User blog:Rgetar/Fundamental sequence system for generalized Veblen function

This blog was inspired by Googology Wiki and Wikipedia articles about Veblen function.

Today I created fundamental sequence system for my generalized Veblen function.

New definition
Also, I made up simpler definition of this function:

φ(X) is ordinal α such as α is (1 + leo(X))-th common fixed point of all functions α = φ(Y), Y ∈ X0{·}α.

(X, Y are arrays of ordinals).

By the way, X0{·}α also may be defined simpler way:

X0{·}α is set of all such arrays of ordinals: lest(X*; β)&lt;X'&gt;(X'; 1; α)&lt;Y&gt;, β < leo(X*), Y ∈ X'{·}α.

(Here I use my new array notation).

Fundamental sequence system
I tried to create a simple fundamental sequence system.

Designations
α[n] is n-th ordinal of fundamental sequence of ordinal α (in some defined before fundamental sequence system, the fundamental sequence system is not designated in α[n], but implied)

ileo means "iterated leo"

ileo(X) = {leo(X)} ∪ ileo(X')

(That is ileo of array X is set of ordinals {leo(X); leo(X'); leo(X")...}).

δ = 0, if leo(X) = 0

δ = φ(X-1) + 1, if leo(X) is successor ordinal

X[n]α = lest(X; leo(X)[n]), if X' = 0

X[n]α = X*[n]α, if X' ≠ 0, leo(X*) is limit ordinal

X[n]α = X*-11, if X' ≠ 0, leo(X*) is successor ordinal, leo(X') is limit ordinal or zero

X[n]α = X*-1α, if leo(X*), leo(X') are successor ordinals

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

2. φ(α-1)[n] = φ(α)·n

3. φ(X)[n] = φ(X[n]0), if leo(X) is limit ordinal

4. φ(X)[n] = φ(lest(X0[n]0; δ)), if leo(X) is successor ordinal or zero, ileo(X0) ∋ limit ordinal

5. φ(X)[n] = φ(X0[n]φ(X)[n-1]) for n > 0 and φ(X)[n] = δ, if leo(X) is successor ordinal or zero, ileo(X0) ∌ limit ordinal

Note: in Rules 3, 4 subscript 0 does not matter and may be replaced with any ordinal.

Сomparison
Let 1st system is fundamental sequence system from Veblen function article, 2nd system is my fundamental sequence system.

They are similar, but somewhat different from each other. Let's compare them.

Finitely many variables
2.1) \((\varphi(s_1)+\varphi(s_2)+\cdots+\varphi(s_k))[n]=\varphi(s_1)+\varphi(s_2)+\cdots+\varphi(s_k)[n]\),

(same)

2.2) \(\varphi(\gamma)[n]=\left\{\begin{array}{lcr} n \quad \text{if} \quad \gamma=1\\ \varphi(\gamma-1)\cdot n \quad \text{if} \quad \gamma \quad \text{is a successor ordinal}\\ \varphi(\gamma[n]) \quad \text{if} \quad \gamma \quad \text{is a limit ordinal}\\ \end{array}\right. \),

(same)

2.3) \(\varphi(s,\beta,z,\gamma)[0]=0\) and \(\varphi(s,\beta,z,\gamma)[n+1]=\varphi(s,\beta-1,\varphi(s,\beta,z,\gamma)[n],z)\) if \(\gamma=0\) and \(\beta\) is a successor ordinal,

(same)

2.4) \(\varphi(s,\beta,z,\gamma)[0]=\varphi(s,\beta,z,\gamma-1)+1\) and \(\varphi(s,\beta,z,\gamma)[n+1]=\varphi(s,\beta-1,\varphi(s,\beta,z,\gamma)[n],z)\) if \(\gamma\) and \(\beta\) are successor ordinals,

(same)

2.5) \(\varphi(s,\beta,z,\gamma)[n]=\varphi(s,\beta,z,\gamma[n])\) if \(\gamma\) is a limit ordinal,

(same)

2.6) \(\varphi(s,\beta,z,\gamma)[n]=\varphi(s,\beta[n],z,\gamma)\) if \(\gamma=0\) and \(\beta\) is a limit ordinal,

(same)

2.7) \(\varphi(s,\beta,z,\gamma)[n]=\varphi(s,\beta[n],\varphi(s,\beta,z,\gamma-1)+1,z)\) if \(\gamma\) is a successor ordinal and \(\beta\) is a limit ordinal.

