User blog comment:Edwin Shade/Is Pi ''Really'' Transcendental ?/@comment-7484840-20170902115429

This is a nice idea! I've got a couple of comments:

Firstly, I assume that all of the functions in $$F_N$$ must have integer coefficients/ powers/ hyper-powers etc, in the same way that the traditional definition of trascendental-ness requires integer coefficients and powers.

Secondly, you're completely right that the set numbers which are class N for some N is countable. This can be shown by using the result that a countable union of countable sets is countable. For polynomials, you show that the set of polynomials of degree d is countable for any natural d, and then take a (countable) union over all d. Showing that the set of degree d polynomials is countable isn't too hard; it's easy to show the size of this set is the same as $$\mathbb(N)^{d+1}$$ by looking at the coefficients. For the hyper-operator functions, we can do something similar; just show that $$F_N$$ is countable for each N, which can be done with basically the same trick as polynomials, except taking a union over the 'degree' in each hyper-operator (e.g. x^3+2x^^4) could be said to have 'degree' (3,4) or something similar, so the sets are all countable, so there have to be uncountably many TT-class numbers.