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ハイパー原始数列は ゆきとによって考案された数列表記である。[1][2] バシク行列システムを1行に限定した原始数列システムを、階差数列を考えることによって拡張している。この表記は、 急増加関数ブーフホルツのψ関数を用いて $$f_{\psi_0(\Omega_{\omega})}(n)$$ と近似される。これはちょうどペア数列システムの極限と同じ大きさである。

この表記はまた、ブーフホルツのψ関数と互換性のある形で展開される。

## 定義

この表記における項は自然数の有限列 $$S = (S_i)_{i=1}^{m}$$ にカッコで囲った自然数 $$n$$ を並べた式 $$(S_1,\ldots,S_m)[n]$$ である。これを次の方法で自然数として評価する：

1. $$m = 0$$ ならば、$$S[n] = n$$ である。
2. $$m > 0$$ とする。
1. $$i \leq m$$ を満たす各 $$i \in \mathbb{N}$$ に対して、$$P_i = \{j \in \mathbb{N} \mid 1 \leq j < i \land S_j < S_i\}$$ と置く。
2. まず $$k = 0$$ と置く。（以下 $$k$$ の値をアップデートする）
3. $$k = 0$$ ならば、$$m_k = m$$ と置く。
4. $$k > 0$$ ならば、$$m_k = \max P_{m_{k-1}}$$ と置く。
5. $$P_{m_k} \neq \emptyset$$ ならば、$$k$$ を自身に$$1$$足した値に置き換え、行番号2-3へ戻る。
6. 有限列 $$\textrm{Expand}(S,n)$$ を以下の方法で定める：
1. $$k = 0$$ ならば、$$\textrm{Expand}(S,n) = (S_1,\ldots,S_{m-1})$$ と置く。
2. $$k > 0$$ とする。
1. $$(S_{m_i})_{i=0}^{k}$$ の階差数列 $$(S_{m_i}-S_{m_{i+1}})_{i=0}^{k-1}$$ を $$N = (N_i)_{i=0}^{k-1}$$ と置く。
2. 悪い部分と呼ばれる数列 $$r$$ と 上昇レベル と呼ばれる自然数 $$\delta$$ を以下のように定める：
1. $$N_0 = 1$$ とする。
1. $$r = m_1$$ と定める。
2. $$\delta = 0$$ と定める。
2. $$N_0 \neq 1$$ とする。
1. $$0 < i \leq k-1$$ かつ $$N_i < N_0$$ を満たす $$i \in \mathbb{N}$$ が存在しないとする。
1. $$r = 1$$ と定める。
2. $$\delta = S_m - 1$$ と定める。
2. そのような $$i$$ が存在するとする。
1. そのような $$i$$ の最小値を $$p$$ と置く。
2. $$r = m_p$$ と定める。
3. $$\delta = S_m - S_r - 1$$ と定める。
3. $$G = (S_i)_{i=1}^{r-1}$$ と置く。（$$r = 1$$ である場合、これは空列である）
4. 各 $$h \in \mathbb{N}$$ に対し、$$B(h) = (S_i + h \delta)_{i=r}^{m-1}$$ と置く。
5. $$\textrm{Expand}(S,n)$$ は $$G,B(0),\ldots,B(n)$$ の結合で得られる数列である。
7. $$S[n] = \textrm{Expand}(S,n)[n+1]$$ と定める。

## Standard form

The set $$OT$$ of finite sequences of standard forms in hyper primitive sequence system is defined in the following recursive way:

1. For any $$n \in \mathbb{N}$$, $$(0,n) \in OT$$.
2. For any $$(S,n) \in (OT \setminus \{()\}) \times \mathbb{N}$$, $$\textrm{Expand}(S,n) \in OT$$.

Then $$OT$$ is a recursive subset of the set $$T$$ of finite sequences of natural numbers, and $$\textrm{Expand}$$ gives a primitive recursive map $$(T \setminus \{()\}) \times \mathbb{N} \to T$$, which preserves standard forms.

## Explanation

Follow the convention and the terminology in the article on difference sequences. Hyper primitive sequence system employs the bad root searching rule $$\textrm{Parent} \colon \textrm{FinSeq} \times \mathbb{N} \to \textrm{FinSeq}$$ for the primitive sequence system explained here. Indeed $$(m_i)_{i=0}^{k}$$ and $$N$$ in the definition of $$S[n]$$ precisely coincide with $$\textrm{Ancestors}(S)$$ and $$\textrm{Kaiser}(S)$$ respectively.

If $$\textrm{Ancestors}(S)$$ is of length $$1$$, then $$S$$ is a successor term, and $$\textrm{Expand}(S,n)$$ is obtained by deleting the rightmost entry of $$S$$. If $$\textrm{Ancestors}(S)$$ is of length $$> 1$$, then apply an expansion rule to $$\textrm{Kaiser}(S)$$ quite similar to the one in primitive sequence system; if $$\textrm{Ancestors}(\textrm{Kaiser}(S))$$ is of length $$1$$, then decrement the rightmost entry of $$\textrm{Kaiser}(S)$$, and otherwise, copy the bad part of $$\textrm{Kaiser}(S)$$ with respect to $$\textrm{Parent}$$. Denote by $$\textrm{Kaiser}(S)(n)$$ the resulting finite sequence. As $$\textrm{Kaiser}(S)$$ is the differece sequence of $$\textrm{RightNodes}(S)$$, $$\textrm{Kaiser}(S)(n)$$ is the difference sequence of a unique finite sequence $$\textrm{RightNodes}(S)(n)$$ whose first entry is $$\textrm{RightNodes}(S)_0$$. Finally, $$\textrm{Expand}(S,n)$$ is given by replacing the subsequence $$\textrm{RightNodes}(S)$$ of $$S$$ by $$\textrm{RightNodes}(S)(n)$$ and interpolating the intermidiate entries by the copies of the corresponding entries of $$S$$ modified by the ascention level.

