日本の巨大数界隈に於いて順序数崩壊関数 を用いる人物は少ない。順序数 や順序数表記 など、ある種の高等な概念を用いることや、単純に難しいことが原因と言われている。確かにそれは正しい。しかし私は日本語で書かれた順序数崩壊関数の記事の個数や入門の記事が少ないことも大きな原因になっていると推測する。よって以下では順序数崩壊関数の入門記事を書くために、その記事で用いられる順序数崩壊関数、順序数表記、そして基本列 を定義する。巨大数研究wikiのルールに於いて参考文献が明示されている必要がある。よって本記事を参考文献として提示するためだ。速攻で作ったので多分間違っているところがあると思います。指摘お願いしたいです。
順序数崩壊関数 [ ]
ψ
:
O
n
→
O
n
{\displaystyle \psi\colon\mathrm{On}\to\mathrm{On}}
C
(
α
)
{\displaystyle \mathrm{C}(\alpha)}
ψ
(
α
)
:=
min
{
ξ
∈
O
n
∣
ξ
∉
C
(
α
)
}
{\displaystyle \psi(\alpha):=\min\{\xi\in\mathrm{On}\mid\xi\notin\mathrm{C}(\alpha)\}}
C
(
α
)
{\displaystyle \mathrm{C}(\alpha)}
{
0
,
Ω
}
{\displaystyle \{0,\Omega\}}
λ
ξ
λ
ζ
.
ξ
+
ζ
,
λ
ξ
.
ω
¯
ξ
,
λ
ξ
<
α
.
ψ
(
ξ
)
{\displaystyle \lambda \xi\lambda\zeta.\xi+\zeta,\lambda\xi.\overline{\omega}^\xi,\lambda\xi<\alpha.\psi(\xi)}
ここで構文解析に失敗 (不明な関数「\begin{cases}」): {\displaystyle \overline{\omega}^\xi:=\begin{cases}\omega^{\xi+1} & \text{if $\xi=\gamma+n$ for some $\gamma\in\mathrm{On},n\in\omega$ such that $\omega^\gamma=\gamma$}\\ \omega^\xi & \text{otherwise}\end{cases}}
とする。
順序数表記 [ ] 相互再帰で文字列の集合
O
T
,
L
,
A
P
,
E
{\displaystyle \mathsf{OT},\mathsf{L},\mathsf{AP},\mathsf{E}}
K
{\displaystyle \mathsf{K}}
≺
{\displaystyle \prec}
E
⊆
A
P
⊆
O
T
{\displaystyle \mathsf{E}\subseteq\mathsf{AP}\subseteq\mathsf{OT}}
E
⊆
A
P
∖
{
ω
¯
0
_
}
⊆
L
⊆
O
T
{\displaystyle \mathsf{E}\subseteq\mathsf{AP}\setminus\{\overline{\omega}^\underline{0}\}\subseteq\mathsf{L}\subseteq\mathsf{OT}}
0
_
∈
O
T
{\displaystyle \underline{0}\in\mathsf{OT}}
Ω
_
∈
E
{\displaystyle \underline{\Omega}\in\mathsf{E}}
s
⪯
t
:⇔
s
≺
t
∨
s
=
t
{\displaystyle s\preceq t:\Leftrightarrow s\prec t\lor s=t}
n
>
0
{\displaystyle n > 0}
s
0
,
…
,
s
n
∈
A
P
{\displaystyle s_0,\ldots,s_n\in\mathsf{AP}}
s
0
⪰
⋯
⪰
s
n
{\displaystyle s_0\succeq\cdots\succeq s_n}
s
0
+
⋯
+
s
n
∈
O
T
{\displaystyle s_0+\cdots +s_n\in\mathsf{OT}}
s
n
∈
L
{\displaystyle s_n\in\mathsf{L}}
s
0
+
⋯
+
s
n
∈
L
{\displaystyle s_0+\cdots +s_n\in\mathsf{L}}
s
∈
O
T
{\displaystyle s\in\mathsf{OT}}
ω
¯
s
∈
A
P
{\displaystyle \overline{\omega}^s\in\mathsf{AP}}
s
∈
O
T
{\displaystyle s\in\mathsf{OT}}
t
∈
K
(
s
)
{\displaystyle t\in\mathsf{K}(s)}
t
≺
s
{\displaystyle t\prec s}
ψ
(
s
)
∈
E
{\displaystyle \psi(s)\in\mathsf{E}}
K
(
0
_
)
:=
K
(
Ω
_
)
:=
∅
{\displaystyle \mathsf{K}(\underline{0}):=\mathsf{K}(\underline{\Omega}):=\emptyset}
K
(
s
0
+
⋯
+
s
n
)
=
⋃
i
<
n
+
1
K
(
s
i
)
