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 !

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)=ω

C(0,C(C(0,0),0))=ω+1

C(0,C(0,C(C(0,0),0)))=ω+2

C(0,C(0,C(0,C(C(0,0),0))))=ω+3

C(0,C(0,C(0,C(0,C(C(0,0),0)))))=ω+4

C(0,C(0,C(0,C(0,C(0,C(C(0,0),0))))))=ω+5

C(0,C(0,C(0,C(0,C(0,C(0,C(C(0,0),0)))))))=ω+6

C(0,C(0,C(0,C(0,C(0,C(0,C(0,C(C(0,0),0))))))))=ω+7

C(1,C(1,0))=ω2

C(0,C(1,C(1,0)))=ω2+1

C(0,C(0,C(1,C(1,0))))=ω2+2

C(0,C(0,C(0,C(1,C(1,0)))))=ω2+3

C(0,C(0,C(0,C(0,C(1,C(1,0))))))=ω2+4

C(0,C(0,C(0,C(0,C(0,C(1,C(1,0)))))))=ω2+5

C(0,C(0,C(0,C(0,C(0,C(0,C(1,C(1,0))))))))=ω2+6

C(0,C(0,C(0,C(0,C(0,C(0,C(0,C(1,C(1,0)))))))))=ω2+7

C(1,C(1,C(1,0)))=ω3

C(0,C(1,C(1,C(1,0))))=ω3+1

C(0,C(0,C(1,C(1,C(1,0)))))=ω3+2

C(0,C(0,C(0,C(1,C(1,C(1,0))))))=ω3+3

C(0,C(0,C(0,C(0,C(1,C(1,C(1,0)))))))=ω3+4

C(0,C(0,C(0,C(0,C(0,C(1,C(1,C(1,0))))))))=ω3+5

C(0,C(0,C(0,C(0,C(0,C(0,C(1,C(1,C(1,0)))))))))=ω3+6

C(0,C(0,C(0,C(0,C(0,C(0,C(0,C(1,C(1,C(1,0))))))))))=ω3+7

C(1,C(1,C(1,C(1,0))))=ω4

C(1,C(1,C(1,C(1,C(1,0)))))=ω5

C(1,C(1,C(1,C(1,C(1,C(1,0))))))=ω6

C(1,C(1,C(1,C(1,C(1,C(1,C(1,0)))))))=ω7

C(1,C(1,C(1,C(1,C(1,C(1,C(1,C(1,0))))))))=ω8

C(1,C(1,C(1,C(1,C(1,C(1,C(1,C(1,C(1,0)))))))))=ω9

C(1,C(1,C(1,C(1,C(1,C(1,C(1,C(1,C(1,C(1,0))))))))))=ω10

C(2,0)=ω2

C(1,C(2,0))=ω2+ω

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

\(C(1,C(1,C(1,C(2,0))))=\omega^2+\omega3\)

\(C(1,C(1,C(1,C(1,C(2,0)))))=\omega^2+\omega4\)

\(C(1,C(1,C(1,C(1,C(1,C(2,0))))))=\omega^2+\omega5\)

\(C(1,C(1,C(1,C(1,C(1,C(1,C(2,0)))))))=\omega^2+\omega6\)

\(C(1,C(1,C(1,C(1,C(1,C(1,C(1,C(2,0))))))))=\omega^2+\omega7\)

\(C(2,C(2,0))=\omega^22\)

\(C(1,C(2,C(2,0)))=\omega^22+\omega\)

\(C(1,C(1,C(2,C(2,0))))=\omega^22+\omega2\)

\(C(1,C(1,C(1,C(2,C(2,0)))))=\omega^22+\omega3\)

\(C(1,C(1,C(1,C(1,C(2,C(2,0))))))=\omega^22+\omega4\)

\(C(1,C(1,C(1,C(1,C(1,C(2,C(2,0)))))))=\omega^22+\omega5\)

\(C(1,C(1,C(1,C(1,C(1,C(1,C(2,C(2,0))))))))=\omega^22+\omega6\)