Through the interpretation above, hyper primitive sequence system can be regarded as an analog of the system of two-lined hydra diagrams, which corresponds to pair sequence system.

## Termination

For the convention and the terminology, see the following:

• [Buc1] W. Buchholz, A new system of proof-theoretic ordinal functions, Annals of Pure and Applied Logic, Volume 32, 1986, pp. 195–207.
• [Buc2] W. Buchholz, Relating ordinals to proofs in a perspicious way, Reflections on the Foundations of Mathematics, Essays in Honor of Solomon Feferman, Lecture Notes in Logic, vol. 15, 2002, pp. 37–59.

The termination of hyper primitive sequence system has been proved by a Japanese Googology Wiki user p進大好きbot in the following way:

Let $$T_B$$ denote the set of terms in Buchholz's notation, in which $$D_{\omega}$$ does not appear.[3] Define a total recursive map \begin{eqnarray*} \textrm{Trans} \colon T_B & \to & T \\ t & \mapsto & \textrm{Trans}(t) \end{eqnarray*} in the following recursive way:

1. If $$t = 0$$, then set $$\textrm{Trans}(t) = ()$$.
2. Suppose $$t = t_0 + D_u t_1$$ for some $$(u,t_0,t_1) \in \mathbb{N} \times T_B \times T_B$$.[4]
1. If $$t_1 = D_0 0 \cdot m$$ for some $$m \in \mathbb{N}$$, then $$\textrm{Trans}(t)$$ is the concatenation of $$\textrm{Trans}(t_0)$$ and $$(u,\underbrace{u+1,\ldots,u+1}_{m})$$.[5]
2. Otherwise, $$\textrm{Trans}(t)$$ is the concatenation of $$\textrm{Trans}(t_0)$$, $$(u)$$, and the finite sequence obtained by adding $$u+1$$ to each entry of $$\textrm{Trans}(t_1)$$.

Let $$OT_B \subset T_B$$ denote the subset of ordinal terms below $$D_1 0$$.[6]

Lemma (Compatibility of $$\textrm{Trans}$$)
The restriction of $$\textrm{Trans}$$ to $$OT_B$$ satisfies the following:
(1) It is a bijective map onto $$OT$$.
(2) For any $$(t,n) \in (OT_B \setminus \{0\}) \times \mathbb{N}$$, $$\textrm{Trans}(t) \neq ()$$, and there exists a $$t' \in OT_B$$ such that $$\textrm{Trans}(t') = \textrm{Expand}(\textrm{Trans}(t),n)$$ and $$t' < t$$. If $$n > 0$$ then such a $$t'$$ can be taken so that $$t[n-1] < t' \leq t[n]$$.[7]
proof
For an $$m \in \mathbb{N}$$, define $$\Sigma_m \subset OT_B$$ in the following recursive way:
1. If $$m = 0$$, then set $$\Sigma_m = \{D_0 D_u 0 \mid u \in \mathbb{N} \land u > 0\}$$.
2. If $$m > 0$$, then set $$\Sigma_m = \Sigma_{m-1} \cup \{t[n] \mid (t,n) \in \Sigma_{m-1} \times \mathbb{N}\}$$.
By [Buc1] 2.2 Lemma (c) and 2.3 Lemma (b), we have $$\bigcup_{m \in \mathbb{N}} \Sigma_m = OT_B$$.
Let $$t \in OT_B$$. By the argument above, there exists an $$m \in \mathbb{N}$$ such that $$t \in \Sigma_m$$. Denote by $$\mu$$ the minimum of such an $$m$$. We show $$\textrm{Trans}(t) \in OT$$ by induction of $$\mu$$.
If $$\mu = 0$$, then we have $$t = D_0 D_u 0$$ for some $$u \in \mathbb{N}$$ satisfying $$u > 0$$, and hence $$\textrm{Trans}(t) = (0,u) \in OT$$. Suppose $$\mu > 0$$ in the following. Take a $$(t',n) \in \Sigma_{\mu-1} \times \mathbb{N}$$ satisfying $$t = t'[n]$$. By the induction hypothesis, we have $$\textrm{Trans}(t') \in OT_B$$. By $$t \notin \Sigma_{\mu-1}$$, we have $$t \neq t'$$, and hence $$t' \neq 0$$. It implies $$t = t'[n] < t'$$ by [Buc1] 3.2 Lemma (a).
Take a unique $$(u,t_0,t_1) \in \mathbb{N} \times OT_B \times OT_B$$ satisfying $$t' = t_0 + D_u t_1$$. By the definition of $$\textrm{Expand}$$ and $$\textrm{Trans}$$, we have $$\textrm{Trans}(t) = \textrm{Expand}(\textrm{Trans}(t'),n) \in OT_B$$ unless $$\textrm{dom}(t_1) = T_v$$ for some $$v \in \mathbb{N}$$ satifying $$u \leq v$$.[8] Therefore we may assume $$\textrm{dom}(t_1) = T_v$$ for some $$v \in \mathbb{N}$$ satifying $$u \leq v$$.
For each $$m \in \mathbb{N}$$, denote by $$t_2(m)$$ a unique ordinal term satisfying $$o(t_2(m)) = o(t'(m))$$, where $$t'(m)$$ is the term obtained by replacing the rightmost appearrence of $$D_v 0$$ in $$t'[m]$$ by $$0$$.[9] Again by the definition of $$\textrm{Expand}$$ and $$\textrm{Trans}$$, $$\textrm{Trans}(t_2(m))$$ and $$\textrm{Trans}(t'(m))$$ coincide with $$\textrm{Expand}(\textrm{Trans}(t'),m)$$.
We have $$t_2(n) \leq t'(n) < t \leq t_2(n+1) \leq t'(n+1)$$, and $$t$$ coincides with the term obtained by replacing the righmost appearrence of $$0$$ in $$t'(n)$$ by $$D_v 0$$ because of $$t = t'[n]$$. Therefore $$\textrm{Trans}(t)$$ coincides with the concatenation of $$\textrm{Trans}(t'(n))$$ and $$(a+v+1)$$, where $$a$$ is the next rightmost entry of $$\textrm{RightNodes}(\textrm{Trans}(t))$$ with respect to the bad root searching rule $$\textrm{Parent}$$, and $$\textrm{Trans}(t'(n+1))$$ coincides with the concatenation of $$\textrm{Trans}(t)$$ and a finite sequence. Since the deletion of the rightmost entry is given by the application of $$\textrm{Expand}(-,0)$$, it implies $$t \in OT$$.
Thus the restriction of $$\textrm{Trans}$$ to $$OT_B$$ gives a map to $$OT$$. The assertions (1) and (2) immediately follow from the argument above, because of $$\textrm{Trans}(D_0 0 + D_0 0) = (0,0)$$ and $$\textrm{Trans}(D_u 0) = (0,u)$$ for any $$u \in \mathbb{N}$$ satisfying $$u > 0$$. □

