θψ function

Japanese version: https://googology.fandom.com/ja/wiki/%E3%83%A6%E3%83%BC%E3%82%B6%E3%83%BC%E3%83%96%E3%83%AD%E3%82%B0:Kanrokoti/%CE%B8%CF%88%E9%96%A2%E6%95%B0

Overview

We define θψ function. This notation is based on 三関数 (三 function) by p進大好きbot.

The following contains terms of OCF for convenience although the notation is not OCF.

We can roughly expect following correspondences by analyzing Analysis of 三関数.

• \(\textrm{三}_1(1) = M \times \omega\)
• \(\textrm{三}_1(0)+\textrm{三}_1(0) = M+M\)
• \(\textrm{三}_1(0) = M\)
• \(\textrm{三}_0(\textrm{三}_1(1)) = \Omega_{\omega}\)
• \(\textrm{三}_0(\textrm{三}_1(0)+\textrm{三}_1(0)) = \Omega_2\)
• \(\textrm{三}_0(\textrm{三}_1(0)) = \Omega\)
• \(\textrm{三}_0(\textrm{三}_0(\textrm{三}_1(1))) = \textrm{BO}\)
• \(\textrm{三}_0(\textrm{三}_0(\textrm{三}_1(0)+\textrm{三}_1(0))) = \textrm{BHO}\)
• \(\textrm{三}_0(\textrm{三}_0(\textrm{三}_1(0))) = \varepsilon_0\)

From these, we can think that 三 function has:

1. Large regular cardinals in the group of \(\textrm{三}_1(0)\) are collapsed to uncountable regular cardinals in the group of \(\textrm{三}_0(\textrm{三}_1(0))\).
2. Uncountable regular cardinals in the group of \(\textrm{三}_0(\textrm{三}_1(0))\) are collapsed to countable ordinals in the group of \(\textrm{三}_0(\textrm{三}_0(\textrm{三}_1(0)))\).

these two collapsing layers.

θψ function aims to construct clearer system of these two collapsing layers which can be seen in 三 function. Specifically, we think following two collapsing layers:

1. \(\theta\) function which collapses large regular cardinals to regular cardinals.
2. \(\psi\) function which collapses regular cardinals to countable ordinals.

When we think these kind of two collapsing layers, how \(\psi\) handles \(\theta\) is very important. There are many possible ways to configure it, but θψ function does it in the following way:

1. Buchholz-type collapsing way, but \(\psi_{\alpha}(0) \neq \Omega_{\alpha}\) and \(\psi_{\psi_0(0)}(0)\) is nonstandard.
2. \(\theta(\alpha,\beta)\) may correspond to some kind of cardinal and \(\psi_{\theta(\alpha,\beta)}(0)\) is standard, but \(\psi_{\theta(\alpha,\beta)}(\theta(\alpha,\beta))\) is nonstandard unlike Rathjen-type \(\psi_{\Omega}(\Omega)\) being standard.

Furthermore, 2-var \(θ\) function has following two standard forms:

1. \(\theta(1,0) = \theta(0,\theta(0,\theta(0,...)))\)
2. \(\theta(0,\theta(1,0)) = \theta(0,\theta(0,\theta(0,...)))\)

We call the former-type "Weak 2-var θψ function" and the latter-type "2-var θψ function". Weak 2-var θψ function has simple standard form and calculation rules, whereas 2-var θψ function may have the strength which represents the limit of Weak 2-var θψ function as \(\psi_0(\psi_{\theta(0,\theta(1,\theta(1,0)))}(0))\). In addition, if we assume Weak multi-var θψ function, 2-var θψ function may represent the limit of it as \(\psi_0(\psi_{\theta(0,\theta(1,\theta(1,1)))}(0))\).

We formalize 1-var θψ function and 2-var θψ function separately for the sake of simplicity.

