Theorem 14.2: How do I love thee? Let me count the ways.

Proof: My love for thee is eternal, so let us express my love as the (countable or otherwise) infinite set L. Now assume that my love is uncountable. In this case we can form an injection

f:R -> L
from the real numbers into my love. But the real numbers contain the irrationals, which means that there must be ways in which I love thee that can be expressed as f(y), where y is an irrational. But there is no way in which my love for thee can be irrational, as to be irrational would imply that there is no reason behind it. Therefore by contradiction we may conclude that L is countable, that is, we /can/ indeed count the ways in which I love thee.

So let us denote the ways in which I love thee as {l(i)}, i taking the values 1,2,3, et cetera. Now we have shown that it is possible count the ways in which I love thee, let us begin by ordering them by magnitude. We know that no way may have infinite magnitude, for that would reduce all other ways to insignificant (and to love thee with infinite magnitude in one way would be simply irrational, which we have already concluded is impossible). So indeed every way does have a finite magnitude, and we may order them in the obvious manner. WLOG assume that we have

l(n) > l(n-1)
for all n.

So we have now reduced my love to the set L, consisting of countably many finite elements l(i), with

l(n) > l(n-1)
for all n. Clearly we may count them, and so the proof is done. But for fun let us analyse the ways in a minor amount of detail.

We will do this by induction. Assume that we may analyse l(i) (if not, simply insert a dummy element

l(0) = "I love the colour of thy hair,"
which may be easily analysed). So we now require a method to analyse l(n) given a thorough understanding of all elements l(m), 0 <= m < n.

Express our element l(n) as part of the subset V(eng), the English language in standard notation (we know we may do this: again, if we may not, then it would imply that the way cannot be expressed in words, which would make it thoroughly irrational - contradiction). So we have V(eng) containing a specific point v(l(n)), and we wish to determine the precise nature of this point. Let us do this by the method of 'lion-hunting.'

Consider the first word of v(l(n)) - denote this as w(1) for ease of notation. Now set

a(0) = "A"
b(0) = "Zymurgy".
c(0) = (a(0) + b(0))/2
(use a dictionary if necessary). Now iterate to find w(1) in the following manner:

If w(1) < c(0), set a(1) = a(0), b(1) = c(0).
If w(1) > c(0), set a(1) = c(0), b(1) = b(0).
If w(1) = c(0), then we have successfully found w(1) and we are done.

Continue this process in the obvious manner until we do indeed find w(1). Now repeat for w(2), w(3) and so on until we have successfully isolated all words of v(l(n)). It is a matter of trivial algebra now to combine all these words, thus producing element l(n) of my love, expressed in a suitable basis of the English language. This may then be analysed by linguists while the mathematicians repeat the process for l(n+1).

Thus, not only have we counted the ways in which I love thee, but we have also provided an algorithm for expressing each in words. This should be more than enough to satisfy thee this Valentine's day.


DISCLAIMER: The methods used above are solely those of a first-year undergraduate. These methods in no way represent those of the maths faculty or of the university in general. The university will not be held responsibile for any results derived using these methods. 1