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Not to be confused with strong array notation.

$$\newcommand{\W}{\Omega} \newcommand{\w}{\omega} \newcommand{\p}{\psi} \newcommand{\a}{\alpha} \newcommand{\b}{\beta} \newcommand{\g}{\gamma}$$

The limit function $$g(n)$$ of 段階配列表記 is approximated to $$f_{\p_0(\W_\w)}(n)$$ using a specific system of fundamental sequences explained later.

Using the iteration of $$g(n)$$, mrna coined a large number $$g^{100}(100)$$ as 段階配列数.

## Definition

### Notation system $$S$$

We denote by $$\mathbb{N}$$ the set of non-negative integers. We define a set $$S$$ of formal strings of non-negative integers, $$($$, $$)$$, and $$+$$ in the following recursive way:

• Every $$n \in \mathbb{N}$$ belongs to $$S$$.
• For any $$\a, \b \in S$$, the formal string "$$\a+\b$$" belongs to $$S$$.
• For any $$\a\in S$$ and $$n \in \mathbb{N}$$, the formal string "$$n(\a)$$" belongs to $$S$$.

### Set $$P$$

We denote by $$P$$ the recursive subset of all $$\g \in S$$ satisfying the following:

• For any $$\a, \b \in S$$, $$\g \neq \a+\b$$.
• $$\g \neq 0$$.

### Set $$S_n$$

For each $$n \in \mathbb{N}$$, we define a set $$S$$ of formal strings of of non-negative integers, $$($$, $$)$$, $$+$$, and a fixed letter $$k$$ in the following recursive way:

• The letter $$k$$ belongs to $$S_n$$.
• For any $$\a \in S,~\b \in S_n$$, the formal string "$$\a+\b$$" belongs to $$S_n$$.
• For any $$\a \in S_n$$ and $$m \in \mathbb{N}$$, if $$n\leq m$$, then the formal string "$$m(\a)$$" belongs to $$S_n$$.

### Formal string $$S_{n,\a}(\b)$$

For any $$n \in \mathbb{N}$$, $$\a \in S_n$$, and formal string $$\b$$, we denote by $$S_{n,\a}(\b)$$ the formal string given by replacing all the occurrence of $$k$$ in $$\a$$ by $$\b$$.

### Map $$f(\a,n)$$

We denote by $$\mathbb{N}_+$$ the set of all positive integers. We define a map \begin{eqnarray*} f \colon S\times \mathbb{N}_+&\rightarrow& S\\ (\a,n) &\mapsto& f(\a,n) \end{eqnarray*} in the following recursive way:

• If $$\a\in\mathbb{N}$$, then
$$f(\a,n)=0$$.
• If there exists some $$\b \in S$$ such that $$\a=\b+0$$, then
$$f(\a,n)=\b$$.
• If there exist some $$\b\in S$$ and $$\g\in P$$ such that $$\a=\b+\g$$, then
$$f(\a,n)=\b+f(\g,n)$$.
• If there exist some $$a,b \in \mathbb{N}_+$$ such that $$a\geq b$$, $$\b\in S_0$$, and $$\a=a(S_{0,\b}(b))$$, then
$$f(\a,n)=0(S_{0,\b}(b))$$.
• If there exist some $$m \in \mathbb{N}$$ and $$\b \in S_0$$ such that $$\a=S_{0,\b}(m(0))$$, then
$$f(\a,n)=S_{0,\b}(\underbrace{m+m+\cdots+m}_{n})$$.
• If there exist some $$m \in \mathbb{N}$$, $$\b\in S_0$$, and $$\g\in S$$ such that $$\a=S_{0,\b}(m(\g+0))$$, then
$$f(\a,n)=S_{0,\b}(\underbrace{m(\g)+m(\g)+\cdots+m(\g)}_{n})$$.
• If there exist some $$a \in \mathbb{N}$$, $$b \in \mathbb{N}_+$$, and $$\b\in S_b$$ such that $$a\lt b$$ and $$\a=a(S_{b,\b}(b))$$, then
$$f(\a,n)=a(\underbrace{S_{b,\b}(b'(S_{b,\b}(b'(\cdots S_{b,\b}(b'}_{n~S_{b,\b}(b'{\rm s}}~\underbrace{)))\cdots)}_{n})$$, where $$b'$$ is the predecessor of $$b$$.
• If there exist some $$a \in \mathbb{N}$$, $$b \in \mathbb{N}_+$$, $$\b\in S_0$$, and $$\g\in S_b$$ such that $$a\lt b$$ and $$\a=S_{0,\b}(a(S_{b,\g}(b)))$$, then
$$f(\a,n)=S_{0,\b}(f(a(S_{b,\g}(b)),n))$$.