1-var θψ function calculator by Naruyoko

Weak 2-var θψ function calculator by Naruyoko

2-var θψ function calculator by Naruyoko

Special thanks: p進大好きbot Naruyoko Mrna den

1-var θψ function

Notation

Let \(T_{\theta}\), \(PT_{\theta}\), \(T_{\psi}\), \(PT_{\psi}\), \(T\), and \(PT\) are sets of formal strings consisting of \(0\), \(+\), \(\theta\), \(\psi\), \((\), \()\), and a comma "\(,\)", which are simultaneously defined in the following recursive way:

1. \(0 \in T_{\theta} \cap T_{\psi} \cap T\).
2. For any \((a,b) \in PT_{\theta} \times (T \setminus \{0\})\), \(a+b \in T_{\theta} \cap T\).
3. For any \((a,b) \in PT_{\psi} \times (T_{\psi} \setminus \{0\})\), \(a+b \in T_{\psi} \cap T\).
4. For any \(a \in T\), \(\theta(a) \in PT_{\theta} \cap T_{\theta} \cap PT \cap T\).
5. For any \((a,b) \in T_{\theta} \times T_{\psi}\), \(\psi(a,b) \in PT_{\psi} \cap T_{\psi} \cap PT \cap T\).

For each \((a,b) \in T_{\theta} \times T_{\psi}\), \(\psi(a,b)\) is abbreviated as \(\psi_a(b)\).

We use \(\omega\) as a formal string in order to keep the whole system computable, and we define \(\mathbb{N}^+ := \mathbb{N} \cup \{\omega\}\).

A computable total map \begin{eqnarray*} $ \colon \mathbb{N}^+ & \to & T \\ n & \mapsto & $n \end{eqnarray*} is defined in the following recursive way:

1. If \(n = 0\), then \($n := 0\).
2. If \(n = 1\), then \($n := \psi_0(0)\).
3. If \(n = \omega\), then \($n := \psi_0($1)\).
4. If \(n \notin \{0,1,\omega\}\), then \($n := $(n-1)+$1\).

Ordering

\(2\)-ary relations \(s \le t\) and \(s \lt t\) on \(T\) are simultaneously defined in the following recursive way:

Definition of \(s \le t\)

1. If \(s = t\), then \(s \le t\) is true.
2. If \(s \neq t\), then \(s \le t\) is equivalent to \(s \lt t\).

Definition of \(s \lt t\)

1. If \(t = 0\), then \(s \lt t\) is false.
2. If \(t \neq 0\) and \(s = 0\), then \(s \lt t\) is true.
3. Suppose that \(t \neq 0\) and \(s = a+b\) for some \((a,b) \in PT \times (T \setminus \{0\})\).
   1. If \(t = c+d\) for some \((c,d) \in PT \times (T \setminus \{0\})\), then \(s \lt t\) is equivalent to that either one of the following holds:
      1. \(a \lt c\).
      2. \(a = c\) and \(b \lt d\).
   2. If \(t = \theta(c)\) for some \(c \in T\), then \(s \lt t\) is equivalent to \(a \lt t\).
   3. If \(t = \psi_c(d)\) for some \((c,d) \in T_{\theta} \times T_{\psi}\), then \(s \lt t\) is equivalent to \(a \lt t\).
4. Suppose that \(t \neq 0\) and \(s = \theta(a)\) for some \(a \in T\).
   1. If \(t = b+c\) for some \((b,c) \in PT \times (T \setminus \{0\})\), then \(s \lt t\) is equivalent to \(s \le b\).
   2. If \(t = \theta(b)\) for some \(b \in T\), then \(s \lt t\) is equivalent to \(a \lt b\).
   3. If \(t = \psi_b(c)\) for some \((b,c) \in T_{\theta} \times T_{\psi}\), then \(s \lt t\) is false.
5. Suppose that \(t \neq 0\) and \(s = \psi_a(b)\) for some \((a,b) \in T_{\theta} \times T_{\psi}\).
   1. If \(t = c+d\) for some \((c,d) \in PT \times (T \setminus \{0\})\), then \(s \lt t\) is equivalent to \(s \le c\).
   2. If \(t = \theta(c)\) for some \(c \in T\), then \(s \lt t\) is true.
   3. If \(t = \psi_c(d)\) for some \((c,d) \in T_{\theta} \times T_{\psi}\), then \(s \lt t\) is equivalent to that either one of the following holds:
      1. \(a \lt c\).
      2. \(a = c\) and \(b \lt d\).

