4.
Let
S
be the set of all natural numbers that are describable in English words using
no more than 50 characters (so, 240 is in
S
since we can describe it as “two hundred forty”,
which requires fewer than 50 characters). Assuming that we are allowed to use only the 27
standard characters (the 26 letters of the alphabet and the space character), show that there
are only finitely many numbers contained in
S
. (In fact, perhaps you can show that there
can be no more than 27
50
elements in
S
.) Now, let the set
T
be all those natural numbers
not in
S
. Show that there are infinitely many elements in
T
. Next, since
T
is a collection of
natural numbers, show that it must contain a smallest number. Finally, consider the smallest
number contained in
T
. Prove that this number must simultaneously be an element of
S
and not an element of
S
– a paradox! (B+S 3.4.20)
Solution.
There are only 27
50
strings consisting of fifty characters or less; furthermore
most of these aren’t actually legitimate strings of English, and sometimes two describe the
same number (for example, “eight” and “two cubed”). Thus there are only finitely many
numbers contained in
S
. So there are infinitely many elements in
T
, since
T
consists of all
the natural numbers except for a finite set, and removing a finite set from a countable set
gives a countable set. A collection of natural numbers contains a smallest number, as we can
see by checking if it contains 1, then 2, then 3, and so on. The smallest number contained in
T
is not contained in
S
, since it is contained in
T
. But it can be described as “the smallest
number contained in
T
”, which is a phrase containing less than fifty characters, so it is
contained in
S
.
If you object that “the smallest number contained in
T
” isn’t really a description of
2
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that number in less than fifty characters, because we’d have to define
T
, then we could
put a definition of
T
in the string. Unfortunately it’s hard to describe
T
in less than fifty
characters! So we can describe
T
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 Summer '09
 Lugo
 Math, Natural Numbers, Georg Cantor, Cantor

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