(in 1st system)

2.7) \(\varphi(s,\beta,z,\gamma)[n]=\varphi(s,\beta[n],z,\varphi(s,\beta,z,\gamma-1)+1)\) if \(\gamma\) is a successor ordinal and \(\beta\) is a limit ordinal.

(in 2nd system)

Transfinitely many variables
3.1) \(\begin{pmatrix}\cdots & \alpha+1 \\ \cdots & \beta+1 \end{pmatrix}[0]=0\)

and \(\begin{pmatrix}\cdots & \alpha+1 \\ \cdots & \beta+1 \end{pmatrix}[n+1]=\begin{pmatrix}\cdots & \alpha & \begin{pmatrix}\cdots & \alpha+1 \\ \cdots & \beta+1 \end{pmatrix}[n] \\ \cdots & \beta+1 & \beta \end{pmatrix}\),

(same)

3.2) \(\begin{pmatrix}\cdots & \alpha+1 & \gamma+1 \\ \cdots & \beta+1 & 0 \end{pmatrix}[0]=\begin{pmatrix}\cdots & \alpha+1 & \gamma \\ \cdots & \beta+1 & 0 \end{pmatrix}+1\)

and \(\begin{pmatrix}\cdots & \alpha+1 & \gamma+1 \\ \cdots & \beta+1 & 0 \end{pmatrix}[n+1]=\begin{pmatrix}\cdots & \alpha & \begin{pmatrix}\cdots & \alpha+1 & \gamma+1 \\ \cdots & \beta+1 & 0 \end{pmatrix}[n] \\ \cdots & \beta+1 & \beta \end{pmatrix}\),

(same)

3.3) \(\begin{pmatrix}\cdots & \alpha & \gamma \\ \cdots & \beta & 0 \end{pmatrix}[n]=\begin{pmatrix}\cdots & \alpha & \gamma [n] \\ \cdots & \beta & 0 \end{pmatrix}\) if \(\gamma\) is a limit ordinal,

(same)

3.4) \(\begin{pmatrix}\cdots & \alpha & \\ \cdots & \beta+1 \end{pmatrix}[n]=\begin{pmatrix}\cdots & \alpha [n] & \\ \cdots & \beta+1 \end{pmatrix}\) if \(\alpha\) is a limit ordinal,

(same)

3.5) \(\begin{pmatrix}\cdots & \alpha & \gamma+1 \\ \cdots & \beta+1 & 0 \end{pmatrix}[n]=\begin{pmatrix}\cdots & \alpha [n] & \begin{pmatrix}\cdots & \alpha & \gamma \\ \cdots & \beta+1 & 0 \end{pmatrix}+1 \\ \cdots & \beta+1 & \beta \end{pmatrix}\) if \(\alpha\) is a limit ordinal,

(in 1st system)

3.5) \(\begin{pmatrix}\cdots & \alpha & \gamma+1 \\ \cdots & \beta+1 & 0 \end{pmatrix}[n]=\begin{pmatrix}\cdots & \alpha [n] & \begin{pmatrix}\cdots & \alpha & \gamma \\ \cdots & \beta+1 & 0 \end{pmatrix}+1 \\ \cdots & \beta+1 & 0 \end{pmatrix}\) if \(\alpha\) is a limit ordinal,

(in 2nd system)

3.6) \(\begin{pmatrix}\cdots & \alpha+1\\ \cdots & \beta\end{pmatrix}[n]=\begin{pmatrix}\cdots & \alpha & 1 \\ \cdots & \beta& \beta [n]\end{pmatrix}\) if \(\beta\) is a limit ordinal,

(same)

3.7) \(\begin{pmatrix}\cdots & \alpha+1 & \gamma+1 \\ \cdots & \beta & 0 \end{pmatrix}[n]=\begin{pmatrix}\cdots & \alpha & \begin{pmatrix}\cdots & \alpha+1 & \gamma \\ \cdots & \beta & 0 \end{pmatrix}+1 \\ \cdots & \beta & \beta[n] \end{pmatrix}\) if \(\beta\) is a limit ordinal,

(in 1st system)

3.7) \(\begin{pmatrix}\cdots & \alpha+1 & \gamma+1 \\ \cdots & \beta & 0 \end{pmatrix}[n]=\begin{pmatrix}\cdots & \alpha & 1 & \begin{pmatrix}\cdots & \alpha+1 & \gamma \\ \cdots & \beta & 0 \end{pmatrix}+1 \\ \cdots & \beta & \beta[n] & 0 \end{pmatrix}\) if \(\beta\) is a limit ordinal,