Cofinality

A computable total map \begin{eqnarray*} \textrm{dom} \colon T & \to & T \\ s & \mapsto & \textrm{dom}(s) \end{eqnarray*} is defined in the following recursive way:

1. If \(s = 0\), then \(\textrm{dom}(s) := 0\).
2. If \(s = a+b\) for some \((a,b) \in PT \times (T \setminus \{0\})\), then \(\textrm{dom}(s) := \textrm{dom}(b)\).
3. Suppose that \(s = \theta(a)\) for some \(a \in T\).
   1. If \(\textrm{dom}(a) = 0\), then \(\textrm{dom}(s) := s\).
   2. If \(\textrm{dom}(a) = $1\), then \(\textrm{dom}(s) := $\omega\).
   3. If \(\textrm{dom}(a) = \theta(0)\), then \(\textrm{dom}(s) := \theta(\theta(0))\).
   4. If \(\textrm{dom}(a) \notin \{0,$1,\theta(0)\}\), then \(\textrm{dom}(s) := \textrm{dom}(a)\).
4. Suppose that \(s = \psi_a(b)\) for some \((a,b) \in T_{\theta} \times T_{\psi}\).
   1. Suppose that \(\textrm{dom}(b) = 0\).
      1. If \(\textrm{dom}(a) \in \{0,\theta(0),\theta(\theta(0))\}\), then \(\textrm{dom}(s) := s\).
      2. Suppose that \(\textrm{dom}(a) \notin \{0,\theta(0),\theta(\theta(0))\}\).
         1. If \(\textrm{dom}(a) \lt s\), then \(\textrm{dom}(s) := \textrm{dom}(a)\).
         2. If \(s \le \textrm{dom}(a)\), then \(\textrm{dom}(s) := $\omega\).
   2. If \(\textrm{dom}(b) = $1\), then \(\textrm{dom}(s) := $\omega\).
   3. Suppose that \(\textrm{dom}(b) \notin \{0,$1\}\).
      1. If \(\textrm{dom}(b) \lt s\), then \(\textrm{dom}(s) := \textrm{dom}(b)\).
      2. If \(s \le \textrm{dom}(b)\), then \(\textrm{dom}(s) := $\omega\).

Search function

A computable total map \begin{eqnarray*} \textrm{BP} \colon T & \to & T \\ s & \mapsto & \textrm{BP}(s) \end{eqnarray*} is defined in the following recursive way:

1. If \(s = 0\), then \(\textrm{BP}(s) := 0\).
2. Suppose that \(s = a+b\) for some \((a,b) \in PT \times (T \setminus \{0\})\).
   1. If \(a \in PT_{\theta}\), then \(\textrm{BP}(s) := \textrm{BP}(b)\).
   2. If \(a \notin PT_{\theta}\), then \(\textrm{BP}(s) := s\).
3. If \(s = \theta(a)\) for some \(a \in T\), then \(\textrm{BP}(s) := \textrm{BP}(a)\).
4. If \(s = \psi_a(b)\) for some \((a,b) \in T_{\theta} \times T_{\psi}\), then \(\textrm{BP}(s) := s\).

Fundamental sequence

A computable total map \begin{eqnarray*} [ \ ] \colon T^2 & \to & T \\ (s,t) & \mapsto & s[t] \end{eqnarray*} is defined in the following recursive way:

1. If \(s = 0\), then \(s[t] := 0\).
2. If \(s = a+b\) for some \((a,b) \in PT \times (T \setminus \{0\})\), then put \(b' := b[t]\).
   1. If \(b' = 0\), then \(s[t] := a\).
   2. If \(b' \neq 0\), then \(s[t] := a+b'\).
3. Suppose that \(s = \theta(a)\) for some \(a \in T\).
   1. If \(\textrm{dom}(a) = 0\), then \(s[t] := t\).
   2. Suppose that \(\textrm{dom}(a) = $1\).
      1. If \(t = t[0]+$1\), then \(s[t] := s[t[0]]+s[$1]\).
      2. If \(t \neq t[0]+$1\), then \(s[t] := \theta(a[0])\).
   3. If \(\textrm{dom}(a) \notin \{0,$1\}\), then \(s[t] := \theta(a[t])\).
4. Suppose that \(s = \psi_a(b)\) for some \((a,b) \in T_{\theta} \times T_{\psi}\).
   1. Suppose that \(\textrm{dom}(b) = 0\).
      1. If \(\textrm{dom}(a) \in \{0,\theta(0)\}\), then \(s[t] := t\).
      2. If \(\textrm{dom}(a) = \theta(\theta(0))\), then \(s[t] := \psi_{a[t]}(b)\).
      3. Suppose that \(\textrm{dom}(a) \notin \{0,\theta(0),\theta(\theta(0))\}\).
         1. If \(\textrm{dom}(a) \lt s\), then \(s[t] := \psi_{a[t]}(b)\).
         2. If \(s \le \textrm{dom}(a)\), then there is a \(c \in T_{\theta}\) such that \(\textrm{dom}(a) = \psi_c(0)\).
            1. Suppose that \(\textrm{dom}(c) = \theta(0)\).
               1. If \(t = $i\) for some \(i \in \mathbb{N} \setminus \{0\}\) and \(s[t[0]] = \psi_{\Gamma}(b)\) for some \(\Gamma \in T\), then \(s[t] := \psi_{a[\psi_{c[0]}(\textrm{BP}(\Gamma))]}(b)\).
               2. Otherwise, then \(s[t] := \psi_a(b[\psi_{c[0]}(0)])\).
            2. Suppose that \(\textrm{dom}(c) \neq \theta(0)\).
               1. If \(t = $i\) for some \(i \in \mathbb{N} \setminus \{0\}\) and \(s[t[0]] = \psi_{\Gamma}(b)\) for some \(\Gamma \in T\), then \(s[t] := \psi_{a[\textrm{BP}(\Gamma)]}(b)\).
               2. Otherwise, then \(s[t] := \psi_{a[0]}(b)\).
   2. Suppose that \(\textrm{dom}(b) = $1\).
      1. If \(t = t[0]+$1\), then \(s[t] := s[t[0]]+s[$1]\).
      2. If \(t \neq t[0]+$1\), then \(s[t] := \psi_a(b[0])\).
   3. Suppose that \(\textrm{dom}(b) \notin \{0,$1\}\).
      1. If \(\textrm{dom}(b) \lt s\), then \(s[t] := \psi_a(b[t])\).
      2. If \(s \le \textrm{dom}(b)\), then there is a \(c \in T_{\theta}\) such that \(\textrm{dom}(b) = \psi_c(0)\).
         1. Suppose that \(\textrm{dom}(c) = \theta(0)\).
            1. If \(t = $i\) for some \(i \in \mathbb{N} \setminus \{0\}\) and \(s[t[0]] = \psi_a(\Gamma)\) for some \(\Gamma \in T\), then \(s[t] := \psi_a(b[\psi_{c[0]}(\Gamma)])\).
            2. Otherwise, then \(s[t] := \psi_a(b[\psi_{c[0]}(0)])\).
         2. Suppose that \(\textrm{dom}(c) \neq \theta(0)\).
            1. If \(t = $i\) for some \(i \in \mathbb{N} \setminus \{0\}\) and \(s[t[0]] = \psi_a(\Gamma)\) for some \(\Gamma \in T\), then \(s[t] := \psi_a(b[\Gamma])\).
            2. Otherwise, then \(s[t] := \psi_a(b[0])\).

FGH

A computable partial map \begin{eqnarray*} f \colon T \times \mathbb{N}^2 & \to & \mathbb{N} \\ (s,m,n) & \mapsto & f_s^m(n) \end{eqnarray*} is defined in the following recursive way:

1. If \(m = 0\), then \(f_s^m(n) := n\).
2. Suppose that \(m = 1\).
   1. If \(\textrm{dom}(s) = 0\), then \(f_s^m(n) := n+1\).
   2. If \(\textrm{dom}(s) = $1\), then \(f_s^m(n) := f_{s[0]}^n(n)\).
   3. If \(\textrm{dom}(s) \notin \{0,$1\}\), then \(f_s^m(n) := f_{s[$n]}^1(n)\).
3. If \(m \notin \{0,1\}\), then \(f_s^m(n) := f_s^1(f_s^{m-1}(n))\).

Limit function

A computable total map \begin{eqnarray*} \Lambda \colon \mathbb{N} & \to & PT_{\theta} \\ n & \mapsto & \Lambda(n) \end{eqnarray*} is defined in the following recursive way:

1. If \(n = 0\), then \(\Lambda(n) := \theta(0)\).
2. If \(n \neq 0\), then \(\Lambda(n) := \theta(\Lambda(n-1))\).

Standard form

A subset \(OT \subset T\) is defined in the following recursive way:

1. For any \(n \in \mathbb{N}\), \(\psi_0(\psi_{\Lambda(n)}(0)) \in OT\).
2. For any \((s,n) \in OT \times \mathbb{N}\), \(s[$n] \in OT\).