\(C(1,C(1,C(1,C(1,C(1,C(1,C(1,C(2,C(2,0)))))))))=\omega^22+\omega7\)

\(C(2,C(2,C(2,0)))=\omega^23\)

\(C(2,C(2,C(2,C(2,0))))=\omega^24\)

\(C(2,C(2,C(2,C(2,C(2,0)))))=\omega^25\)

\(C(2,C(2,C(2,C(2,C(2,C(2,0))))))=\omega^26\)

\(C(2,C(2,C(2,C(2,C(2,C(2,C(2,0)))))))=\omega^27\)

\(C(2,C(2,C(2,C(2,C(2,C(2,C(2,C(2,0))))))))=\omega^28\)

\(C(2,C(2,C(2,C(2,C(2,C(2,C(2,C(2,C(2,0)))))))))=\omega^29\)

\(C(2,C(2,C(2,C(2,C(2,C(2,C(2,C(2,C(2,C(2,0))))))))))=\omega^210\)

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

\(C(2,C(3,0))=\omega^3+\omega^2\)

\(C(2,C(2,C(3,0)))=\omega^3+\omega^22\)

\(C(2,C(2,C(2,C(3,0))))=\omega^3+\omega^23\)

\(C(2,C(2,C(2,C(2,C(3,0)))))=\omega^3+\omega^24\)

\(C(2,C(2,C(2,C(2,C(2,C(3,0))))))=\omega^3+\omega^25\)

\(C(2,C(2,C(2,C(2,C(2,C(2,C(3,0)))))))=\omega^3+\omega^26\)

\(C(2,C(2,C(2,C(2,C(2,C(2,C(2,C(3,0))))))))=\omega^3+\omega^27\)

\(C(3,C(3,0))=\omega^32\)

\(C(2,C(3,C(3,0)))=\omega^32+\omega^2\)

\(C(2,C(2,C(3,C(3,0))))=\omega^32+\omega^22\)

\(C(2,C(2,C(2,C(3,C(3,0)))))=\omega^32+\omega^23\)

\(C(2,C(2,C(2,C(2,C(3,C(3,0))))))=\omega^32+\omega^24\)

\(C(2,C(2,C(2,C(2,C(2,C(3,C(3,0)))))))=\omega^32+\omega^25\)

\(C(2,C(2,C(2,C(2,C(2,C(2,C(3,C(3,0))))))))=\omega^32+\omega^26\)

\(C(2,C(2,C(2,C(2,C(2,C(2,C(2,C(3,C(3,0)))))))))=\omega^32+\omega^27\)

\(C(3,C(3,C(3,0)))=\omega^33\)

\(C(3,C(3,C(3,C(3,0))))=\omega^34\)

\(C(3,C(3,C(3,C(3,C(3,0)))))=\omega^35\)

\(C(3,C(3,C(3,C(3,C(3,C(3,0))))))=\omega^36\)

\(C(3,C(3,C(3,C(3,C(3,C(3,C(3,0)))))))=\omega^37\)

\(C(3,C(3,C(3,C(3,C(3,C(3,C(3,C(3,0))))))))=\omega^38\)

\(C(3,C(3,C(3,C(3,C(3,C(3,C(3,C(3,C(3,0)))))))))=\omega^39\)

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

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

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

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

\(C(8,0)=\omega^8\)

\(C(9,0)=\omega^9\)

\(C(10,0)=\omega^{10}\)

\(C(C(1,0),0)=\omega^{\omega}\)

\(C(0,C(C(1,0),0))=\omega^{\omega}+1\)

\(C(0,C(0,C(C(1,0),0)))=\omega^{\omega}+2\)

\(C(0,C(0,C(0,C(C(1,0),0))))=\omega^{\omega}+3\)

\(C(0,C(0,C(0,C(0,C(C(1,0),0)))))=\omega^{\omega}+4\)