(in 2nd system)

3.8) \(\begin{pmatrix}\cdots & \alpha\\ \cdots & \beta\end{pmatrix}[n]=\begin{pmatrix}\cdots & \alpha [n] \\ \cdots & \beta \end{pmatrix}\) if \(\alpha\) and \(\beta\) are limit ordinals,

(same)

3.9) \(\begin{pmatrix}\cdots & \alpha & \gamma+1 \\ \cdots & \beta & 0 \end{pmatrix}[n]=\begin{pmatrix}\cdots & \alpha [n]& \begin{pmatrix}\cdots & \alpha & \gamma \\ \cdots & \beta & 0 \end{pmatrix}+1 \\ \cdots & \beta & \beta [n] \end{pmatrix}\) if \(\alpha\) and \(\beta\) are limit ordinals.

(in 1st system)

3.9) \(\begin{pmatrix}\cdots & \alpha & \gamma+1 \\ \cdots & \beta & 0 \end{pmatrix}[n]=\begin{pmatrix}\cdots & \alpha [n]& \begin{pmatrix}\cdots & \alpha & \gamma \\ \cdots & \beta & 0 \end{pmatrix}+1 \\ \cdots & \beta & \0 \end{pmatrix}\) if \(\alpha\) and \(\beta\) are limit ordinals.

(in 2nd system)

Note: in 1st system Rules 3.4, 3.8 may be united, and in 2nd system Rules 3.4, 3.8 and Rules 3.5, 3.9 may be united.

3.4, 3.8) \(\begin{pmatrix}\cdots & \alpha\\ \cdots & \beta\end{pmatrix}[n]=\begin{pmatrix}\cdots & \alpha [n] \\ \cdots & \beta \end{pmatrix}\) if \(\alpha\) is a limit ordinal, \(\beta\) ≠ 0.

(in both systems)

3.5, 3.9) \(\begin{pmatrix}\cdots & \alpha & \gamma+1 \\ \cdots & \beta & 0 \end{pmatrix}[n]=\begin{pmatrix}\cdots & \alpha [n]& \begin{pmatrix}\cdots & \alpha & \gamma \\ \cdots & \beta & 0 \end{pmatrix}+1 \\ \cdots & \beta & \0 \end{pmatrix}\) if \(\alpha\) is a limit ordinal.

(in 2nd system)

Rules 3.5 / 2.7
φ(ω,0,1)[n]

In 1st system

φ(ω,0,1)[n] = φ(n,φ(ω,0,0)+1,0)

Fundamental sequence:

φ(φ(ω,0,0)+1,0)

φ(1,φ(ω,0,0)+1,0)

φ(2,φ(ω,0,0)+1,0)

φ(3,φ(ω,0,0)+1,0)

φ(4,φ(ω,0,0)+1,0)

φ(5,φ(ω,0,0)+1,0)

...

In 2nd system

φ(ω,0,1)[n] = φ(n,0,φ(ω,0,0)+1)

Fundamental sequence:

φ(φ(ω,0,0)+1)

φ(1,0,φ(ω,0,0)+1)

φ(2,0,φ(ω,0,0)+1)

φ(3,0,φ(ω,0,0)+1)

φ(4,0,φ(ω,0,0)+1)

φ(5,0,φ(ω,0,0)+1)

...

Rule 3.7
To simplify writing I will use my new array notation. It is equivalent to the Schutte Klammersymbolen, but in one row:

φ(α1<β1>α2<β2>α3<β3>α4<β4>α5<β5>) = \(\begin{pmatrix}\alpha_1 & \alpha_2 & \alpha_3 & \alpha_4 & \alpha_5 \\\beta_1 & \beta_2 & \beta_3 & \beta_4 & \beta_5 \end{pmatrix}\)

\(\begin{pmatrix} 1 & 1 \\ \omega & 0 \end{pmatrix}\) = φ(1&lt;ω&gt;1&lt;0&gt;) = φ(1&lt;ω&gt;1)

(<0> may be omitted)

In 1st system

\(\begin{pmatrix} 1 & 1 \\ \omega & 0 \end{pmatrix}[n]=\begin{pmatrix} \begin{pmatrix} 1 \\ \omega \end{pmatrix}+1 \\ n \end{pmatrix}\)

φ(1&lt;ω&gt;1)[n] = φ(φ(1&lt;ω&gt;)+1&lt;n&gt;)

Fundamental sequence:

φ(φ(1&lt;ω&gt;)+1)

φ(φ(1&lt;ω&gt;)+1,0)

φ(φ(1&lt;ω&gt;)+1,0,0)

φ(φ(1&lt;ω&gt;)+1,0,0,0)

φ(φ(1&lt;ω&gt;)+1,0,0,0,0)

φ(φ(1&lt;ω&gt;)+1,0,0,0,0,0)

...

