Paradoxical Proof?

[BAndrew]

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). 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

This is a valid proof? Because it seems to me like yet another attempt to apply the concept of infinity to a system designed only to work with finite numbers.

What is wrong with the proof? I am not saying that this is the most elegant proof you can find, but it's valid.
What infinity has to do with it?

A practical way to see that there is a smallest natural number N is the following:
does 1 belong to the set S? NO? Then
does 2 belong to the set S? No? Then
................................................
does N belong to the set S? Finally the first yes? Then this is your smallest number.