Paradoxical Proof?

[BAndrew]

Actually there has to be a smallest natural number N (which we of course don't know).

I do not see how it is even remotely possible to describe something indefinable.

Wait, what? All I meant was that there has to be a smallest natural number N which can be described with fourteen words or fewer.

There very well may be, but at the present moment it is impossible to know this "smallest number" - therefore it strikes me as completely impossible to consider it. About as impossible as mapping the coordinates of the edge of the universe.

It's indeed impossible to know this number, but we can easily prove that such a number exists.

Spoiler below!

Assume S the set of all natural numbers that can't be described with 14 words or less (which is not an empty set and is a subset of N). This set must have a minimum natural number N because:

Let another number X of the same set.

If X<N then there is a smallest number (it's just not N)
If X>=N then N is the smallest number

But I think I missed the point you are making.