{\displaystyle \mathsf{K}(s_0+\cdots +s_n)=\bigcup_{i<n+1}\mathsf{K}(s_i)}
K
(
ω
¯
s
)
=
K
(
s
)
{\displaystyle \mathsf{K}(\overline{\omega}^s)=\mathsf{K}(s)}
K
(
ψ
(
s
)
)
=
{
s
}
∪
K
(
s
)
{\displaystyle \mathsf{K}(\psi(s))=\{s\}\cup\mathsf{K}(s)}
s
≺
t
{\displaystyle s\prec t}
s
=
0
_
{\displaystyle s=\underline{0}}
t
≠
0
_
{\displaystyle t\neq\underline{0}}
s
=
s
0
+
⋯
+
s
m
{\displaystyle s=s_0+\cdots +s_m}
t
=
t
0
+
⋯
+
t
n
{\displaystyle t=t_0+\cdots +t_n}
i
<
m
{\displaystyle i<m}
j
≤
i
{\displaystyle j\leq i}
s
j
=
t
j
{\displaystyle s_j=t_j}
s
i
+
1
≺
t
i
+
1
{\displaystyle s_{i+1}\prec t_{i+1}}
m
<
n
{\displaystyle m < n}
j
≤
m
{\displaystyle j\leq m}
s
j
=
i
j
{\displaystyle s_j=i_j}
s
=
s
0
+
⋯
+
s
m
{\displaystyle s=s_0+\cdots +s_m}
t
∈
A
P
{\displaystyle t\in\mathsf{AP}}
s
0
≺
t
{\displaystyle s_0\prec t}
s
∈
A
P
{\displaystyle s\in\mathsf{AP}}
t
=
t
0
+
⋯
+
t
n
{\displaystyle t=t_0+\cdots +t_n}
s
⪯
t
0
{\displaystyle s\preceq t_0}
s
=
ω
¯
s
0
{\displaystyle s=\overline{\omega}^{s_0}}
t
=
ω
¯
t
0
{\displaystyle t=\overline{\omega}^{t_0}}
s
0
≺
t
0
{\displaystyle s_0\prec t_0}
s
=
ω
¯
s
0
{\displaystyle s=\overline{\omega}^{s_0}}
t
∈
E
{\displaystyle t\in\mathsf{E}}
s
0
≺
t
{\displaystyle s_0\prec t}
s
∈
E
{\displaystyle s\in\mathsf{E}}
t
=
ω
¯
t
0
{\displaystyle t=\overline{\omega}^{t_0}}
s
⪯
t
0
{\displaystyle s\preceq t_0}
s
=
ψ
(
s
0
)
{\displaystyle s=\psi(s_0)}
t
=
ψ
(
t
0
)
{\displaystyle t=\psi(t_0)}
s
0
≺
t
0
{\displaystyle s_0\prec t_0}
s
=
ψ
(
s
0
)
{\displaystyle s=\psi(s_0)}
t
=
Ω
_
{\displaystyle t=\underline{\Omega}}
基本列 [ ]
s
∈
O
T
{\displaystyle s\in\mathsf{OT}}
t
p
:
O
T
→
O
T
{\displaystyle \mathrm{tp}\colon\mathsf{OT}\to\mathsf{OT}}
x
≺
t
p
(
s
)
{\displaystyle x\prec\mathrm{tp}(s)}
s
[
x
]
{\displaystyle s[x]}
t
p
(
0
_
)
:=
0
_
{\displaystyle \mathrm{tp}(\underline{0}):=\underline{0}}
t
p
(
1
_
)
:=
1
_
{\displaystyle \mathrm{tp}(\underline{1}):=\underline{1}}
1
_
[
0
]
:=
0
_
{\displaystyle \underline{1}[0]:=\underline{0}}
t
p
(
Ω
_
)
:=
Ω
_
{\displaystyle \mathrm{tp}(\underline{\Omega}):=\underline{\Omega}}
Ω
_
[
x
]
:=
x
{\displaystyle \underline{\Omega}[x]:=x}
t
p
(
s
0
+
⋯
+
s
m
)
=
t
p
(
s
m
)
{\displaystyle \mathrm{tp}(s_0+\cdots +s_m)=\mathrm{tp}(s_m)}
(
s
0
+
⋯
+
s
m
)
[
x
]
=
s
0
+
⋯
+
s
m
[
x
]
{\displaystyle (s_0+\cdots +s_m)[x]=s_0+\cdots +s_m[x]}
t
p
(
ω
¯
s
0
+
1
)
=
ω
_
{\displaystyle \mathrm{tp}(\overline{\omega}^{s_0+1})=\underline{\omega}}
ω
¯
s
0
+
1
_
[