In 2nd system

\(\begin{pmatrix} 1 & 1 \\ \omega & 0 \end{pmatrix}[n]=\begin{pmatrix} 1 & \begin{pmatrix} 1 \\ \omega \end{pmatrix}+1 \\ n & 0 \end{pmatrix}\)

φ(1&lt;ω&gt;1)[n] = φ(1&lt;n&gt;φ(1&lt;ω&gt;)+1)

Fundamental sequence:

φ(φ(1&lt;ω&gt;)+1)

φ(1,φ(1&lt;ω&gt;)+1)

φ(1,0,φ(1&lt;ω&gt;)+1)

φ(1,0,0,φ(1&lt;ω&gt;)+1)

φ(1,0,0,0,φ(1&lt;ω&gt;)+1)

φ(1,0,0,0,0,φ(1&lt;ω&gt;)+1)

...

Rule 3.9
\(\begin{pmatrix} \omega & 1 \\ \omega & 0 \end{pmatrix}\) = φ(ω&lt;ω&gt;1&lt;0&gt;) = φ(ω&lt;ω&gt;1)

In 1st system

\(\begin{pmatrix} \omega & 1 \\ \omega & 0 \end{pmatrix}\[n]=\begin{pmatrix} n & \begin{pmatrix} \omega \\ \omega \end{pmatrix}+1 \\ \omega & n \end{pmatrix}\)

φ(ω&lt;ω&gt;1)[n] = φ(n&lt;ω&gt;φ(ω&lt;ω&gt;)+1&lt;n&gt;)

Fundamental sequence:

φ(φ(ω&lt;ω&gt;)+1)

φ(1&lt;ω&gt;φ(ω&lt;ω&gt;)+1,0)

φ(2&lt;ω&gt;φ(ω&lt;ω&gt;)+1,0,0)

φ(3&lt;ω&gt;φ(ω&lt;ω&gt;)+1,0,0,0)

φ(4&lt;ω&gt;φ(ω&lt;ω&gt;)+1,0,0,0,0)

φ(5&lt;ω&gt;φ(ω&lt;ω&gt;)+1,0,0,0,0,0)

...

In 2nd system

\(\begin{pmatrix} \omega & 1 \\ \omega & 0 \end{pmatrix}\[n]=\begin{pmatrix} n & \begin{pmatrix} \omega \\ \omega \end{pmatrix}+1 \\ \omega & 0 \end{pmatrix}\)

φ(ω&lt;ω&gt;1)[n] = φ(n&lt;ω&gt;φ(ω&lt;ω&gt;)+1)

Fundamental sequence:

φ(φ(ω&lt;ω&gt;)+1)

φ(1&lt;ω&gt;φ(ω&lt;ω&gt;)+1)

φ(2&lt;ω&gt;φ(ω&lt;ω&gt;)+1)

φ(3&lt;ω&gt;φ(ω&lt;ω&gt;)+1)

φ(4&lt;ω&gt;φ(ω&lt;ω&gt;)+1)

φ(5&lt;ω&gt;φ(ω&lt;ω&gt;)+1)

...

Conclusion
I think that 2nd system is slightly simpler and more "symmetric" than 1st system, and here's why.

1. Rules 3.5 and 3.9 can be united in 2nd system, and cannot in 1st system, so, 2nd system has fewer rules.

2. Rules are in pairs: 3.4 and 3.5, 3.6 and 3.7, 3.8 and 3.9. In 1st system in locations where in 3.5, 3.7, 3.9 is φ(X-1) + 1, in other rules is 0 except of Rule 3.6, where is 1. In 2nd system in corresponding location in Rule 3.6 is also 0.

3. In Rules 3.5 and 3.9 0 is simpler than β and β[n]. In Rule 3.9 in 1st system n in right part appears two times, but in 2nd system n in right part always appears no more one time.