User blog comment:Edwin Shade/Is Pi ''Really'' Transcendental ?/@comment-1605058-20170901195921/@comment-32876686-20170901213239

Thank you for your feedback.

 Although I didn't make this clear in my post, I would not consider S(S(x))=0 to be a valid Class-0 equation, because it can be simplified to x+2=0, which is a Class-1 equation. Likewise, I would not consider x+x+(-1)=0 to be a Class-1 equation because it can be simplified to 2x+(-1)=0, which is a Class-2 equation. In general, if a Class-N equation cannot be simplified to a higher class equation then it is strictly a Class-N equation.

 As for the matter of defining hyperoperaters for the entire set of real numbers, it is quite easy. First consider the relation x^(a/b)=(x^a)^(1/b), where a and b are integers. With this relation we can evaluate fractional exponents even though it is impossible to visualize what "a number multiplied by itself a fractional number of times" might look like. This is because we are already able to calculate whole numbered exponents and roots. The relation will now be extended into tetration, so that we can define x^^(a/b) as b/(x^^a)\. b/(x^^a)\ means "the bth hyper-root of x tetrated with a". Hyper-roots are like normal roots, except that instead of defining what number you must multiply by itself p times to yeild q, you are defining what number you must raise to the power of itself p times to yeild q. For instance, 3/16\, or "the 3rd hyper-root of 16" means the number that when you raise it to the power of itself three times, (in other words, just tetrate it with 3), yeilds 16; that number is 2 because 2^(2^2)=16. Hyper-roots are the inverse of tetration just as roots are the inverse of exponentiation. a/x\b means "the ath hyper-root of x according to b-ation. 3/16\4 means "the 3rd hyper root of 16 according to tetration". These hyper-roots can be calculated by approximation.

 With this knowledge we can write a general formula that allows us to evaluate any value of x^^^...{any positive number of arrows}...^^^(a/b). It is as follows: x[n+2](p/q)=q/x[n+2]p\(n+2), where [n+2] refers to the n+2nd hyper-operator beginning from n=1 and continuing arbitrarily high. Okay, now let's demonstrate this practically. How would we calculate, let's say, (2^^^1.5) ? First we plug it into the aforementioned formula giving 2[5](3/2)=2/2[5]3\5. Since we know how to calculate whole-numbered hyper-powers and hyper-roots we can calculate  2/2[5]3\5. 2/2[5]3\5=2/65536\5, and 2/65536\5 is between 2 and 3, (as we now know how to evaluate fractional pentation we could get a better answer through succesive approximation, though a computer would be needed).

    You are right though when you say that " it's not exactly clear how the proof of countability of rational numbers would extend". It a matter that I had a feeling must be true, but didn't provide a clear cut proof for, so I will attmept one here.

    Basically, you can write out all the fractions possible with integer numerators and denominators within one quadrant of a plane, leaving 3/4's of the space unplotted. Now let's map out all the polynomials with integer coefficients that we can in one quadrant of 3D space, leaving 7/8's of the space unplotted. We can continue plotting all the polynomials we can in Nth dimensional space, leaving ((2^N)-1)/(2^N) of the space unmapped. As the number of dimensions tend towards infinity, the amount of space we have used is increasingly small in relation to the entire space, and so there is enough room to "fit" the higher-order types of equations, because each one can fit into a quadrant of it's own without taking up the entire space.

   I have no proof that every properly defined higher class equation has a finite number of roots, but I'm going to conjecture it's true.

 Lastly, I believe I have asked what I meant to ask, that is, given any number, is there a general way for determining weather it is a N-Class or a TT-Class number ? For instance, how can one go about proving that pi is not the root of Class-N equation when it has been only proved for Class-3 equations ?