My expectation is that \((OT,\lt)\) is an ordinal notation, or \(OT\) is recursive and the restriction of \(\lt\) to \(OT\) is well-ordered.

Naming

A computable partial map \begin{eqnarray*} F \colon \mathbb{N} & \to & \mathbb{N} \\ n & \mapsto & F(n) \end{eqnarray*} is defined as \(F(n) := f_{\psi_0(\psi_{\Lambda(n)}(0))}^1(n)\).

I name \(F^{10^{100}}(10^{100})\) "1-var theta psi number".

I name an ordinal which correspond to the limit of 1-var θψ function "1-var theta psi ordinal" (1-TPO).

Weak 2-var θψ function

Notation

Let \(T_{\theta}\), \(PT_{\theta}\), \(T_{\psi}\), \(PT_{\psi}\), \(T\), and \(PT\) are sets of formal strings consisting of \(0\), \(+\), \(\theta\), \(\psi\), \((\), \()\), and a comma "\(,\)", which are simultaneously defined in the following recursive way:

1. \(0 \in T_{\theta} \cap T_{\psi} \cap T\).
2. For any \((a,b) \in PT_{\theta} \times (T \setminus \{0\})\), \(a+b \in T_{\theta} \cap T\).
3. For any \((a,b) \in PT_{\psi} \times (T_{\psi} \setminus \{0\})\), \(a+b \in T_{\psi} \cap T\).
4. For any \((a,b) \in T^2\), \(\theta(a,b) \in PT_{\theta} \cap T_{\theta} \cap PT \cap T\).
5. For any \((a,b) \in T_{\theta} \times T_{\psi}\), \(\psi(a,b) \in PT_{\psi} \cap T_{\psi} \cap PT \cap T\).

For each \((a,b) \in T_{\theta} \times T_{\psi}\), \(\psi(a,b)\) is abbreviated as \(\psi_a(b)\).

We use \(\omega\) as a formal string in order to keep the whole system computable, and we define \(\mathbb{N}^+ := \mathbb{N} \cup \{\omega\}\).

A computable total map \begin{eqnarray*} $ \colon \mathbb{N}^+ & \to & T \\ n & \mapsto & $n \end{eqnarray*} is defined in the following recursive way:

1. If \(n = 0\), then \($n := 0\).
2. If \(n = 1\), then \($n := \psi_0(0)\).
3. If \(n = \omega\), then \($n := \psi_0($1)\).
4. If \(n \notin \{0,1,\omega\}\), then \($n := $(n-1)+$1\).

Ordering

\(2\)-ary relations \(s \le t\) and \(s \lt t\) on \(T\) are simultaneously defined in the following recursive way:

Definition of \(s \le t\)

1. If \(s = t\), then \(s \le t\) is true.
2. If \(s \neq t\), then \(s \le t\) is equivalent to \(s \lt t\).

Definition of \(s \lt t\)

1. If \(t = 0\), then \(s \lt t\) is false.
2. If \(t \neq 0\) and \(s = 0\), then \(s \lt t\) is true.
3. Suppose that \(t \neq 0\) and \(s = a+b\) for some \((a,b) \in PT \times (T \setminus \{0\})\).
   1. If \(t = c+d\) for some \((c,d) \in PT \times (T \setminus \{0\})\), then \(s \lt t\) is equivalent to that either one of the following holds:
      1. \(a \lt c\).
      2. \(a = c\) and \(b \lt d\).
   2. If \(t = \theta(c,d)\) for some \((c,d) \in T^2\), then \(s \lt t\) is equivalent to \(a \lt t\).
   3. If \(t = \psi_c(d)\) for some \((c,d) \in T_{\theta} \times T_{\psi}\), then \(s \lt t\) is equivalent to \(a \lt t\).
4. Suppose that \(t \neq 0\) and \(s = \theta(a,b)\) for some \((a,b) \in T^2\).
   1. If \(t = c+d\) for some \((c,d) \in PT \times (T \setminus \{0\})\), then \(s \lt t\) is equivalent to \(s \le c\).
   2. If \(t = \theta(c,d)\) for some \((c,d) \in T^2\), then \(s \lt t\) is equivalent to that either one of the following holds:
      1. \(a \lt c\).
      2. \(a = c\) and \(b \lt d\).
   3. If \(t = \psi_c(d)\) for some \((c,d) \in T_{\theta} \times T_{\psi}\), then \(s \lt t\) is false.
5. Suppose that \(t \neq 0\) and \(s = \psi_a(b)\) for some \((a,b) \in T_{\theta} \times T_{\psi}\).
   1. If \(t = c+d\) for some \((c,d) \in PT \times (T \setminus \{0\})\), then \(s \lt t\) is equivalent to \(s \le c\).
   2. If \(t = \theta(c,d)\) for some \((c,d) \in T^2\), then \(s \lt t\) is true.
   3. If \(t = \psi_c(d)\) for some \((c,d) \in T_{\theta} \times T_{\psi}\), then \(s \lt t\) is equivalent to that either one of the following holds:
      1. \(a \lt c\).
      2. \(a = c\) and \(b \lt d\).

