long names with letters 1 through 9 rather than A through Z and 0 instead
of blank. For example,^1 ⁄ 4 .25000.... Cantor worked out the arithmetic of
cardinal numbers, and designated the cardinal number of the positive inte-
gers as aleph-0, and the cardinal number of the continuum as c.
A lot of mileage can be gained from the Cantor diagonal proof. Cantor
used it to show that the set of rational numbers has cardinality aleph-0, as
does the set of algebraic numbers (all those numbers that are roots of
polynomials with integer coefficients). Also, it can be used to show the
infinite analogy of the child’s result that there is no largest (finite) number.
Cantor was able to show that, for any set S, the set of all subsets of S could
not be matched one-to-one with the set S, and so had a larger cardinal
number. As a result, there is no largest cardinal number.
Filling the Gaps
Leopold Kronecker, when he wasn’t making life miserable for Cantor, was
a mathematician of considerable talent, and is also the author of one of
the more famous quotations in mathematics: “God made the integers, all
else is the work of man.”^6 One of the first jobs that man had to do was fill
in the gaps between the integers in the number line. The task of filling
the gaps was to return in the nineteenth century, when mathematicians
encountered the problem of whether there existed cardinal numbers
between aleph-0 and c. As explained above, efforts to show that obvi-
ous sets, such as the set of rational numbers and the set of algebraic
numbers, had different cardinal numbers from aleph-0 and c proved
unsuccessful. Cantor hypothesized that there was no such cardinal
number—every subset of the continuum had cardinality aleph-0 or c; this
conjecture became known as the continuum hypothesis. Proving or dis-
proving the continuum hypothesis was a high priority for the mathemati-
cal community. In a key turn-of-the-century mathematics conference,
David Hilbert listed the solution of this problem as the first on his fa-
mous list of twenty-three problems that would confront mathematicians
in the twentieth century. Solution of just one of these problems would
make the career of any mathematician.
The Axiom of Choice
The axiom of choice is a relatively recent arrival on the mathematical
scene—in fact, it wasn’t until Cantor arrived on the mathematical scene
that anybody even thought that such an axiom was necessary. The axiomThe Mea sure of All Things 21