As is shown in the proof above, hyper primitive sequence system is quite compatible with Buchholz's ordinal notation unlike pair sequence system.

Theorem (The termination of hyper primitive sequence system)
(1) The pair of $$OT$$ and the lexicographic order $$<_{OT}$$ on it is an ordinal notation.
(2) Let $$S \in OT \setminus \{()\}$$. For any map $$f \colon \mathbb{N} \to \mathbb{N}$$, there exists a unique finite sequence $$(X_k)_{k=0}^{N}$$ in $$OT$$ such that $$X_0 = S$$, $$X_k$$ is a non-empty sequence satisfying $$\textrm{Expand}(X_k,f(k)) = X_{k+1}$$ for any $$k \in \mathbb{N}$$ smaller than $$N+1$$, and $$X_N= ()$$.
Proof
(1) By Lemma and [Buc1] 2.2 Lemma (c), the pair of $$OT_B$$ and its canonical ordering $$<_{OT_B}$$, which coincides with the lexicographic ordering for a suitable ordering of letters by definition, is an ordinal notation. Since $$\textrm{Trans}$$ preserves the lexicographic ordering, Lemma (Compatibility of $$\textrm{Trans}$$) (1) implies that $$(OT_B,<_{OT_B})$$ is isomorphic to $$(OT,<_{OT})$$. Thus $$(OT,<_{OT})$$ is an ordinal notation.
(2) Since $$(OT_B,<_{OT_B})$$ is an ordinal notation, the compatibility of the canonical ordering on $$OT_B$$ and $$[ \ ]$$ and Lemma (Compatibility of $$\textrm{Trans}$$) (2) implies the assertion for the case where $$S$$ belongs to the image of $$\textrm{Trans}$$, which coincides with $$OT$$ by Lemma (Compatibility of $$\textrm{Trans}$$) (1).□

## Analysis

By Lemma (2), the limit of hyper primitive sequence is $$\psi_0(\Omega_{\omega})$$ in ハーディ階層 with respect to a minor replacedment of the recursive system of fundamental sequences assciated to Buchholz's ordinal notation. In particular, it is also approximated to $$\psi_0(\Omega_{\omega})$$ in 急増加関数. The composite \begin{eqnarray*} o \circ (\textrm{Trans} |_{OT_B})^{-1} \colon OT \to \psi_0(\Omega_{\omega}) \end{eqnarray*} gives an interpretation of terms in hyper primitive sequence system of standard form into ordinals below $$\psi_0(\Omega_{\omega})$$. The following table exhibits examples of the correspondence:

hyper primitive seqeunce ordinal
$$()$$ $$0$$
$$(0)$$ $$1$$
$$(0,0)$$ $$2$$
$$(0,0,0)$$ $$3$$
$$(0,1)$$ $$\omega$$
$$(0,1,0)$$ $$\omega+1$$
$$(0,1,0,0)$$ $$\omega+2$$
$$(0,1,0,0,0)$$ $$\omega+3$$
$$(0,1,0,1)$$ $$\omega \times 2$$
$$(0,1,0,1,0)$$ $$\omega \times 2+1$$
$$(0,1,0,1,0,0)$$ $$\omega \times 2+2$$
$$(0,1,0,1,0,0,0)$$ $$\omega \times 2+3$$
$$(0,1,0,1,0,1)$$ $$\omega \times 3$$
$$(0,1,1)$$ $$\omega^2$$
$$(0,1,1,0)$$ $$\omega^2+1$$
$$(0,1,1,0,0)$$ $$\omega^2+2$$
$$(0,1,1,0,1)$$ $$\omega^2+\omega$$
$$(0,1,1,0,1,0)$$ $$\omega^2+\omega+1$$
$$(0,1,1,0,1,1)$$ $$\omega^2 \times 2$$
$$(0,1,1,1)$$ $$\omega^3$$
$$(0,1,2)$$ $$\omega^{\omega}$$
$$(0,1,2,1)$$ $$\omega^{\omega+1}$$
$$(0,1,2,1,1)$$ $$\omega^{\omega+2}$$
$$(0,1,2,1,2)$$ $$\omega^{\omega \times 2}$$
$$(0,1,2,2)$$ $$\omega^{\omega^2}$$
$$(0,1,2,2,1)$$ $$\omega^{\omega^2+1}$$
$$(0,1,2,2,1,1)$$ $$\omega^{\omega^2+2}$$
$$(0,1,2,2,1,2)$$ $$\omega^{\omega^2+\omega}$$
$$(0,1,2,2,1,2,1)$$ $$\omega^{\omega^2+\omega+1}$$
$$(0,1,2,2,1,2,1,1)$$ $$\omega^{\omega^2+\omega+2}$$
$$(0,1,2,2,1,2,1,2)$$ $$\omega^{\omega^2+\omega \times 2}$$
$$(0,1,2,2,1,2,2)$$ $$\omega^{\omega^2 \times 2}$$
$$(0,1,2,2,2)$$ $$\omega^{\omega^3}$$
$$(0,1,2,3)$$ $$\omega^{\omega^{\omega}}$$
$$(0,2)$$ $$\varepsilon_0 = \psi_0(\Omega)$$
$$(0,2,1)$$ $$\varepsilon_0 \times \omega = \psi_0(\Omega+1)$$
$$(0,2,1,1)$$ $$\varepsilon_0 \times \omega^2 = \psi_0(\Omega+2)$$
$$(0,2,1,2)$$ $$\varepsilon_0 \times \omega^{\omega} = \psi_0(\Omega+\omega)$$
$$(0,2,1,2,1)$$ $$\varepsilon_0 \times \omega^{\omega+1} = \psi_0(\Omega+\omega+1)$$
$$(0,2,1,2,1,1)$$ $$\varepsilon_0 \times \omega^{\omega+2} = \psi_0(\Omega+\omega+2)$$
$$(0,2,1,2,1,2)$$ $$\varepsilon_0 \times \omega^{\omega \times 2} = \psi_0(\Omega+\omega \times 2)$$
$$(0,2,1,2,2)$$ $$\varepsilon_0 \times \omega^{\omega^2} = \psi_0(\Omega+\omega^2)$$
$$(0,2,1,2,3)$$ $$\varepsilon_0 \times \omega^{\omega^{\omega}} = \psi_0(\Omega+\omega^{\omega})$$
$$(0,2,1,3)$$ $$\varepsilon_0^2 = \psi_0(\Omega+\psi_0(\Omega))$$
$$(0,2,1,3,1)$$ $$\varepsilon_0^2 \times \omega = \psi_0(\Omega+\psi_0(\Omega)+1)$$
$$(0,2,1,3,1,2)$$ $$\varepsilon_0^2 \times \omega^{\omega} = \psi_0(\Omega+\psi_0(\Omega)+\omega)$$
$$(0,2,1,3,1,2,3)$$ $$\varepsilon_0^2 \times \omega^{\omega^{\omega}} = \psi_0(\Omega+\psi_0(\Omega)+\omega^{\omega})$$
$$(0,2,1,3,1,3)$$ $$\varepsilon_0^3 = \psi_0(\Omega+\psi_0(\Omega) \times 2)$$
$$(0,2,1,3,2)$$ $$\varepsilon_0^{\omega} = \psi_0(\Omega+\psi_0(\Omega+1))$$
$$(0,2,1,3,2,1)$$ $$\varepsilon_0^{\omega} \times \omega = \psi_0(\Omega+\psi_0(\Omega+1)+1)$$
$$(0,2,1,3,2,1,2)$$ $$\varepsilon_0^{\omega} \times \omega^{\omega} = \psi_0(\Omega+\psi_0(\Omega+1)+\omega)$$
$$(0,2,1,3,2,1,2,3)$$ $$\varepsilon_0^{\omega} \times \omega^{\omega^{\omega}} = \psi_0(\Omega+\psi_0(\Omega+1)+\omega^{\omega})$$