### Large number notation

We define a function \begin{eqnarray*} [] \colon S \times \mathbb{N}_+ & \to & \mathbb{N}_+ \\ (\a,n) & \mapsto \a[n] \end{eqnarray*} in the following recursive way:

• If $$\a= 0$$, then $$0[n]=n\times 2$$.
• If $$\a \neq 0$$, then $$\a[n]=f(\a,n)[n\times 2]$$.

## Large number

The creator mrna coined $$g^{100}(100)$$ as 段階配列数, where $$g(n)$$ is defined as $$0(n)[n]$$ for any $$n \in \mathbb{N}_+$$.

## Analysis

We explain the result on termination and analysis originally proven by a Japanese Googology Wiki user p進大好きbot.

Let $$T$$ denote Buchholz's notation system, and $$OT \subset T$$ Buchholz's ordinal notation. We define a map \begin{eqnarray*} o \colon S & \to & T \setminus \{0\} \\ \alpha & \mapsto & o(\alpha) \end{eqnarray*} in the following recursive way:

1. If $$\alpha \in \mathbb{N}$$, then $$o(\alpha) := D_{\alpha} 0$$.
2. If there exists some $$\beta \in S$$ and $$\gamma \in P$$ such that $$\alpha = \beta + \gamma$$, then $$o(\alpha) := o(\beta) + o(\gamma)$$.
3. If there exist some $$m \in \mathbb{N}$$ and $$\beta \in S$$ such that $$\alpha = m(\beta)$$, then $$o(\alpha) := D_m o(\beta)$$.

We put $$OS := \{\alpha \in S \mid o(\alpha) \in OT \land o(\alpha) < D_1 0\}$$.

Theorem（Well-foundedness of $$OS$$）
(1) For any $$(a_i)_{i \in \mathbb{N}} \in \mathbb{N}_+^{\mathbb{N}}$$ and $$\alpha \in OS$$, there exists some $$N \in \mathbb{N}$$ such that $$f(\cdots f(\alpha,a_0) \cdots,a_N) = 0$$.
(2) For any $$\alpha \in OS$$ and $$n \in \mathbb{N}_+$$, the computation of $$\alpha[n]$$ terminates and the inequality $$\alpha[n] \geq HH_{o(\alpha)}(n-1)$$ holds. Here, we endow $$C_0(\varepsilon_{\Omega_{\omega}+1}) \cap \Omega$$ with the system of fundamental sequences associated to the restriction of $$[]$$ to $$\{a \in OT \mid a < D_1 0\}$$.

In particular, $$g(n)$$ is a total function on $$\mathbb{N}_+$$ googologically approximated to Hardy hierarchy $$HH_{\psi_0(\Omega_{\omega})}(n-1)$$. We note that the system of fundamental sequences used here is a little different from the most popular one in googology originally introduced by a Googology Wiki user Denis Maksudov.

## Demonstration

\begin{eqnarray*} & & g(1) = 0(1) = 0(0) = 0+0 = 0 = 16 \\ & & g(2) = 0(2) = 0(1(1)) = 0(1(0(1(0(1(0(1(0))))))) = 0(1(0(1(0(1(0(\underbrace{1+\cdots+1}_{8})))))) \\ & = & 0(1(0(1(0(1(0(\underbrace{\underbrace{1+\cdots+1}_{7}+0(\cdots 0(\underbrace{1+\cdots+1}_{7}+0(\underbrace{1+\cdots+1}_{7}+0}_{16}))\cdots))))))) \\ & = & \cdots \end{eqnarray*}

## Extensions

Later, mrna extended 段階配列表記 to a notation up to the countable limit of Extended Buchholz's function called 降下段階配列表記 on 10/09/2020, and to a notation expected to express all ordinals below $$\psi_{\chi_0(0)}(\psi_{\chi_{\varphi_0(\varphi_0(0))}(0)}(0)) = \psi_{\chi_0(0)}(\psi_{\chi_{\omega}(0)}(0))$$ with respect to Rathjen's $$\psi$$ called 多変数段階配列表記 on 20/10/2020 using a similar method.

## Large ordinal

Further, mrna coined the ordinal corresponding to the limit of 多変数段階配列表記 as 1-stage omega ordinal or 1-SOO for short with respect to the correspondence associated to the expansion rule under the assumption of the well-foundedness. If the expectation above is correct, 1-stage omega ordinal coincides with $$\psi_{\chi_0(0)}(\psi_{\chi_{\varphi_0(\varphi_0(0))}(0)}(0)) = \psi_{\chi_0(0)}(\psi_{\chi_{\omega}(0)}(0))$$.

## Sources

1. The user page of mrna in Japanese Googology Wiki.
2. mrna, "段階配列表記", Google Document, 2020.
3. p進大好きbot, 段階配列表記観察日記, Japanese Googology Wiki user blog.
6. mrna, 段階配列表記　解説配信, Google Document.

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