Infinity 155
Corollary 113. Transcendental real numbers exist. Indeed, most real numbers are tran-
scendental, in the sense that there are only countably many algebraic real numbers but
uncountably many transcendental real numbers.
Proof. A real number r is algebraic if it is a root of a nontrivial polynomial in one variable
over the integers. Every such polynomial has only finitely many roots, which are naturally
ordered by the order of the real numbers. For example,
√
2 is the second root of the
polynomial x
2
− 2, the first being −
√
2. In this way, every algebraic number is uniquely
determined by a finite list of integers: the coefficients of a polynomial for which it is
the root and its place among the roots of that polynomial. Since theorem 111 shows that
there are only countably many finite sequences of integers, it follows that there are only
countably many algebraic numbers.
Meanwhile, theorem 112 shows that the set of all real numbers is uncountable. And
so there must be some real numbers that are not algebraic, which is to say that they are
transcendental. Furthermore, since the union of two countable sets is countable by theo-
rem 106, there must be uncountably many transcendental real numbers.
The proof of corollary 113 gives rise to a subtle philosophical issue. Namely, the ques-
tion is whether or not Cantor’s proof of the existence of transcendental numbers is con-
structive or not, whether it provides specific real numbers that it proves are transcendental
or whether it is instead a pure-existence proof of transcendental numbers. Liouville in
1844 had proved that transcendental numbers exist, after all, by exhibiting a specific real
number and proving that it was transcendental. Does Cantor’s argument also do this? Does
Cantor provide a specific transcendental number? Or is his proof merely a pure-existence
proof showing that some real numbers must be transcendental but without exhibiting any
particular such real number?
Sometimes mathematicians claim, wrongly in my view, that Cantor’s argument is not
constructive. The reason, they say, is that Cantor’s proof of the uncountability of the set
of real numbers proceeds by contradiction—it is uncountable because any proposed enu-
meration is inadequate. One may then deduce that there must be transcendental numbers,
because otherwise there would be only countably many real numbers, which is a contradic-
tion. Indeed, this way of arguing appears to be nonconstructive, since it proves that some
real numbers must be transcendental, but it does not seem to exhibit any particular such
transcendental number.
But meanwhile, a closer inspection of Cantor’s argument shows nevertheless that it is
constructive, that it does actually provide specific transcendental numbers. First, Cantor
provides an explicit enumeration of the algebraic numbers, by means of the finite sequences
of integers corresponding to polynomials and the particular roots of each; and second, his
diagonalization method provides a way to construct a specific number—the diagonal real—
which is not on a given enumeration. By simply combining these two steps, therefore, he