$$(0,2,1,3,2,1,3)$$ $$\varepsilon_0^{\omega+1} = \psi_0(\Omega+\psi_0(\Omega+1)+\psi_0(\Omega))$$
$$(0,2,1,3,2,1,3,1)$$ $$\varepsilon_0^{\omega+1} \times \omega = \psi_0(\Omega+\psi_0(\Omega+1)+\psi_0(\Omega)+1)$$
$$(0,2,1,3,2,1,3,2)$$ $$\varepsilon_0^{\omega \times 2} = \psi_0(\Omega+\psi_0(\Omega+1) \times 2)$$
$$(0,2,1,3,2,2)$$ $$\varepsilon_0^{\omega^2} = \psi_0(\Omega+\psi_0(\Omega+2))$$
$$(0,2,1,3,2,2,1)$$ $$\varepsilon_0^{\omega^2} \times \omega = \psi_0(\Omega+\psi_0(\Omega+2)+1)$$
$$(0,2,1,3,2,2,2)$$ $$\varepsilon_0^{\omega^3} = \psi_0(\Omega+\psi_0(\Omega+3))$$
$$(0,2,1,3,2,3)$$ $$\varepsilon_0^{\omega^{\omega}} = \psi_0(\Omega+\psi_0(\Omega+\omega))$$
$$(0,2,1,3,2,3,1)$$ $$\varepsilon_0^{\omega^{\omega}} = \psi_0(\Omega+\psi_0(\Omega+\omega)+1)$$
$$(0,2,1,3,2,3,2)$$ $$\varepsilon_0^{\omega^{\omega} \times \omega} = \psi_0(\Omega+\psi_0(\Omega+\omega+1))$$
$$(0,2,1,3,2,3,3)$$ $$\varepsilon_0^{\omega^{\omega^2}} = \psi_0(\Omega+\psi_0(\Omega+\omega^2))$$
$$(0,2,1,3,2,3,4)$$ $$\varepsilon_0^{\omega^{\omega^{\omega}}} = \psi_0(\Omega+\psi_0(\Omega+\omega^{\omega}))$$
$$(0,2,1,3,2,4)$$ $$\varepsilon_0^{\varepsilon_0} = \psi_0(\Omega+\psi_0(\Omega+\psi_0(\Omega)))$$
$$(0,2,1,3,2,4,1)$$ $$\varepsilon_0^{\varepsilon_0} \times \omega = \psi_0(\Omega+\psi_0(\Omega+\psi_0(\Omega))+1)$$
$$(0,2,1,3,2,4,2)$$ $$\varepsilon_0^{\varepsilon_0 \times \omega} = \psi_0(\Omega+\psi_0(\Omega+\psi_0(\Omega)+1))$$
$$(0,2,1,3,2,4,3)$$ $$\varepsilon_0^{\varepsilon_0^{\omega}} = \psi_0(\Omega+\psi_0(\Omega+\psi_0(\Omega+1)))$$
$$(0,2,2)$$ $$\varepsilon_0 \uparrow \uparrow \omega = \psi_0(\Omega \times 2)$$
$$(0,2,2,1)$$ $$\psi_0(\Omega \times 2+1)$$
$$(0,2,2,2)$$ $$\psi_0(\Omega \times 3)$$
$$(0,2,3)$$ $$\psi_0(\Omega \times \omega) = \psi_0(\psi_1(\psi_0(0)))$$
$$(0,2,3,1)$$ $$\psi_0(\Omega \times \omega + 1) = \psi_0(\psi_1(\psi_0(0))+\psi_0(0))$$
$$(0,2,3,2)$$ $$\psi_0(\Omega \times (\omega + 1)) = \psi_0(\psi_1(\psi_0(0))+\psi_1(0))$$
$$(0,2,3,3)$$ $$\psi_0(\Omega \times \omega^2) = \psi_0(\psi_1(\psi_0(0)+\psi_0(0)))$$
$$(0,2,3,4)$$ $$\psi_0(\Omega \times \omega^{\omega}) = \psi_0(\psi_1(\psi_0(\psi_0(0))))$$
$$(0,2,3,4,1)$$ $$\psi_0(\Omega \times \omega^{\omega}+1) = \psi_0(\psi_1(\psi_0(\psi_0(0)))+\psi_0(0))$$
$$(0,2,3,4,2)$$ $$\psi_0(\Omega \times (\omega^{\omega}+1)) = \psi_0(\psi_1(\psi_0(\psi_0(0)))+\psi_1(0))$$
$$(0,2,3,4,3)$$ $$\psi_0(\Omega \times \omega^{\omega+1}) = \psi_0(\psi_1(\psi_0(\psi_0(0))+\psi_0(0)))$$
$$(0,2,3,4,4)$$ $$\psi_0(\Omega \times \omega^{\omega \times 2}) = \psi_0(\psi_1(\psi_0(\psi_0(0)+\psi_0(0))))$$
$$(0,2,3,4,5)$$ $$\psi_0(\Omega \times \omega^{\omega^{\omega}}) = \psi_0(\psi_1(\psi_0(\psi_0(\psi_0(0)))))$$
$$(0,2,3,5)$$ $$\psi_0(\Omega \times \psi_0(\Omega)) = \psi_0(\psi_1(\psi_0(\psi_1(0))))$$
$$(0,2,4)$$ $$\psi_0(\Omega^2) = \psi_0(\psi_1(\psi_1(0)))$$
$$(0,2,4,1)$$ $$\psi_0(\Omega^2+1) = \psi_0(\psi_1(\psi_1(0))+\psi_0(0))$$
$$(0,2,4,2)$$ $$\psi_0(\Omega^2+\Omega) = \psi_0(\psi_1(\psi_1(0))+\psi_1(0))$$
$$(0,2,4,3)$$ $$\psi_0(\Omega^2 \times \omega) = \psi_0(\psi_1(\psi_1(0)+\psi_0(0)))$$
$$(0,2,4,3,1)$$ $$\psi_0(\Omega^2 \times \omega + 1) = \psi_0(\psi_1(\psi_1(0)+\psi_0(0))+\psi_0(0))$$
$$(0,2,4,3,2)$$ $$\psi_0(\Omega^2 \times \omega + \Omega) = \psi_0(\psi_1(\psi_1(0)+\psi_0(0))+\psi_1(0))$$
$$(0,2,4,3,3)$$ $$\psi_0(\Omega^2 \times \omega^2) = \psi_0(\psi_1(\psi_1(0)+\psi_0(0)+\psi_0(0)))$$
$$(0,2,4,3,4)$$ $$\psi_0(\Omega^2 \times \omega^{\omega}) = \psi_0(\psi_1(\psi_1(0)+\psi_0(\psi_0(0))))$$
$$(0,2,4,3,5)$$ $$\psi_0(\Omega^2 \times \psi_0(\Omega)) = \psi_0(\psi_1(\psi_1(0)+\psi_0(\psi_1(0))))$$
$$(0,2,4,3,5,1)$$ $$\psi_0(\Omega^2 \times \psi_0(\Omega)+1) = \psi_0(\psi_1(\psi_1(0)+\psi_0(\psi_1(0)))+\psi_0(0))$$
$$(0,2,4,3,5,2)$$ $$\psi_0(\Omega^2 \times \psi_0(\Omega)+\Omega) = \psi_0(\psi_1(\psi_1(0)+\psi_0(\psi_1(0)))+\psi_1(0))$$
$$(0,2,4,3,5,3)$$ $$\psi_0(\Omega^2 \times \psi_0(\Omega+1)) = \psi_0(\psi_1(\psi_1(0)+\psi_0(\psi_1(0))+\psi_0(0)))$$
$$(0,2,4,3,5,4)$$ $$\psi_0(\Omega^2 \times \psi_0(\Omega+\omega)) = \psi_0(\psi_1(\psi_1(0)+\psi_0(\psi_1(0)+\psi_0(0))))$$
$$(0,2,4,3,5,5)$$ $$\psi_0(\Omega^2 \times \psi_0(\Omega \times 2)) = \psi_0(\psi_1(\psi_1(0)+\psi_0(\psi_1(0)+\psi_1(0))))$$
$$(0,2,4,3,5,6)$$ $$\psi_0(\Omega^2 \times \psi_0(\Omega \times \omega)) = \psi_0(\psi_1(\psi_1(0)+\psi_0(\psi_1(\psi_0(0)))))$$
$$(0,2,4,3,5,6,1)$$ $$\psi_0(\Omega^2 \times \psi_0(\Omega \times \omega)+1) = \psi_0(\psi_1(\psi_1(0)+\psi_0(\psi_1(\psi_0(0))))+\psi_0(0))$$
$$(0,2,4,3,5,6,2)$$ $$\psi_0(\Omega^2 \times \psi_0(\Omega \times \omega)+\Omega) = \psi_0(\psi_1(\psi_1(0)+\psi_0(\psi_1(\psi_0(0))))+\psi_1(0))$$
$$(0,2,4,3,5,6,3)$$ $$\psi_0(\Omega^2 \times \psi_0(\Omega \times \omega+1)) = \psi_0(\psi_1(\psi_1(0)+\psi_0(\psi_1(\psi_0(0)))+\psi_0(0)))$$
$$(0,2,4,3,5,6,4)$$ $$\psi_0(\Omega^2 \times \psi_0(\Omega \times \omega+\omega))= \psi_0(\psi_1(\psi_1(0)+\psi_0(\psi_1(\psi_0(0))+\psi_0(0))))$$
$$(0,2,4,3,5,6,5)$$ $$\psi_0(\Omega^2 \times \psi_0(\Omega \times (\omega+1))) = \psi_0(\psi_1(\psi_1(0)+\psi_0(\psi_1(\psi_0(0))+\psi_1(0))))$$
$$(0,2,4,3,5,6,6)$$ $$\psi_0(\Omega^2 \times \psi_0(\Omega \times \omega^2)) = \psi_0(\psi_1(\psi_1(0)+\psi_0(\psi_1(\psi_0(0)+\psi_0(0)))))$$
$$(0,2,4,3,5,6,7)$$ $$\psi_0(\Omega^2 \times \psi_0(\Omega \times \omega^{\omega})) = \psi_0(\psi_1(\psi_1(0)+\psi_0(\psi_1(\psi_0(\psi_0(0))))))$$
$$(0,2,4,3,5,6,8)$$ $$\psi_0(\Omega^2 \times \psi_0(\Omega \times \psi_0(\Omega))) = \psi_0(\psi_1(\psi_1(0)+\psi_0(\psi_1(\psi_0(\psi_1(0))))))$$
$$(0,2,4,3,5,7)$$ $$\psi_0(\Omega^2 \times \psi_0(\Omega^2)) = \psi_0(\psi_1(\psi_1(0)+\psi_0(\psi_1(\psi_1(0)))))$$
$$(0,2,4,3,5,7,1)$$ $$\psi_0(\Omega^2 \times \psi_0(\Omega^2)+1) = \psi_0(\psi_1(\psi_1(0)+\psi_0(\psi_1(\psi_1(0))))+\psi_0(0))$$