\(C(0,C(0,C(0,C(0,C(0,C(C(1,0),0))))))=\omega^{\omega}+5\)

\(C(0,C(0,C(0,C(0,C(0,C(0,C(C(1,0),0)))))))=\omega^{\omega}+6\)

\(C(0,C(0,C(0,C(0,C(0,C(0,C(0,C(C(1,0),0))))))))=\omega^{\omega}+7\)

\(C(1,C(C(1,0),0))=\omega^{\omega}+\omega\)

\(C(2,C(C(1,0),0))=\omega^{\omega}+\omega^2\)

\(C(3,C(C(1,0),0))=\omega^{\omega}+\omega^3\)

\(C(4,C(C(1,0),0))=\omega^{\omega}+\omega^4\)

\(C(5,C(C(1,0),0))=\omega^{\omega}+\omega^5\)

\(C(6,C(C(1,0),0))=\omega^{\omega}+\omega^6\)

\(C(7,C(C(1,0),0))=\omega^{\omega}+\omega^7\)

\(C(C(1,0),C(C(1,0),0))=\omega^{\omega}2\)

\(C(1,C(C(1,0),C(C(1,0),0)))=\omega^{\omega}2+\omega\)

\(C(2,C(C(1,0),C(C(1,0),0)))=\omega^{\omega}2+\omega^2\)

\(C(3,C(C(1,0),C(C(1,0),0)))=\omega^{\omega}2+\omega^3\)

\(C(4,C(C(1,0),C(C(1,0),0)))=\omega^{\omega}2+\omega^4\)

\(C(5,C(C(1,0),C(C(1,0),0)))=\omega^{\omega}2+\omega^5\)

\(C(6,C(C(1,0),C(C(1,0),0)))=\omega^{\omega}2+\omega^6\)

\(C(7,C(C(1,0),C(C(1,0),0)))=\omega^{\omega}2+\omega^7\)

\(C(C(1,0),C(C(1,0),C(C(1,0),0)))=\omega^{\omega}3\)

\(C(C(1,0),C(C(1,0),C(C(1,0),C(C(1,0),0))))=\omega^{\omega}4\)

\(C(C(1,0),C(C(1,0),C(C(1,0),C(C(1,0),C(C(1,0),0)))))=\omega^{\omega}5\)

\(C(C(1,0),C(C(1,0),C(C(1,0),C(C(1,0),C(C(1,0),C(C(1,0),0))))))=\omega^{\omega}6\)

\(C(C(1,0),C(C(1,0),C(C(1,0),C(C(1,0),C(C(1,0),C(C(1,0),C(C(1,0),0)))))))=\omega^{\omega}7\)

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

\(=\omega^{\omega}8\)

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

\(,0)))))))))=\omega^{\omega}9\)

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

\(,C(C(1,0),0))))))))))=\omega^{\omega}10\)

\(C(C(0,C(1,0)),0)=\omega^{\omega+1}\)

\(C(C(0,C(0,C(1,0)),0)=\omega^{\omega+2}\)

\(C(C(0,C(0,C(0,C(1,0))),0)=\omega^{\omega+3}\)

\(C(C(0,C(0,C(0,C(0,C(1,0)))),0)=\omega^{\omega+4}\)

\(C(C(0,C(0,C(0,C(0,C(0,C(1,0))))),0)=\omega^{\omega+5}\)

\(C(C(0,C(0,C(0,C(0,C(0,C(0,C(1,0)))))),0)=\omega^{\omega+6}\)

\(C(C(0,C(0,C(0,C(0,C(0,C(0,C(0,C(1,0))))))),0)=\omega^{\omega+7}\)

\(C(C(1,C(1,0)),0)=\omega^{\omega2}\)

\(C(C(0,C(1,C(1,0))),0)=\omega^{\omega2+1}\)

\(C(C(0,C(0,C(1,C(1,0)))),0)=\omega^{\omega2+2}\)

\(C(C(0,C(0,C(0,C(1,C(1,0))))),0)=\omega^{\omega2+3}\)

