Those are rookie numbers

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 the real, non-rational numbers?

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.