Meta-morphosis

There is a cute mathematical paradox that goes as follows:

The vocabulary of the English language, while undoubtedly vast, has finitely many words; hundreds of thousands perhaps, but still only finitely many. The number of English phrases with less than seventeen words is therefore finite as well. But there are infinitely many natural numbers (i.e. numbers of the type 0, 1, 2, 3, 4, ...) so not all of them can be described by English phrases of less than seventeen words. Some can be --- e.g. 6 can be described by the phrase "six" or by the short phrase "one plus five" --- but some just can't. The principle of mathematical induction tells us then that among all the natural numbers which cannot be expressed in less than seventeen words there is one which is the smallest. That is, there is the smallest natural number which cannot be described by English phrases of less than seventeen words. But wait! Hold on... Counting the words in italics we see that we just managed to describe that number in sixteen words!

How did that happen? The culprits here are the manners in which we used the English language. Indeed, "manners" in plural because we used English in two ways: in a low-level to talk about numbers and in a higher meta-level to discuss the system of numbers.

Other mathematical paradoxes (e.g. Russell's famous paradox) stem from similar dual usage of languages, in low- and meta-levels.

Mankind has long ceased to participate passively in the process of evolution. We have been breeding dogs, genetically modifying plants, transplanting hearts, and much more. Our reach is no longer confined to within the bounds of the evolution system; we have increasingly been manipulating the evolution system itself.

I don't know about you but I hear the music of paradoxes. And at least in Math this music rings trouble.

Zzzzz...