\(C(C(0,C(0,C(0,C(0,C(1,C(1,0)))))),0)=\omega^{\omega2+4}\)

\(C(C(0,C(0,C(0,C(0,C(0,C(1,C(1,0))))))),0)=\omega^{\omega2+5}\)

\(C(C(0,C(0,C(0,C(0,C(0,C(0,C(1,C(1,0)))))))),0)=\omega^{\omega2+6}\)

\(C(C(0,C(0,C(0,C(0,C(0,C(0,C(0,C(1,C(1,0))))))))),0)=\omega^{\omega2+7}\)

\(C(C(1,C(1,C(1,0))),0)=\omega^{\omega3}\)

\(C(C(1,C(1,C(1,C(1,0)))),0)=\omega^{\omega4}\)

\(C(C(1,C(1,C(1,C(1,C(1,0))))),0)=\omega^{\omega5}\)

\(C(C(1,C(1,C(1,C(1,C(1,C(1,0)))))),0)=\omega^{\omega6}\)

\(C(C(1,C(1,C(1,C(1,C(1,C(1,C(1,0))))))),0)=\omega^{\omega7}\)

\(C(C(1,C(1,C(1,C(1,C(1,C(1,C(1,C(1,0)))))))),0)=\omega^{\omega8}\)

\(C(C(1,C(1,C(1,C(1,C(1,C(1,C(1,C(1,C(1,0))))))))),0)=\omega^{\omega9}\)

\(C(C(1,C(1,C(1,C(1,C(1,C(1,C(1,C(1,C(1,C(1,0)))))))))),0)=\omega^{\omega10}\)

\(C(C(2,0),0)=\omega^{\omega^2}\)

\(C(C(3,0),0)=\omega^{\omega^3}\)

\(C(C(4,0),0)=\omega^{\omega^4}\)

\(C(C(5,0),0)=\omega^{\omega^5}\)

\(C(C(6,0),0)=\omega^{\omega^6}\)

\(C(C(7,0),0)=\omega^{\omega^7}\)

\(C(C(8,0),0)=\omega^{\omega^8}\)

\(C(C(C(1,0),0),0)=\omega^{\omega^{\omega}}\)

\(C(C(C(C(1,0),0),0),0)=\omega^{\omega^{\omega^{\omega}}}\)

\(C(C(C(C(C(1,0),0),0),0),0)=\omega^{\omega^{\omega^{\omega^{\omega}}}}\)

\(C(C(C(C(C(C(1,0),0),0),0),0),0)=\omega^{\omega^{\omega^{\omega^{\omega^{\omega}}}}}\)

\(C(C(C(C(C(C(C(1,0),0),0),0),0),0),0)=\omega^{\omega^{\omega^{\omega^{\omega^{\omega^{\omega}}}}}}\)

\(C(C(C(C(C(C(C(C(1,0),0),0),0),0),0),0),0)=\omega^{\omega^{\omega^{\omega^{\omega^{\omega^{\omega^{\omega}}}}}}}\)

\(C(C(C(C(C(C(C(C(C(1,0),0),0),0),0),0),0),0),0)=\omega^{\omega^{\omega^{\omega^{\omega^{\omega^{\omega^{\omega^{\omega}}}}}}}}\)

\(C(\Omega,0)=\varepsilon_0\)

\(C(0,C(\Omega,0))=\varepsilon_0+1\)

\(C(0,C(0,C(\Omega,0)))=\varepsilon_0+2\)

\(C(0,C(0,C(0,C(\Omega,0))))=\varepsilon_0+3\)

\(C(0,C(0,C(0,C(0,C(\Omega,0)))))=\varepsilon_0+4\)

\(C(0,C(0,C(0,C(0,C(0,C(\Omega,0))))))=\varepsilon_0+5\)

\(C(0,C(0,C(0,C(0,C(0,C(0,C(\Omega,0)))))))=\varepsilon_0+6\)