Cofinality

A computable total map \begin{eqnarray*} \textrm{dom} \colon T & \to & T \\ s & \mapsto & \textrm{dom}(s) \end{eqnarray*} is defined in the following recursive way:

1. If \(s = 0\), then \(\textrm{dom}(s) := 0\).
2. If \(s = a+b\) for some \((a,b) \in PT \times (T \setminus \{0\})\), then \(\textrm{dom}(s) := \textrm{dom}(b)\).
3. Suppose that \(s = \theta(a,b)\) for some \((a,b) \in T^2\).
   1. Suppose that \(\textrm{dom}(b) = 0\).
      1. If \(\textrm{dom}(a) = 0\), then \(\textrm{dom}(s) := s\).
      2. If \(\textrm{dom}(a) = $1\), then \(\textrm{dom}(s) := $\omega\).
      3. If \(\textrm{dom}(a) = \theta(0,0)\), then \(\textrm{dom}(s) := \theta(0,\theta(0,0))\).
      4. If \(\textrm{dom}(a) \notin \{0,$1,\theta(0,0)\}\), then \(\textrm{dom}(s) := \textrm{dom}(a)\).
   2. If \(\textrm{dom}(b) = $1\), then \(\textrm{dom}(s) := $\omega\).
   3. If \(\textrm{dom}(b) = \theta(0,0)\), then \(\textrm{dom}(s) := \theta(0,\theta(0,0))\).
   4. If \(\textrm{dom}(b) \notin \{0,$1,\theta(0,0)\}\), then \(\textrm{dom}(s) := \textrm{dom}(b)\).
4. Suppose that \(s = \psi_a(b)\) for some \((a,b) \in T_{\theta} \times T_{\psi}\).
   1. Suppose that \(\textrm{dom}(b) = 0\).
      1. If \(\textrm{dom}(a) \in \{0,\theta(0,0),\theta(0,\theta(0,0))\}\), then \(\textrm{dom}(s) := s\).
      2. Suppose that \(\textrm{dom}(a) \notin \{0,\theta(0,0),\theta(0,\theta(0,0))\}\).
         1. If \(\textrm{dom}(a) \lt s\), then \(\textrm{dom}(s) := \textrm{dom}(a)\).
         2. If \(s \le \textrm{dom}(a)\), then \(\textrm{dom}(s) := $\omega\).
   2. If \(\textrm{dom}(b) = $1\), then \(\textrm{dom}(s) := $\omega\).
   3. Suppose that \(\textrm{dom}(b) \notin \{0,$1\}\).
      1. If \(\textrm{dom}(b) \lt s\), then \(\textrm{dom}(s) := \textrm{dom}(b)\).
      2. If \(s \le \textrm{dom}(b)\), then \(\textrm{dom}(s) := $\omega\).

