User blog:Rgetar/New fundamental sequence system for generalized Veblen function

Today I made up new fundamental sequence system for generalized Veblen function. It can be named "6th system". Designations see here.

6th system
\(δ = \left\{\begin{array}{lcr} 0 \qquad \qquad \qquad \, \text{if} \; leo(X) = 0\\ φ(X-1) + 1 \quad \text{if} \; leo(X) ≠ 0\\ \end{array}\right. \)

\(X[n]_α = \left\{\begin{array}{lcr} lbest(X; lbeo(X)[n]) \quad \text{if} \; lbeo(X) \; \text{- l.}\\ \left.\begin{array}{lcr} lbest(X; lbeo(X)-1), \langle X'[n]_α\rangle 1 \quad \text{if} \; leo(X') \; \text{- not s.}\\ lbest(X; lbeo(X)-1), \langle X'-1\rangle α \quad \text{if} \; leo(X') \; \text{- s.}\\ \end{array}\right\} \; \text{if} \; lbeo(X) \; \text{- s.}\\ \end{array}\right. \)

To get fundamental sequence of Cantor normal form, replace its last term with fundamental sequence of the last term.

Fundamental sequence for Cantor normal form term:

\(φ(X)[n] = \left\{\begin{array}{lcr} φ(X-1)·n \quad \text{if} \; X \; \text{- s.}\\ \left.\begin{array}{lcr} φ(X[n]_0) \quad \text{if} \; leo(X) \; \text{- l.}\\ \left.\begin{array}{lcr} φ(lest(X^0[n]_0; δ)) \quad \text{if} \; ilbeo(X_0) ∋ \; \text{l.} \; or \; n=0\\ φ(X^0[n]_{φ(X)[n-1]}) \quad \text{if} \; ilbeo(X_0) ∌ \; \text{l.} \; and \; n>0\\ \end{array}\right\} \; \text{if} \; leo(X) \; \text{- not l.}\\ \end{array}\right\} \; \text{if} \; X \; \text{- not s.}\\ \end{array}\right. \)

2nd system
The old (2nd) system:

\(δ = \left\{\begin{array}{lcr} 0 \qquad \qquad \qquad \, \text{if} \; leo(X) = 0\\ φ(X-1) + 1 \quad \text{if} \; leo(X) ≠ 0\\ \end{array}\right. \)

\(X[n]_α = \left\{\begin{array}{lcr} lbest(X; lbeo(X)[n]) \quad \text{if} \; lbeo(X) \; \text{- l.}\\ \left.\begin{array}{lcr} lbest(X; lbeo(X)-1), \langle X'[n]_α\rangle 1 \quad \text{if} \; leo(X') \; \text{- not s.}\\ lbest(X; lbeo(X)-1), \langle X'-1\rangle α \quad \text{if} \; leo(X') \; \text{- s.}\\ \end{array}\right\} \; \text{if} \; lbeo(X) \; \text{- s.}\\ \end{array}\right. \)

To get fundamental sequence of Cantor normal form, replace its last term with fundamental sequence of the last term.

Fundamental sequence for Cantor normal form term:

\(φ(X)[n] = \left\{\begin{array}{lcr} φ(X-1)·n \quad \text{if} \; X \; \text{- s.}\\ \left.\begin{array}{lcr} φ(X[n]_0) \quad \text{if} \; leo(X) \; \text{- l.}\\ \left.\begin{array}{lcr} φ(lest(X^0[n]_0; δ)) \quad \text{if} \; ilbeo(X_0) ∋ \; \text{l.}\\ \left.\begin{array}{lcr} φ(X^0[n]_{φ(X)[n-1]}) \quad \text{if} \; n>0\\ φ(X^0[n]_δ) \qquad \qquad \text{if} \; n=0\\ \end{array}\right\} \; \text{if} \; ilbeo(X_0) ∌ \; \text{l.}\\ \end{array}\right\} \; \text{if} \; leo(X) \; \text{- not l.}\\ \end{array}\right\} \; \text{if} \; X \; \text{- not s.}\\ \end{array}\right. \)

Difference between the systems
Example of difference between the systems:

φ(8, 0, 5)

Fundamental sequence in the 6th system:

φ(8, 0, 6)[0] = φ(7, 0, φ(8, 0, 5) + 1)

φ(8, 0, 6)[1] = φ(7, φ(7, 0, φ(8, 0, 5) + 1), 0)

φ(8, 0, 6)[2] = φ(7, φ(7, φ(7, 0, φ(8, 0, 5) + 1), 0), 0)

φ(8, 0, 6)[3] = φ(7, φ(7, φ(7, φ(7, 0, φ(8, 0, 5) + 1), 0), 0), 0)

φ(8, 0, 6)[4] = φ(7, φ(7, φ(7, φ(7, φ(7, 0, φ(8, 0, 5) + 1), 0), 0), 0), 0)

φ(8, 0, 6)[5] = φ(7, φ(7, φ(7, φ(7, φ(7, φ(7, 0, φ(8, 0, 5) + 1), 0), 0), 0), 0), 0)

...

Fundamental sequence in the 2nd system:

φ(8, 0, 6)[0] = φ(7, φ(8, 0, 5) + 1, 0)

φ(8, 0, 6)[1] = φ(7, φ(7, φ(8, 0, 5) + 1, 0), 0)

φ(8, 0, 6)[2] = φ(7, φ(7, φ(7, φ(8, 0, 5) + 1, 0), 0), 0)

φ(8, 0, 6)[3] = φ(7, φ(7, φ(7, φ(7, φ(8, 0, 5) + 1, 0), 0), 0), 0)

φ(8, 0, 6)[4] = φ(7, φ(7, φ(7, φ(7, φ(7, φ(8, 0, 5) + 1, 0), 0), 0), 0), 0)

φ(8, 0, 6)[5] = φ(7, φ(7, φ(7, φ(7, φ(7, φ(7, φ(8, 0, 5) + 1, 0), 0), 0), 0), 0), 0)

...

In my opinion, the 6th system is simpler than the 2nd, and from now on I'll use the 6th system.

ilbeo
And, I found an error in my definitions of fundamental sequence systems: I wrote "ileo", but should be "ilbeo", that is "iterated last base element of".

ilbeo(X) = lbeo(X) ∪ ilbeo(X')

that is ilbeo(X) is set

{lbeo(X), lbeo(X'), lbeo(X"), ...}

(whereas ileo(X) is set {leo(X), leo(X'), leo(X"), ...})

I already fixed this error in my previous blogs.