\(C(0,C(0,C(0,C(0,C(0,C(0,C(0,C(\Omega,0))))))))=\varepsilon_0+7\)

\(C(1,C(\Omega,0))=\varepsilon_0+\omega\)

\(C(1,C(1,C(\Omega,0)))=\varepsilon_0+\omega2\)

\(C(1,C(1,C(1,C(\Omega,0))))=\varepsilon_0+\omega3\)

\(C(1,C(1,C(1,C(1,C(\Omega,0)))))=\varepsilon_0+\omega4\)

\(C(1,C(1,C(1,C(1,C(1,C(\Omega,0))))))=\varepsilon_0+\omega5\)

\(C(1,C(1,C(1,C(1,C(1,C(1,C(\Omega,0)))))))=\varepsilon_0+\omega6\)

\(C(1,C(1,C(1,C(1,C(1,C(1,C(1,C(\Omega,0))))))))=\varepsilon_0+\omega7\)

\(C(2,C(\Omega,0))=\varepsilon_0+\omega^2\)

\(C(2,C(2,C(\Omega,0)))=\varepsilon_0+\omega^22\)

\(C(2,C(2,C(2,C(\Omega,0))))=\varepsilon_0+\omega^23\)

\(C(2,C(2,C(2,C(2,C(\Omega,0)))))=\varepsilon_0+\omega^24\)

\(C(2,C(2,C(2,C(2,C(2,C(\Omega,0))))))=\varepsilon_0+\omega^25\)

\(C(2,C(2,C(2,C(2,C(2,C(2,C(\Omega,0)))))))=\varepsilon_0+\omega^26\)

\(C(2,C(2,C(2,C(2,C(2,C(2,C(2,C(\Omega,0))))))))=\varepsilon_0+\omega^27\)

\(C(3,C(\Omega,0))=\varepsilon_0+\omega^3\)

\(C(3,C(3,C(\Omega,0)))=\varepsilon_0+\omega^32\)

\(C(3,C(3,C(3,C(\Omega,0))))=\varepsilon_0+\omega^33\)

\(C(3,C(3,C(3,C(3,C(\Omega,0)))))=\varepsilon_0+\omega^34\)

\(C(3,C(3,C(3,C(3,C(3,C(\Omega,0))))))=\varepsilon_0+\omega^35\)

\(C(3,C(3,C(3,C(3,C(3,C(3,C(\Omega,0)))))))=\varepsilon_0+\omega^36\)

\(C(3,C(3,C(3,C(3,C(3,C(3,C(3,C(\Omega,0))))))))=\varepsilon_0+\omega^37\)

\(C(4,C(\Omega,0))=\varepsilon_0+\omega^4\)

\(C(4,C(4,C(\Omega,0)))=\varepsilon_0+\omega^42\)

\(C(4,C(4,C(4,C(\Omega,0))))=\varepsilon_0+\omega^43\)

\(C(4,C(4,C(4,C(4,C(\Omega,0)))))=\varepsilon_0+\omega^44\)

\(C(4,C(4,C(4,C(4,C(4,C(\Omega,0))))))=\varepsilon_0+\omega^45\)

\(C(4,C(4,C(4,C(4,C(4,C(4,C(\Omega,0)))))))=\varepsilon_0+\omega^46\)

\(C(4,C(4,C(4,C(4,C(4,C(4,C(4,C(\Omega,0))))))))=\varepsilon_0+\omega^47\)

\(C(5,C(\Omega,0))=\varepsilon_0+\omega^5\)

\(C(6,C(\Omega,0))=\varepsilon_0+\omega^6\)

\(C(7,C(\Omega,0))=\varepsilon_0+\omega^7\)

\(C(8,C(\Omega,0))=\varepsilon_0+\omega^8\)

\(C(9,C(\Omega,0))=\varepsilon_0+\omega^9\)

\(C(10,C(\Omega,0))=\varepsilon_0+\omega^{10}\)

\(C(11,C(\Omega,0))=\varepsilon_0+\omega^{11}\)