$$(0,2,4,3,5,7,2)$$ $$\psi_0(\Omega^2 \times \psi_0(\Omega^2)+\Omega) = \psi_0(\psi_1(\psi_1(0)+\psi_0(\psi_1(\psi_1(0))))+\psi_1(0))$$
$$(0,2,4,3,5,7,3)$$ $$\psi_0(\Omega^2 \times \psi_0(\Omega^2+1)) = \psi_0(\psi_1(\psi_1(0)+\psi_0(\psi_1(\psi_1(0)))+\psi_0(0)))$$
$$(0,2,4,3,5,7,4)$$ $$\psi_0(\Omega^2 \times \psi_0(\Omega^2)^{\omega}) = \psi_0(\psi_1(\psi_1(0)+\psi_0(\psi_1(\psi_1(0))+\psi_0(0))))$$
$$(0,2,4,3,5,7,5)$$ $$\psi_0(\Omega^2 \times \psi_0(\Omega^2+\Omega)) = \psi_0(\psi_1(\psi_1(0)+\psi_0(\psi_1(\psi_1(0))+\psi_1(0))))$$
$$(0,2,4,3,5,7,6)$$ $$\psi_0(\Omega^2 \times \psi_0(\Omega^2 \times \omega)) = \psi_0(\psi_1(\psi_1(0)+\psi_0(\psi_1(\psi_1(0)+\psi_0(0)))))$$
$$(0,2,4,4)$$ $$\psi_0(\Omega^3) = \psi_0(\psi_1(\psi_1(0)+\psi_1(0)))$$
$$(0,2,4,4,1)$$ $$\psi_0(\Omega^3+1) = \psi_0(\psi_1(\psi_1(0)+\psi_1(0))+\psi_0(0))$$
$$(0,2,4,4,2)$$ $$\psi_0(\Omega^3+\Omega) = \psi_0(\psi_1(\psi_1(0)+\psi_1(0))+\psi_1(0))$$
$$(0,2,4,4,3)$$ $$\psi_0(\Omega^3 \times \omega) = \psi_0(\psi_1(\psi_1(0)+\psi_1(0)+\psi_0(0)))$$
$$(0,2,4,4,4)$$ $$\psi_0(\Omega^4) = \psi_0(\psi_1(\psi_1(0)+\psi_1(0)+\psi_1(0)))$$
$$(0,2,4,5)$$ $$\psi_0(\Omega^{\omega}) = \psi_0(\psi_1(\psi_1(\psi_0(0))))$$
$$(0,2,4,5,1)$$ $$\psi_0(\Omega^{\omega}+1) = \psi_0(\psi_1(\psi_1(\psi_0(0)))+\psi_0(0))$$
$$(0,2,4,5,2)$$ $$\psi_0(\Omega^{\omega}+\Omega) = \psi_0(\psi_1(\psi_1(\psi_0(0)))+\psi_1(0))$$
$$(0,2,4,5,3)$$ $$\psi_0(\Omega^{\omega} \times \omega) = \psi_0(\psi_1(\psi_1(\psi_0(0))+\psi_0(0)))$$
$$(0,2,4,5,4)$$ $$\psi_0(\Omega^{\omega+1}) = \psi_0(\psi_1(\psi_1(\psi_0(0))+\psi_1(0)))$$
$$(0,2,4,5,5)$$ $$\psi_0(\Omega^{\omega^2}) = \psi_0(\psi_1(\psi_1(\psi_0(0)+\psi_0(0))))$$
$$(0,2,4,5,6)$$ $$\psi_0(\Omega^{\omega^{\omega}}) = \psi_0(\psi_1(\psi_1(\psi_0(\psi_0(0)))))$$
$$(0,2,4,5,7)$$ $$\psi_0(\Omega^{\psi_0(\Omega)}) = \psi_0(\psi_1(\psi_1(\psi_0(\psi_1(0)))))$$
$$(0,2,4,5,7,1)$$ $$\psi_0(\Omega^{\psi_0(\Omega)}+1) = \psi_0(\psi_1(\psi_1(\psi_0(\psi_1(0))))+\psi_0(0))$$
$$(0,2,4,5,7,2)$$ $$\psi_0(\Omega^{\psi_0(\Omega)}+\Omega) = \psi_0(\psi_1(\psi_1(\psi_0(\psi_1(0))))+\psi_1(0))$$
$$(0,2,4,5,7,3)$$ $$\psi_0(\Omega^{\psi_0(\Omega)} \times \omega) = \psi_0(\psi_1(\psi_1(\psi_0(\psi_1(0)))+\psi_0(0)))$$
$$(0,2,4,5,7,4)$$ $$\psi_0(\Omega^{\psi_0(\Omega)+1}) = \psi_0(\psi_1(\psi_1(\psi_0(\psi_1(0)))+\psi_1(0)))$$
$$(0,2,4,5,7,5)$$ $$\psi_0(\Omega^{\psi_0(\Omega+1)}) = \psi_0(\psi_1(\psi_1(\psi_0(\psi_1(0))+\psi_0(0))))$$
$$(0,2,4,5,7,6)$$ $$\psi_0(\Omega^{\psi_0(\Omega)^{\omega}}) = \psi_0(\psi_1(\psi_1(\psi_0(\psi_1(0)+\psi_0(0)))))$$
$$(0,2,4,5,7,7)$$ $$\psi_0(\Omega^{\psi_0(\Omega \times 2)}) = \psi_0(\psi_1(\psi_1(\psi_0(\psi_1(0)+\psi_1(0)))))$$
$$(0,2,4,5,7,8)$$ $$\psi_0(\Omega^{\psi_0(\Omega \times \omega)}) = \psi_0(\psi_1(\psi_1(\psi_0(\psi_1(\psi_0(0))))))$$
