Chapter 7 Infinite Sets214
Larger Infinities
There are lots of different sizes of infinite sets. For example, starting with the
infinite set,N, of nonnegative integers, we can build the infinite sequence of sets
Nstrict pow.N/strict pow.pow.N//strict pow.pow.pow.N///strict::::By Cantor’s Theorem 7.1.11, each of these sets is strictly bigger than all the pre-
ceding ones. But that’s not all: the union of all the sets in the sequence is strictly
bigger than each set in the sequence (see Problem 7.23). In this way you can keep
going indefinitely, building “bigger” infinities all the way.
7.1.4 Diagonal Argument
Theorem 7.1.11 and similar proofs are collectively known as “diagonal arguments”
because of a more intuitive version of the proof described in terms of on an infinite
square array. Namely, suppose there was a bijection betweenNandf0;1g!. If such
a relation existed, we would be able to display it as a list of the infinite bit strings
in some countable order or another. Once we’d found a viable way to organize
this list, any given string inf0;1g!would appear in a finite number of steps, just
as any integer you can name will show up a finite number of steps from 0. This
hypothetical list would look something like the one below, extending to infinity
both vertically and horizontally:
A 0 D 1 0 0 0 1 1
A 1 D 0 1 1 1 0 1
A 2 D 1 1 1 1 1 1
A 3 D 0 1 0 0 1 0
A 4 D 0 0 1 0 0 0
A 5 D 1 0 0 1 1 1
::
:::
:
::
:
::
:
::
:
::
:
::
:
:::
But now we can exhibit a sequence that’s missing from our allegedly complete list
of all the sequences. Look at the diagonal in our sample list:
A 0 D 1 0 0 0 1 1
A 1 D 0 1 1 1 0 1
A 2 D 1 1 1 1 1 1
A 3 D 0 1 0 0 1 0
A 4 D 0 0 1 0 0 0
A 5 D 1 0 0 1 1 1
::
: