User blog:B1mb0w/Fundamental Sequences

Fundamental Sequences
This blog will map out much of the standard definitions on Fundamental Sequences for Ordinals. It will attempt to simplify the rule-set where possible.

Basics (Cantor's Normal Form)
Let \(\gamma\) and \(\delta\) be two arbitrary transfinite ordinals, and \(n\) is a finite integer. Then:

\((\gamma + 1)[n] = \gamma\)

\((\gamma + \delta)[n] = \gamma + \delta[n]\) when \(\gamma >> \delta\)

\(\gamma.(\delta + 1)[n] = \gamma.\delta + \gamma[n]\)

\(\gamma.\delta[n] = \gamma.(\delta[n])\) when \(\gamma >> \delta\)

\(\gamma^{\delta + 1}[n] = \gamma^{\delta}.(\gamma[n])\)

and

\(\gamma^{\delta}[n] = \gamma^{\delta[n]}\)

Some Common Transfinite Ordinals
\(\omega[n] = n\)

\(\epsilon_0[n] = \omega\uparrow\uparrow n\)

\(\epsilon_1[n] = \epsilon_0\uparrow\uparrow n\)

\(\epsilon_{j+1}[n] = \epsilon_j\uparrow\uparrow n\)

and

\(\epsilon_{\omega}[n] = \epsilon_{\omega[n]} = \epsilon_n\)

Veblen Hierarchy
Continuing into Veblen Hierarchy and the \(\varphi\) function. Lets start with these equations which are equivalent to those in the Common Transfinite Ordinal section.

\(\varphi(1)[n] = \omega[n] = n\)

\(\varphi(1,0)[n] = \epsilon_0[n] = \varphi(n) = \omega\uparrow\uparrow n\)

\(\varphi(1,1)[n] = \epsilon_1[n] = \varphi(1,0)\uparrow\uparrow n\)

\(\varphi(1,j + 1)[n] = \epsilon_{j + 1}[n] = \varphi(1,j)\uparrow\uparrow n\)

and

\(\varphi(1,\omega)[n] = \varphi(1,\omega[n]) = \varphi(1,n)\)

Rule-set
At this point, we can focus on the commonly used rule-sets and generalise all the above and more. Let's allow \(\alpha\) and \(\beta\) to be transfinite ordinals. Then the following rules seem to apply but they do not appear to be self-consistent.

\(\varphi(\alpha,\beta)[n] = \varphi(\alpha,\beta[n])\)

\(\varphi(\alpha,\beta + 1)[n] = \varphi(\alpha[n],\varphi(\alpha,\beta)+1)\)

\(\varphi(\alpha,0)[n] = \varphi(\alpha[n],0)\)

and

\(\varphi(\alpha + 1,0)[0] = 0\)

\(\varphi(\alpha + 1,0)[n + 1] = \varphi(\alpha,\varphi(\alpha + 1,0)[n] + c)\) where \(c = 0\) or \(1\) ???

and

\(\varphi(\alpha + 1,\beta + 1)[0] = \varphi(\alpha + 1,\beta) + 1\) but why ???

\(\varphi(\alpha + 1,\beta + 1)[n + 1] = \varphi(\alpha,\varphi(\alpha + 1,\beta + 1)[n])\)

And, the following rules do not seem to be consistent with the general rules just given.

\(\varphi(0,\beta)[n] = \varphi(0,\beta[n]) = \varphi(\beta[n])\)

\(\varphi(0,\beta + 1)[n] = \varphi(\beta + 1)[n] = \varphi(\beta)^n\)

Calculated Example
What is the fundamental sequence for \(\zeta_0[2]\) ?

\(\zeta_0[2] = \varphi(2,0)[2] = \varphi(1,\varphi(2,0)[1] + c)\)

\(= \varphi(1,\varphi(1,\varphi(2,0)[0] + c) + c) = \varphi(1,\varphi(1,0 + c) + c)\)

then if c = 0

\(\zeta_0[2] = \varphi(2,0)[2] = \varphi(1,\varphi(1,0)) = \epsilon_{\epsilon_0}\)

of if c = 1

\(\zeta_0[2] = \varphi(2,0)[2] = \varphi(1,\varphi(1,1) + 1) = \epsilon_{\epsilon_1+1}\)