$$(0,2,4,5,7,8,1)$$ $$\psi_0(\Omega^{\psi_0(\Omega \times \omega)}+1) = \psi_0(\psi_1(\psi_1(\psi_0(\psi_1(\psi_0(0)))))+\psi_0(0))$$
$$(0,2,4,5,7,8,2)$$ $$\psi_0(\Omega^{\psi_0(\Omega \times \omega)}+\Omega) = \psi_0(\psi_1(\psi_1(\psi_0(\psi_1(\psi_0(0)))))+\psi_1(0))$$
$$(0,2,4,5,7,8,3)$$ $$\psi_0(\Omega^{\psi_0(\Omega \times \omega)} \times \omega) = \psi_0(\psi_1(\psi_1(\psi_0(\psi_1(\psi_0(0))))+\psi_0(0)))$$
$$(0,2,4,5,7,8,4)$$ $$\psi_0(\Omega^{\psi_0(\Omega \times \omega)+1}) = \psi_0(\psi_1(\psi_1(\psi_0(\psi_1(\psi_0(0))))+\psi_1(0)))$$
$$(0,2,4,5,7,8,5)$$ $$\psi_0(\Omega^{\psi_0(\Omega \times \omega+1)}) = \psi_0(\psi_1(\psi_1(\psi_0(\psi_1(\psi_0(0)))+\psi_0(0))))$$
$$(0,2,4,5,7,8,6)$$ $$\psi_0(\Omega^{\psi_0(\Omega \times \omega)^{\omega}}) = \psi_0(\psi_1(\psi_1(\psi_0(\psi_1(\psi_0(0))+\psi_0(0)))))$$
$$(0,2,4,5,7,8,7)$$ $$\psi_0(\Omega^{\psi_0(\Omega \times (\omega+1))}) = \psi_0(\psi_1(\psi_1(\psi_0(\psi_1(\psi_0(0))+\psi_1(0)))))$$
$$(0,2,4,5,7,8,8)$$ $$\psi_0(\Omega^{\psi_0(\Omega \times \omega^2)}) = \psi_0(\psi_1(\psi_1(\psi_0(\psi_1(\psi_0(0)+\psi_0(0))))))$$
$$(0,2,4,5,7,8,9)$$ $$\psi_0(\Omega^{\psi_0(\Omega \times \omega^{\omega})}) = \psi_0(\psi_1(\psi_1(\psi_0(\psi_1(\psi_0(\psi_0(0)))))))$$
$$(0,2,4,5,7,8,10)$$ $$\psi_0(\Omega^{\psi_0(\Omega \times \psi_0(\Omega))}) = \psi_0(\psi_1(\psi_1(\psi_0(\psi_1(\psi_0(\psi_1(0)))))))$$
$$(0,2,4,5,7,9)$$ $$\psi_0(\Omega^{\psi_0(\Omega^2)}) = \psi_0(\psi_1(\psi_1(\psi_0(\psi_1(\psi_1(0))))))$$
$$(0,2,4,6)$$ $$\psi_0(\Omega^{\Omega}) = \psi_0(\psi_1(\psi_1(\psi_1(0))))$$
$$(0,3)$$ $$\psi_0(\Omega_2) = \psi_0(\psi_2(0))$$
$$(0,3,1)$$ $$\psi_0(\Omega_2+1) = \psi_0(\psi_2(0)+\psi_0(0))$$
$$(0,3,2)$$ $$\psi_0(\Omega_2+\Omega) = \psi_0(\psi_2(0)+\psi_1(0))$$
$$(0,3,3)$$ $$\psi_0(\Omega_2 \times 2) = \psi_0(\psi_2(0)+\psi_2(0))$$
$$(0,3,4)$$ $$\psi_0(\Omega_2 \times \omega) = \psi_0(\psi_2(\psi_0(0)))$$
$$(0,3,5)$$ $$\psi_0(\Omega_2 \times \Omega) = \psi_0(\psi_2(\psi_1(0)))$$
$$(0,3,6)$$ $$\psi_0(\Omega_2^2) = \psi_0(\psi_2(\psi_2(0)))$$
$$(0,4)$$ $$\psi_0(\Omega_3) = \psi_0(\psi_3(0))$$
$$(0,n+1)$$ $$\psi_0(\Omega_n) = \psi_0(\psi_n(0))$$

## Sources

1. ゆきとの巨大数研究wikiのユーザーページ
2. ゆきと, ハイパー原始数列
3. The notation is defined in [Buc1] p. 200.
4. The addition is defined in [Buc1] p. 203.
5. The scalar multiplication is defined in [Buc1] p. 203.
6. The notion of an ordinal term is defined in [Buc1] p. 201.
7. The recursive system of fundamental sequences is defined in [Buc1] p. 203--204 except for the case of ([].4) (ii). Replacing ([].4) (ii) by the rule 6 in Definition in [Buc2] p. 6 applied to the convention $$\Omega_0 = 1$$, we obtain the full definition of the recursive system of fundamental sequences.
8. The map $$\textrm{dom}$$ is defined in [Buc1] p. 203--204.
9. The map $$o$$ is defined in [Buc1] p. 201.

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