Search function

A computable total map \begin{eqnarray*} \textrm{BP} \colon T & \to & T \\ s & \mapsto & \textrm{BP}(s) \end{eqnarray*} is defined in the following recursive way:

1. If \(s = 0\), then \(\textrm{BP}(s) := 0\).
2. Suppose that \(s = a+b\) for some \((a,b) \in PT \times (T \setminus \{0\})\).
   1. If \(a \in PT_{\theta}\), then \(\textrm{BP}(s) := \textrm{BP}(b)\).
   2. If \(a \notin PT_{\theta}\), then \(\textrm{BP}(s) := s\).
3. Suppose that \(s = \theta(a,b)\) for some \((a,b) \in T^2\).
   1. If \(b = 0\), then \(\textrm{BP}(s) := \textrm{BP}(a)\).
   2. If \(b \neq 0\), then \(\textrm{BP}(s) := \textrm{BP}(b)\).
4. If \(s = \psi_a(b)\) for some \((a,b) \in T_{\theta} \times T_{\psi}\), then \(\textrm{BP}(s) := s\).

Fundamental sequence

A computable total map \begin{eqnarray*} [ \ ] \colon T^2 & \to & T \\ (s,t) & \mapsto & s[t] \end{eqnarray*} is defined in the following recursive way:

1. If \(s = 0\), then \(s[t] := 0\).
2. If \(s = a+b\) for some \((a,b) \in PT \times (T \setminus \{0\})\), then put \(b' := b[t]\).
   1. If \(b' = 0\), then \(s[t] := a\).
   2. If \(b' \neq 0\), then \(s[t] := a+b'\).
3. Suppose that \(s = \theta(a,b)\) for some \((a,b) \in T^2\).
   1. Suppose that \(\textrm{dom}(b) = 0\).
      1. If \(\textrm{dom}(a) = 0\), then \(s[t] := t\).
      2. Suppose that \(\textrm{dom}(a) = $1\).
         1. If \(t = $i\) for some \(i \in \mathbb{N} \setminus \{0\}\) and \(s[t[0]] = \Gamma\) for some \(\Gamma \in T\), then \(s[t] := \theta(a[0],\Gamma)\).
         2. Otherwise, then \(s[t] := \theta(a[0],b)\).
      3. If \(\textrm{dom}(a) \notin \{0,$1\}\), then \(s[t] := \theta(a[t],b)\).
   2. Suppose that \(\textrm{dom}(b) = $1\).
      1. If \(t = t[0]+$1\), then \(s[t] := s[t[0]]+s[$1]\).
      2. If \(t \neq t[0]+$1\), then \(s[t] := \theta(a,b[0])\).
   3. If \(\textrm{dom}(b) \notin \{0,$1\}\), then \(s[t] := \theta(a,b[t])\).
4. Suppose that \(s = \psi_a(b)\) for some \((a,b) \in T_{\theta} \times T_{\psi}\).
   1. Suppose that \(\textrm{dom}(b) = 0\).
      1. If \(\textrm{dom}(a) \in \{0,\theta(0,0)\}\), then \(s[t] := t\).
      2. If \(\textrm{dom}(a) = \theta(0,\theta(0,0))\), then \(s[t] := \psi_{a[t]}(b)\).
      3. Suppose that \(\textrm{dom}(a) \notin \{0,\theta(0,0),\theta(0,\theta(0,0))\}\).
         1. If \(\textrm{dom}(a) \lt s\), then \(s[t] := \psi_{a[t]}(b)\).
         2. If \(s \le \textrm{dom}(a)\), then there is a \(c \in T_{\theta}\) such that \(\textrm{dom}(a) = \psi_c(0)\).
            1. Suppose that \(\textrm{dom}(c) = \theta(0,0)\).
               1. If \(t = $i\) for some \(i \in \mathbb{N} \setminus \{0\}\) and \(s[t[0]] = \psi_{\Gamma}(b)\) for some \(\Gamma \in T\), then \(s[t] := \psi_{a[\psi_{c[0]}(\textrm{BP}(\Gamma))]}(b)\).
               2. Otherwise, then \(s[t] := \psi_a(b[\psi_{c[0]}(0)])\).
            2. Suppose that \(\textrm{dom}(c) \neq \theta(0,0)\).
               1. If \(t = $i\) for some \(i \in \mathbb{N} \setminus \{0\}\) and \(s[t[0]] = \psi_{\Gamma}(b)\) for some \(\Gamma \in T\), then \(s[t] := \psi_{a[\textrm{BP}(\Gamma)]}(b)\).
               2. Otherwise, then \(s[t] := \psi_{a[0]}(b)\).
   2. Suppose that \(\textrm{dom}(b) = $1\).
      1. If \(t = t[0]+$1\), then \(s[t] := s[t[0]]+s[$1]\).
      2. If \(t \neq t[0]+$1\), then \(s[t] := \psi_a(b[0])\).
   3. Suppose that \(\textrm{dom}(b) \notin \{0,$1\}\).
      1. If \(\textrm{dom}(b) \lt s\), then \(s[t] := \psi_a(b[t])\).
      2. If \(s \le \textrm{dom}(b)\), then there is a \(c \in T_{\theta}\) such that \(\textrm{dom}(b) = \psi_c(0)\).
         1. Suppose that \(\textrm{dom}(c) = \theta(0,0)\).
            1. If \(t = $i\) for some \(i \in \mathbb{N} \setminus \{0\}\) and \(s[t[0]] = \psi_a(\Gamma)\) for some \(\Gamma \in T\), then \(s[t] := \psi_a(b[\psi_{c[0]}(\Gamma)])\).
            2. Otherwise, then \(s[t] := \psi_a(b[\psi_{c[0]}(0)])\).
         2. Suppose that \(\textrm{dom}(c) \neq \theta(0,0)\).
            1. If \(t = $i\) for some \(i \in \mathbb{N} \setminus \{0\}\) and \(s[t[0]] = \psi_a(\Gamma)\) for some \(\Gamma \in T\), then \(s[t] := \psi_a(b[\Gamma])\).
            2. Otherwise, then \(s[t] := \psi_a(b[0])\).