\(C(12,C(\Omega,0))=\varepsilon_0+\omega^{12}\)

\(C(C(1,0),C(\Omega,0))=\varepsilon_0+\omega^{\omega}\)

\(C(1,C(C(1,0),C(\Omega,0)))=\varepsilon_0+\omega^{\omega}+\omega\)

\(C(2,C(C(1,0),C(\Omega,0)))=\varepsilon_0+\omega^{\omega}+\omega^2\)

\(C(3,C(C(1,0),C(\Omega,0)))=\varepsilon_0+\omega^{\omega}+\omega^3\)

\(C(4,C(C(1,0),C(\Omega,0)))=\varepsilon_0+\omega^{\omega}+\omega^4\)

\(C(5,C(C(1,0),C(\Omega,0)))=\varepsilon_0+\omega^{\omega}+\omega^5\)

\(C(6,C(C(1,0),C(\Omega,0)))=\varepsilon_0+\omega^{\omega}+\omega^6\)

\(C(7,C(C(1,0),C(\Omega,0)))=\varepsilon_0+\omega^{\omega}+\omega^7\)

\(C(C(1,0),C(C(1,0),C(\Omega,0)))=\varepsilon_0+\omega^{\omega}2\)

\(C(1,C(C(1,0),C(C(1,0),C(\Omega,0))))=\varepsilon_0+\omega^{\omega}2+\omega\)

\(C(2,C(C(1,0),C(C(1,0),C(\Omega,0))))=\varepsilon_0+\omega^{\omega}2+\omega^2\)

\(C(3,C(C(1,0),C(C(1,0),C(\Omega,0))))=\varepsilon_0+\omega^{\omega}2+\omega^3\)

\(C(4,C(C(1,0),C(C(1,0),C(\Omega,0))))=\varepsilon_0+\omega^{\omega}2+\omega^4\)

\(C(5,C(C(1,0),C(C(1,0),C(\Omega,0))))=\varepsilon_0+\omega^{\omega}2+\omega^5\)

\(C(6,C(C(1,0),C(C(1,0),C(\Omega,0))))=\varepsilon_0+\omega^{\omega}2+\omega^6\)

\(C(7,C(C(1,0),C(C(1,0),C(\Omega,0))))=\varepsilon_0+\omega^{\omega}2+\omega^7\)

\(C(C(1,0),C(C(1,0),C(C(1,0),C(\Omega,0))))=\varepsilon_0+\omega^{\omega}3\)

\(C(1,C(C(1,0),C(C(1,0),C(C(1,0),C(\Omega,0)))))=\varepsilon_0+\omega^{\omega}3+\omega\)

\(C(2,C(C(1,0),C(C(1,0),C(C(1,0),C(\Omega,0)))))=\varepsilon_0+\omega^{\omega}3+\omega^2\)

\(C(3,C(C(1,0),C(C(1,0),C(C(1,0),C(\Omega,0)))))=\varepsilon_0+\omega^{\omega}3+\omega^3\)

\(C(4,C(C(1,0),C(C(1,0),C(C(1,0),C(\Omega,0)))))=\varepsilon_0+\omega^{\omega}3+\omega^4\)

\(C(5,C(C(1,0),C(C(1,0),C(C(1,0),C(\Omega,0)))))=\varepsilon_0+\omega^{\omega}3+\omega^5\)

\(C(6,C(C(1,0),C(C(1,0),C(C(1,0),C(\Omega,0)))))=\varepsilon_0+\omega^{\omega}3+\omega^6\)

\(C(7,C(C(1,0),C(C(1,0),C(C(1,0),C(\Omega,0)))))=\varepsilon_0+\omega^{\omega}3+\omega^7\)

\(C(C(1,0),C(C(1,0),C(C(1,0),C(C(1,0),C(\Omega,0)))))=\varepsilon_0+\omega^{\omega}4\)

