My favorite version of this proof:

Let S be the subset of natural numbers that are not interesting. Suppose by way of contradiction that S is inhabited. Then by the well ordering principle of natural numbers, there is a least such element, s in S. In virtue of being the least non interesting number, s is in fact interesting. Hence s is not in S. Since s is in S and not in S, we have derived a contradiction. Therefore our assumption that S is inhabited must be false. Thus S is empty and there are no non interesting numbers.

Counterproof:

n+1 is not the smallest non-interesting number... n-1 is.

What about 31? That's the smallest non-interesting number so if we take that as the first n, then every n+1 is either interesting or the second-smallest non-interesting number, and the second smallest non-interesting number is still not interesting.

what about the real, non-rational numbers?

The number of particles in the observable universe. For all we know, the universe may be infinite.

Start counting in binary. Gets you to 31 on one hand, 1023 on two.

True, You can only induce natural numbers from this.

However, you could extend it to the positive reals by saying [0,1) is a small number. And building induction on all of those.

You could cover negative and even complex numbers if “small” is a reference to magnitude of a vector, but that is a slippery slope…

In a very not rigorous way, you can cover combinations of ordinal numbers and even non-numbers if you treat them as orthogonal “unit vectors” and the composite “number” as a vector in an infinite vector space which again allows you to specify smallness as a reference to magnitude like we did for the complex numbers.

If you multiply two not really numbers, just count the product as a new dimension for the vector. Same with exponentiation. Same with non math shit like a cow or the color orange. Count all unique things as a unique dimension to a vector then by our little vector magnitude hack, everything is a small number, even things that aren’t numbers. QED.

This proof is a joke, broken in many ways, but the most interesting is the question of if you can actually have a vector with an uncountably infinite (or higher ordinals) of dimensions and what the hell that even means.

Sure you can. Proof:

0 is a number.

If n is a number, n+1 is also a number.

Therefore, by mathematical induction, we can induce all numbers.