User blog:Edwin Shade/A Complete Analysis of Taranovsky's Notation

First, a disclaimer:

I'm sorry, but I really don't feel like I can understand all significant googological notations by the end of 2018. I'll try, but there's just so many of them and I believe I've gotten myself in too deep when I made that goal for the year of 2018. As PsiCubed2 rightfully pointed out, I haven't been the most focused on googological matters lately, and feeling like I have to always present something that will impress and make top blog post here is starting to become tiresome. I'd rather feel at ease writing blog posts, so I will. I know it sounds like I'm a quitter, but I sincerely don't believe I made a reasonable goal, which is a habit I often have, so I apologize if I got anyone's hopes up. I feel this is best for me though, to take it easier. I've decided on a compromise though, which is to understand at least one major googological notation, thus I've chosen Taranovsky's notation because being at the very threshold of my comprehension, it offers a challenge, yet at it's lower levels offers easy problems to work through. Perhaps if I devote my time to this one thing I'll crack the standing question of the strength of \(f_{C(C(\cdots C(\Omega_n2,0)\cdots,0),0)}(n)\), or Taranovsky's notation in general, (okay, probably not, but it would be nice).

So with out further ado, let's begin !

These are the approximations that I have come up in a more or less random way.

\(C(0,0)=1\)

\(C(0,C(0,0))=2\)

\(C(0,C(0,C(0,0)))=3\)

\(C(0,C(0,C(0,C(0,0))))=4\)

\(C(0,C(0,C(0,C(0,C(0,0)))))=5\)

\(C(0,C(0,C(0,C(0,C(0,C(0,0))))))=6\)

\(C(0,C(0,C(0,C(0,C(0,C(0,C(0,0)))))))=7\)

\(C(C(0,0),0)=\omega\)

\(C(0,C(C(0,0),0))=\omega+1\)

\(C(0,C(0,C(C(0,0),0)))=\omega+2\)

\(C(0,C(0,C(0,C(C(0,0),0))))=\omega+3\)

\(C(0,C(0,C(0,C(0,C(C(0,0),0)))))=\omega+4\)

\(C(0,C(0,C(0,C(0,C(0,C(C(0,0),0))))))=\omega+5\)

\(C(0,C(0,C(0,C(0,C(0,C(0,C(C(0,0),0)))))))=\omega+6\)

\(C(0,C(0,C(0,C(0,C(0,C(0,C(0,C(C(0,0),0))))))))=\omega+7\)

\(C(1,C(1,0))=\omega2\)

\(C(0,C(1,C(1,0)))=\omega2+1\)

\(C(0,C(0,C(1,C(1,0))))=\omega2+2\)

\(C(0,C(0,C(0,C(1,C(1,0)))))=\omega2+3\)

\(C(0,C(0,C(0,C(0,C(1,C(1,0))))))=\omega2+4\)

\(C(0,C(0,C(0,C(0,C(0,C(1,C(1,0)))))))=\omega2+5\)

\(C(0,C(0,C(0,C(0,C(0,C(0,C(1,C(1,0))))))))=\omega2+6\)

\(C(0,C(0,C(0,C(0,C(0,C(0,C(0,C(1,C(1,0)))))))))=\omega2+7\)

''This is going to be in progress for a while. Please tell me though if I've made any mistakes, so I can improve in my understanding of the notation ! Also, I might need help when dealing with the higher cardinals.''