\(C(C(1,0),C(C(1,0),C(C(1,0),C(C(1,0),C(C(1,0),C(\Omega,0))))))=\varepsilon_0+\omega^{\omega}5\)

\(C(C(1,0),C(C(1,0),C(C(1,0),C(C(1,0),C(C(1,0),C(C(1,0),C(\Omega,0)))))))=\varepsilon_0+\omega^{\omega}6\)

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

\(\varepsilon_0+\omega^{\omega}7\)

\(C(C(0,C(1,0)),C(\Omega,0))=\varepsilon_0+\omega^{\omega+1}\)

\(C(C(0,C(0,C(1,0))),C(\Omega,0))=\varepsilon_0+\omega^{\omega+2}\)

\(C(C(0,C(0,C(0,C(1,0)))),C(\Omega,0))=\varepsilon_0+\omega^{\omega+3}\)

\(C(C(0,C(0,C(0,C(0,C(1,0))))),C(\Omega,0))=\varepsilon_0+\omega^{\omega+4}\)

\(C(C(0,C(0,C(0,C(0,C(0,C(1,0)))))),C(\Omega,0))=\varepsilon_0+\omega^{\omega+5}\)

\(C(C(0,C(0,C(0,C(0,C(0,C(0,C(1,0))))))),C(\Omega,0))=\varepsilon_0+\omega^{\omega+6}\)

\(C(C(0,C(0,C(0,C(0,C(0,C(0,C(0,C(1,0)))))))),C(\Omega,0))=\varepsilon_0+\omega^{\omega+7}\)

\(C(C(1,C(1,0)),C(\Omega,0))=\varepsilon_0+\omega^{\omega2}\)

\(C(C(1,C(1,C(1,0))),C(\Omega,0))=\varepsilon_0+\omega^{\omega3}\)

\(C(C(1,C(1,C(1,C(1,0)))),C(\Omega,0))=\varepsilon_0+\omega^{\omega4}\)

\(C(C(1,C(1,C(1,C(1,C(1,0))))),C(\Omega,0))=\varepsilon_0+\omega^{\omega5}\)

\(C(C(1,C(1,C(1,C(1,C(1,C(1,0)))))),C(\Omega,0))=\varepsilon_0+\omega^{\omega6}\)

\(C(C(1,C(1,C(1,C(1,C(1,C(1,C(1,0))))))),C(\Omega,0))=\varepsilon_0+\omega^{\omega7}\)

\(C(C(2,0),C(\Omega,0))=\varepsilon_0+\omega^{\omega^2}\)

\(C(C(1,C(2,0)),C(\Omega,0))=\varepsilon_0+\omega^{\omega^2+\omega}\)

\(C(C(1,C(1,C(2,0))),C(\Omega,0))=\varepsilon_0+\omega^{\omega^2+\omega2}\)

\(C(C(1,C(1,C(1,C(2,0)))),C(\Omega,0))=\varepsilon_0+\omega^{\omega^2+\omega3}\)

\(C(C(1,C(1,C(1,C(1,C(2,0))))),C(\Omega,0))=\varepsilon_0+\omega^{\omega^2+\omega4}\)

\(C(C(1,C(1,C(1,C(1,C(1,C(2,0)))))),C(\Omega,0))=\varepsilon_0+\omega^{\omega^2+\omega5}\)

\(C(C(1,C(1,C(1,C(1,C(1,C(1,C(2,0))))))),C(\Omega,0))=\varepsilon_0+\omega^{\omega^2+\omega6}\)

\(C(C(1,C(1,C(1,C(1,C(1,C(1,C(1,C(2,0)))))))),C(\Omega,0))=\varepsilon_0+\omega^{\omega^2+\omega7}\)

\(C(C(2,C(2,0)),C(\Omega,0))=\varepsilon_0+\omega^{\omega^22}\)

\(C(C(2,C(2,C(2,0))),C(\Omega,0))=\varepsilon_0+\omega^{\omega^23}\)

\(C(C(2,C(2,C(2,C(2,0)))),C(\Omega,0))=\varepsilon_0+\omega^{\omega^24}\)

