair (you could
choose the left shoe), but if you have an infinite number of pairs of socks,
there is no way to distinguish one sock from another, and so you can’t
explicitly write out a way to choose one from each pair.
The great majority of mathematicians favor the existence formulation—
a choice exists (possibly in some abstract never-neverland in which we
cannot specify how), and an incredible amount of fascinating mathemat-
ics has resulted from incorporating the axiom of choice. Far and away the
most intriguing of the results is the Banach-Tarski paradox,^8 the state-
ment of which usually results in people feeling that mathematicians have
lost their collective minds. This theorem states that it is possible to de-
compose a three-dimensional sphere into a finite number of pieces and
rearrange them by rotations and translations (moving from one point of
space to another by pushing or pulling, but not rotating) into a sphere
with twice the radius of the original. Tempting though it may be to buy a
small golden sphere for a few hundred bucks, Banach-Tarskify it to dou-
ble its radius, and do so repeatedly until you have enough gold to retire to
your seaside villa, not even Charles Ponzi can help you with this one. Un-
fortunately, the pieces into which the sphere can be decomposed (notice
that I did not use the word cut, which is an actual physical process), exist
only in the abstract never-neverland of what are called “nonmeasurable
sets.” No one has ever seen a nonmeasurable set and no one ever will—if
you can make it, then it is not nonmeasurable, but if you accept the axiom
of choice in the existence sense, there is an abundance of these sets in
that never-neverland.
Consistent Sets of Axioms
I’m not sure that other mathematicians would agree with me, but I think
of mathematicians as those who make deductions from sets of axioms,
and mathematical logicians as those who make deductions about sets of
axioms. On one point, though, mathematicians and mathematical logi-
cians are in agreement—a set of axioms from which contradictory re-
sults can be deduced is a bad set of axioms. A set of axioms from which
no contradictory results can be deduced is called consistent. Mathemati-
cians generally work with axiom sets that the community feels are con-
sistent (even though this may not have been proven), whereas among
the goals of the mathematical logicians are to prove that axiom sets are
consistent.The Mea sure of All Things 23