User blog:Rgetar/Clested fundamental sequence systems

I wrote a program to make lists of ordinals such as this, and it turned out that in that list was used not modified 2nd fundamental sequence system, but a slightly different system, which I named "clested" modified 2nd system.

Also, in this blog I described special case not for 6th system, but for clested 6th system.

The difference between "ordinary" and "clested" systems is that in rules of "clested" systems in expressions for φ(X)[n] "lest" is replaced by "clest". (Only in φ(X)[n], in X[n]α still "lest").

Definition of clest
clest means "conditionally last element set to"

\(clest(X; α) = \left\{\begin{array}{lcr} X \qquad \qquad \, \text{if} \; α = 0\\ lest(X; α) \quad \text{if} \; α ≠ 0\\ \end{array}\right. \)

(I already added clest to my list of definitions).

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

X[n]α
(Here and below "l." means "limit ordinal", and "s." means "successor ordinal").

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

φ(X)[n]
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:

2nd system
\(φ(X)[n] = \left\{\begin{array}{lcr} φ(X-1)·n \quad \text{if} \; X \; \text{- s.}\\ \left.\begin{array}{lcr} φ(X[n]_0) \quad \text{if} \; leo(X) \; \text{- l.}\\ \left.\begin{array}{lcr} φ(lest(X^0[n]_0; δ)) \quad \text{if} \; ilbeo(X_0) ∋ \; \text{l.}\\ \left.\begin{array}{lcr} φ(X^0[n]_{φ(X)[n-1]}) \quad \text{if} \; n>0\\ φ(X^0[n]_δ) \qquad \qquad \text{if} \; n=0\\ \end{array}\right\} \; \text{if} \; ilbeo(X_0) ∌ \; \text{l.}\\ \end{array}\right\} \; \text{if} \; leo(X) \; \text{- not l.}\\ \end{array}\right\} \; \text{if} \; X \; \text{- not s.}\\ \end{array}\right. \)

clested 2nd system
\(φ(X)[n] = \left\{\begin{array}{lcr} φ(X-1)·n \quad \text{if} \; X \; \text{- s.}\\ \left.\begin{array}{lcr} φ(X[n]_0) \quad \text{if} \; leo(X) \; \text{- l.}\\ \left.\begin{array}{lcr} φ(clest(X^0[n]_0; δ)) \quad \text{if} \; ilbeo(X_0) ∋ \; \text{l.}\\ \left.\begin{array}{lcr} φ(X^0[n]_{φ(X)[n-1]}) \quad \text{if} \; n>0\\ φ(X^0[n]_δ) \qquad \qquad \text{if} \; n=0\\ \end{array}\right\} \; \text{if} \; ilbeo(X_0) ∌ \; \text{l.}\\ \end{array}\right\} \; \text{if} \; leo(X) \; \text{- not l.}\\ \end{array}\right\} \; \text{if} \; X \; \text{- not s.}\\ \end{array}\right. \)

6th system
\(φ(X)[n] = \left\{\begin{array}{lcr} φ(X-1)·n \quad \text{if} \; X \; \text{- s.}\\ \left.\begin{array}{lcr} φ(X[n]_0) \quad \text{if} \; leo(X) \; \text{- l.}\\ \left.\begin{array}{lcr} φ(lest(X^0[n]_0; δ)) \quad \text{if} \; ilbeo(X_0) ∋ \; \text{l.} \; or \; n=0\\ φ(X^0[n]_{φ(X)[n-1]}) \quad \text{if} \; ilbeo(X_0) ∌ \; \text{l.} \; and \; n>0\\ \end{array}\right\} \; \text{if} \; leo(X) \; \text{- not l.}\\ \end{array}\right\} \; \text{if} \; X \; \text{- not s.}\\ \end{array}\right. \)

clested 6th system
\(φ(X)[n] = \left\{\begin{array}{lcr} φ(X-1)·n \quad \text{if} \; X \; \text{- s.}\\ \left.\begin{array}{lcr} φ(X[n]_0) \quad \text{if} \; leo(X) \; \text{- l.}\\ \left.\begin{array}{lcr} φ(clest(X^0[n]_0; δ)) \quad \text{if} \; ilbeo(X_0) ∋ \; \text{l.} \; or \; n=0\\ φ(X^0[n]_{φ(X)[n-1]}) \quad \text{if} \; ilbeo(X_0) ∌ \; \text{l.} \; and \; n>0\\ \end{array}\right\} \; \text{if} \; leo(X) \; \text{- not l.}\\ \end{array}\right\} \; \text{if} \; X \; \text{- not s.}\\ \end{array}\right. \)

Other systems
There is also clested version for 5th, but there are no different clested versions for 1st, 3rd, 4th systems since they does not contain "lest" in expressions for φ(X)[n].

Modified versions
"Modified" version of a fundamental sequence system is this system, but if fundamental sequence is defined through another fundamental sequence, the modified version is used, and if fundamental sequence contains not limit ordinals, then, beginning from some element, only limit ordinals, then this fundamental sequence starts from this element. (That is this new fundamental sequence contains only limit ordinals).