x
]
:=
ω
s
0
¯
x
{\displaystyle \overline{\omega}^{s_0+\underline{1}}[x]:=\overline{\omega^{s_0}}x}
s
0
∈
L
∖
E
{\displaystyle s_0\in\mathsf{L}\setminus\mathsf{E}}
t
p
(
ω
¯
s
0
)
:=
t
p
(
s
0
)
{\displaystyle \mathrm{tp}(\overline{\omega}^{s_0}):=\mathrm{tp}(s_0)}
ω
¯
s
0
[
x
]
:=
s
0
[
x
]
{\displaystyle \overline{\omega}^{s_0}[x]:=s_0[x]}
s
0
∈
E
{\displaystyle s_0\in\mathsf{E}}
t
p
(
ω
¯
s
0
)
:=
ω
_
{\displaystyle \mathrm{tp}(\overline{\omega}^{s_0}):=\underline{\omega}}
ω
¯
s
0
[
x
]
:=
s
0
x
{\displaystyle \overline{\omega}^{s_0}[x]:=s_0x}
s
0
∉
L
{\displaystyle s_0\notin\mathsf{L}}
t
p
(
ψ
(
s
0
)
)
:=
ω
_
{\displaystyle \mathrm{tp}(\psi(s_0)):=\underline{\omega}}
構文解析に失敗 (不明な関数「\begin{cases}」): {\displaystyle \psi(s_0)[x]:=\begin{cases}\overline{\omega}_x(\underline{0}) & \text{if$s_0=0$}\\ \overline{\omega}_x(\psi(s_0')) & \text{$s_0=s_0'+\underline{1}$}\end{cases}}
s
0
∈
L
{\displaystyle s_0\in\mathsf{L}}
t
p
(
s
0
)
≺
Ω
_
{\displaystyle \mathrm{tp}(s_0)\prec\underline{\Omega}}
t
p
(
ψ
(
s
0
)
)
:=
t
p
(
s
0
)
{\displaystyle \mathrm{tp}(\psi(s_0)):=\mathrm{tp}(s_0)}
ψ
(
s
0
)
[
x
]
:=
ψ
(
s
0
[
x
]
)
{\displaystyle \psi(s_0)[x]:=\psi(s_0[x])}
s
0
∈
L
{\displaystyle s_0\in\mathsf{L}}
t
p
(
s
0
)
=
Ω
_
{\displaystyle \mathrm{tp}(s_0)=\underline{\Omega}}
t
p
(
ψ
(
s
0
)
)
:=
ω
_
{\displaystyle \mathrm{tp}(\psi(s_0)):=\underline{\omega}}
ψ
(
s
0
)
[
x
]
:=
ψ
(
s
0
[
t
x
]
)
{\displaystyle \psi(s_0)[x]:=\psi(s_0[t_x])}
t
0
_
:=
0
_
,
t
n
+
1
_
:=
ψ
(
s
0
[
t
n
_
]
)
{\displaystyle t_\underline{0}:=\underline{0},t_\underline{n+1}:=\psi(s_0[t_\underline{n}])}
ただし
1
_
:=
ω
¯
0
_
,
ω
_
:=
ω
¯
1
_
{\displaystyle \underline{1}:=\overline{\omega}^\underline{0},\underline{\omega}:=\overline{\omega}^\underline{1}}
n
>
1
{\displaystyle n>1}
構文解析に失敗 (構文エラー): {\displaystyle \underline{n}:=\underbrace{\underline{1}+\cdots+\underline{1}}_{\text{$n$ times}}}
とし、構文解析に失敗 (構文エラー): {\displaystyle s\underline{n}:=\underbrace{\underline{s}+\cdots+\underline{s}}_{\text{$n$ times}}}
、
ω
¯
0
_
(
s
)
:=
s
,
ω
¯
n
+
1
_
(
s
)
:=
ω
¯
ω
¯
n
_
(
s
)
{\displaystyle \overline{\omega}_\underline{0}(s):=s,\overline{\omega}_\underline{n+1}(s):=\overline{\omega}^{\overline{\omega}_\underline{n}(s)}}
余談 [ ] 本題には全く関係ないが巨大数の慣習にならいここで一つ巨大数を定義し、命名しよう。まず
ψ
(
ε
Ω
+
1
)
[
−
]
:
N
→
O
T
,
ψ
(
ε
Ω
+
1
)
[
n
]
:=
ψ
(
ω
¯
n
_
(
Ω
_
)
)
