sume that it could be so matched. If so, then every person with an infi-
nitely long name could be assigned a room in Hilbert’s Hotel, and we’ll
assume we’ve done so. We’ll demonstrate a contradiction by showing that
there is a person with an infinitely long name who has no room in the
hotel.
To do this, we’ll construct the name of such a person, whom we’ll call the
mystery guest, letter by letter. Look at the name of the person in room R1, and
choose a letter different from the first letter of that name. That “different let-
ter” is the first letter of our mystery guest’s name. Then look at the name of
the person in room R2, and choose a letter different from the second letter of
that name. That “different letter” is the second letter of our mystery guest’s
name. In general, we look at the nth letter of the name of the guest in room
Rn, and choose a ‘different letter’ from that one as the nth letter of our mystery
guest’s name.
So constructed, our mystery guest is indeed roomless. He’s not in R1, be-
cause the first letter of his name differs from the first letter of the guest in R1.
Our guest is not in R2, because the second letter of his name differs from the
second letter of the guest in R2. And so on. Our mystery guest is nowhere to
be found in Hilbert’s Hotel, and so the collection of people with infinitely long
names cannot be matched one-to-one with the positive integers.
Great results in mathematics have the name of their discoverer attached,
such as the Pythagorean theorem. Mathematical objects worthy of study have
the name of an important contributor affixed, such as “Cantor set.” Brilliant
mathematical proof techniques are similarly immortalized—the above con-
struction is known as a “Cantor diagonal proof ” (if we were to arrange the
names of the hotel guests in a list from top to bottom, with the first letters of
each name comprising the first column, the second letters of each name com-
prising the second column, and so on, the line connecting the first letter of
the first name to the second letter of the second name, thence to the third let-
ter of the third name, and so on, would form the diagonal of the infinite
square that comprises the list). In fact, Cantor is one of the few mathemati-
cians to hit for the cycle, having not only proof techniques named for him, but
theorems and mathematical objects as well.
The Continuum Hypothesis
It is fairly easy to see that the above proof technique shows that the collection
of real numbers between 0 and 1 also has a different cardinal number than
that of the positive integers. The real numbers between 0 and 1 (known as
“the continuum”) are, when written in decimal expansion, simply infinitely
20 How Math Explains the World