154 Chapter 13

Let me try to give you a taste of the kind of objections that I have seen raised. In some

messages, perhaps with some needlessly complex notation, the person proposes that in-

stead of the Cantor’s diagonalization, we should instead define a different diagonal real

number y, whose nth digit agrees with the nth digit of r

n

, rather than disagreeing with it.

With this way of arguing, they say, Cantor’s argument does not work, and so the real num-

ber y could be on the list after all; they conclude that the real numbers could be countable

after all. But this is clearly a baseless objection, and the conclusion does not follow. If a

modified version of a correct argument does not succeed, it does not mean that the original

argument is wrong; it just means that the modified argument does not constitute a proof.

Another objection one sees is that, after having constructed Cantor’s diagonal real num-

ber z, if it does not appear on the original list r

1

, r

2

, r

3

, and so on, which was supposed

to include all real numbers, then one should simply add it to the list! Put the diagonal

real number z in front, they say, and everything is fine. And furthermore, if one should

diagonalize against this new list, then that real also could simply be added to the list. This

objection, of course, is without merit, since the claim was not that the diagonal real num-

ber z could not appear on any list at all; rather, the claim was that the original list did not

already contain all real numbers, since it did not contain the diagonal real z. This contra-

diction shows that it is impossible to make a list containing all real numbers, and that is

precisely what it means to say that the set of real numbers R is uncountable.

Another common objection is that people sometimes try to prove directly that the set of

real numbers is countable. For example, sometimes they start by proving that the set of real

numbers with finite terminating decimal expansions is countable. This part is correct, since

these real numbers are precisely the rational numbers expressible with a denominator that

is a power of 10. Next, they point out that every real number is a limit of a sequence of such

rational numbers, which approximate it ever more closely. And that part also is correct.

Finally, they attempt to conclude, usually without any proof or with erroneous proof, that

this implies that the set of real numbers is countable. But this step of their argument is

simply not correct. The set of real numbers does indeed have a countable dense set, but

this does not mean that the real numbers themselves are countable.

Shall we leave these silly objections behind and aim for more mathematics?

13.4 Transcendental numbers

Cantor used his theorem to prove the following corollary, giving a new proof of Joseph

Liouville’s 1844 theorem that transcendental numbers exist, while also strengthening the

conclusion. Recall from chapter 1 that a real number is algebraic if it is the root of a

polynomial with integer coefficients; a number is transcendental if it is not. It had been

an open question whether there were any transcendental numbers at all, until Liouville’s

celebrated theorem that they exist. Cantor’s goes further, showing that indeed, most real

numbers are transcendental.