User blog comment:Edwin Shade/Is Pi ''Really'' Transcendental ?/@comment-1605058-20170901195921

A very nice couple of ideas! Let me leave a few comments regarding them.

Your descriptions of the number classes aren't completely formal, which is perfectly fine in this case, since it appears to be clear what you mean. However, from your examples, it appears to me that we understand them differently. For example, for class 0 (let me write S for successor to avoid writing +), would you consider S(S(x))=0 a valid class 0 equation? You seem to not allow it, why not? Similarly, for class 1, wouldn't x+x+(-1)=0 be a fine equation with 1/2?

For higher classes, there is a larger problem -- hyperoperators are not naturally defined for arbitrary real numbers. In fact, most are defined only for natural numbers. An example is x^^^2, i.e. x^^x (how do you write a tower of x x's?). For the sake of further discussion, I am going to assume we have agreed on some specific way to extend the hyperoperators.

There are some subtelties one has to deal with in your proof of countability. For example, the set of all polynomials already appears to be an "infinitely-many"-dimensional space, so it's not exactly clear how the proof of countability of rational numbers would extend. Nevertheless, you are right that this set is countable, you just need a slightly different proof for that.

Another subtelty is in deducing, from the fact that there are countably many equations, that there are countably many solutions. This can only be done if we know that every equation has only countably many solutions. In polynomial case this follows from some simple algebra, but I don't know how this could be proven for higher classes (and that would depend on how exactly you extend the hyperoperators -- for some naive definitions, this is not true!).

I'm not sure whether at the end you have asked what you have really meant to ask -- by definition of TT, every number which is not in any N-Class is in TT-Class, so every number is in N or TT-Class.

Hope all of the above help you with improving this or your future posts!