Proof and the Art of Mathematics

Inﬁnity 153

in chapter 15 shows more generally that every nested sequence of closed intervals has a

common real number inside all of them.) Since every r

on the list was eventually excluded

from the intervals, it follows that z  r

for any n, a contradiction.

Although the two proofs may look very different at ﬁrst, one using diagonalization

against the digits of the number and the other using a nested sequence of closed intervals,

my view is that these arguments are fundamentally the same. They are merely different

manners of describing essentially the same underlying construction.

3.9 4.1 4.2 4.3 4.4

4.5 4.6

4.7 4.8 4.9

5.1

real numbers having integer part 4

real numbers with decimal expansion beginning 4.6

real numbers with decimal expansion beginning 4.63

The reason is that to specify the ﬁrst several digits of a real number in the decimal ex-

pansion is exactly to restrict to a certain interval of real numbers, the numbers whose

expansion begins with those digits. The resulting intervals are closed intervals, in light of

the nonuniqueness of the decimal representations, as with 5 = 4.99999 ···, at the right end

of the blue interval above. Because of this, the construction in the two formulations of

Cantor’s argument are essentially identical; the digit diagonalization argument is simply a

more concrete and attractive way to describe the particular intervals that are being chosen.

Cranks

Cantor’s theorem and proof are sometimes misunderstood in certain characteristic ways,

and I sometimes get email, for example, from people who claim that Cantor’s proof is

all wrong or even that all of mathematics is wrong and so forth. Such messages often

come from unfortunate people, who take themselves to be unrecognized geniuses, having

refuted what they view as the established dogma of mathematics. If they were correct, then

I should be amongst the ﬁrst to applaud them, since I would give no favor to dogmatism

and, furthermore, I would sincerely enjoy any show in which high-and-mighty pompous

fools are toppled. But it is simply not the situation here. Upon inspection, one invariably

sees, sadly, that the objections they raise are without any force at all. Usually, they have

simply misunderstood Cantor’s argument.