\(C(C(2,C(2,C(2,C(2,C(2,0))))),C(\Omega,0))=\varepsilon_0+\omega^{\omega^25}\)

\(C(C(2,C(2,C(2,C(2,C(2,C(2,0)))))),C(\Omega,0))=\varepsilon_0+\omega^{\omega^26}\)

\(C(C(2,C(2,C(2,C(2,C(2,C(2,C(2,0))))))),C(\Omega,0))=\varepsilon_0+\omega^{\omega^27}\)

\(C(C(3,0),C(\Omega,0))=\varepsilon_0+\omega^{\omega^3}\)

\(C(C(3,C(3,0)),C(\Omega,0))=\varepsilon_0+\omega^{\omega^32}\)

\(C(C(3,C(3,C(3,0))),C(\Omega,0))=\varepsilon_0+\omega^{\omega^33}\)

\(C(C(3,C(3,C(3,C(3,0)))),C(\Omega,0))=\varepsilon_0+\omega^{\omega^34}\)

\(C(C(3,C(3,C(3,C(3,C(3,0))))),C(\Omega,0))=\varepsilon_0+\omega^{\omega^35}\)

\(C(C(3,C(3,C(3,C(3,C(3,C(3,0)))))),C(\Omega,0))=\varepsilon_0+\omega^{\omega^36}\)

\(C(C(3,C(3,C(3,C(3,C(3,C(3,C(3,0))))))),C(\Omega,0))=\varepsilon_0+\omega^{\omega^37}\)

\(C(C(4,0),C(\Omega,0))=\varepsilon_0+\omega^{\omega^4}\)

\(C(C(5,0),C(\Omega,0))=\varepsilon_0+\omega^{\omega^5}\)

\(C(C(6,0),C(\Omega,0))=\varepsilon_0+\omega^{\omega^6}\)

\(C(C(7,0),C(\Omega,0))=\varepsilon_0+\omega^{\omega^7}\)

\(C(C(8,0),C(\Omega,0))=\varepsilon_0+\omega^{\omega^8}\)

\(C(C(9,0),C(\Omega,0))=\varepsilon_0+\omega^{\omega^9}\)

\(C(C(10,0),C(\Omega,0))=\varepsilon_0+\omega^{\omega^{10}}\)

\(C(C(C(1,0),0),C(\Omega,0))=\varepsilon_0+\omega^{\omega^{\omega}}\)

\(C(C(C(C(1,0),0),0),C(\Omega,0))=\varepsilon_0+\omega^{\omega^{\omega^{\omega}}}\)

\(C(C(C(C(C(1,0),0),0),0),C(\Omega,0))=\varepsilon_0+\omega^{\omega^{\omega^{\omega^{\omega}}}}\)

\(C(C(C(C(C(C(1,0),0),0),0),0),C(\Omega,0))=\varepsilon_0+\omega^{\omega^{\omega^{\omega^{\omega^{\omega}}}}}\)

\(C(C(C(C(C(C(C(1,0),0),0),0),0),0),C(\Omega,0))=\varepsilon_0+\omega^{\omega^{\omega^{\omega^{\omega^{\omega^{\omega}}}}}}\)

\(C(C(C(C(C(C(C(C(1,0),0),0),0),0),0),0),C(\Omega,0))=\varepsilon_0+\omega^{\omega^{\omega^{\omega^{\omega^{\omega^{\omega^{\omega}}}}}}}\)

\(C(C(C(C(C(C(C(C(C(1,0),0),0),0),0),0),0),0),C(\Omega,0))=\varepsilon_0+\omega^{\omega^{\omega^{\omega^{\omega^{\omega^{\omega^{\omega^{\omega}}}}}}}}\)

\(C(C(\omega,0),C(\Omega,0))=\varepsilon_02\)

\(C(0,C(C(\omega,0),C(\Omega,0)))=\varepsilon_02+1\)

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''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.''