{\displaystyle \psi(\varepsilon_{\Omega+1})[{-}]:\mathbb{N}\to\mathsf{OT},\psi(\varepsilon_{\Omega+1})[n]:=\psi(\overline{\omega}_{\underline{n}}({\underline{\Omega}}))}
{
f
s
}
s
≺
Ω
_
{\displaystyle \{f_s\}_{s\prec\underline{\Omega}}}
急成長階層 とする。
f
ψ
(
ε
Ω
+
1
)
[
10
100
]
(
10
100
)
{\displaystyle f_{\psi(\varepsilon_{\Omega+1})[10^{100}]}(10^{100})}
なるべく分かりやすいOCFに伴う基本列による巨大数 (the large number with fundamental sequences of the OCF defined as simple as possible) とする。
![{\displaystyle s[x]}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/a6003c77a63d77f4fc4985cba72970416d51e893)
![{\displaystyle {\underline {1}}[0]:={\underline {0}}}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/ffcdd115cb81ffdea7a869b91e9016e1fa3608d5)
![{\displaystyle {\underline {\Omega }}[x]:=x}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/62fe68a1f5c3f3892da96bee009ec1eee86585e4)
![{\displaystyle (s_{0}+\cdots +s_{m})[x]=s_{0}+\cdots +s_{m}[x]}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/afa0d19de09fb500cd216c4f153420e14914174e)
![{\displaystyle {\overline {\omega }}^{s_{0}+{\underline {1}}}[x]:={\overline {\omega ^{s_{0}}}}x}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/43c8415ac89d1d9f9bcc9543ed5f9cc81faa5423)
![{\displaystyle {\overline {\omega }}^{s_{0}}[x]:=s_{0}[x]}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/1e8890c14a0b2cffa192b355be9641d1fb1a1efc)
![{\displaystyle {\overline {\omega }}^{s_{0}}[x]:=s_{0}x}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/27a6d21292de240a8c4d6c69334c0dbff504a768)
![{\displaystyle \psi (s_{0})[x]:=\psi (s_{0}[x])}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/254309b857f06f20ab2a39affa5e9a47db6fcd9d)
![{\displaystyle \psi (s_{0})[x]:=\psi (s_{0}[t_{x}])}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/5cb9d626b03591d926622bf72ead7b721aaf6171)
![{\displaystyle t_{\underline {0}}:={\underline {0}},t_{\underline {n+1}}:=\psi (s_{0}[t_{\underline {n}}])}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/6f771eaefa917ad1837d1ddb8900c4e32d066bf8)
![{\displaystyle \psi (\varepsilon _{\Omega +1})[{-}]:\mathbb {N} \to {\mathsf {OT}},\psi (\varepsilon _{\Omega +1})[n]:=\psi ({\overline {\omega }}_{\underline {n}}({\underline {\Omega }}))}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/819a208541ee9268c1fb4b18c374aabea7e1842b)
![{\displaystyle f_{\psi (\varepsilon _{\Omega +1})[10^{100}]}(10^{100})}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/1c4d62fb008464b2d98f5ff1ad1b3a5557cf58a3)