FGH

A computable partial map \begin{eqnarray*} f \colon T \times \mathbb{N}^2 & \to & \mathbb{N} \\ (s,m,n) & \mapsto & f_s^m(n) \end{eqnarray*} is defined in the following recursive way:

1. If \(m = 0\), then \(f_s^m(n) := n\).
2. Suppose that \(m = 1\).
   1. If \(\textrm{dom}(s) = 0\), then \(f_s^m(n) := n+1\).
   2. If \(\textrm{dom}(s) = $1\), then \(f_s^m(n) := f_{s[0]}^n(n)\).
   3. If \(\textrm{dom}(s) \notin \{0,$1\}\), then \(f_s^m(n) := f_{s[$n]}^1(n)\).
3. If \(m \notin \{0,1\}\), then \(f_s^m(n) := f_s^1(f_s^{m-1}(n))\).

Limit function

A computable total map \begin{eqnarray*} \Lambda \colon \mathbb{N} & \to & PT_{\theta} \\ n & \mapsto & \Lambda(n) \end{eqnarray*} is defined in the following recursive way:

1. If \(n = 0\), then \(\Lambda(n) := \theta(0,0)\).
2. If \(n \neq 0\), then \(\Lambda(n) := \theta(\Lambda(n-1),0)\).

Standard form

A subset \(OT \subset T\) is defined in the following recursive way:

1. For any \(n \in \mathbb{N}\), \(\psi_0(\psi_{\Lambda(n)}(0)) \in OT\).
2. For any \((s,n) \in OT \times \mathbb{N}\), \(s[$n] \in OT\).

My expectation is that \((OT,\lt)\) is an ordinal notation, or \(OT\) is recursive and the restriction of \(\lt\) to \(OT\) is well-ordered.

Naming

A computable partial map \begin{eqnarray*} F \colon \mathbb{N} & \to & \mathbb{N} \\ n & \mapsto & F(n) \end{eqnarray*} is defined as \(F(n) := f_{\psi_0(\psi_{\Lambda(n)}(0))}^1(n)\).

I name \(F^{10^{100}}(10^{100})\) "Weak 2-var theta psi number".

I name an ordinal which correspond to the limit of Weak 2-var θψ function "Weak 2-var theta psi ordinal" (W2-TPO).

2-var θψ function

Notation

Let \(T_{\theta}\), \(PT_{\theta}\), \(T_{\psi}\), \(PT_{\psi}\), \(T\), and \(PT\) are sets of formal strings consisting of \(0\), \(+\), \(\theta\), \(\psi\), \((\), \()\), and a comma "\(,\)", which are simultaneously defined in the following recursive way:

1. \(0 \in T_{\theta} \cap T_{\psi} \cap T\).
2. For any \((a,b) \in PT_{\theta} \times (T \setminus \{0\})\), \(a+b \in T_{\theta} \cap T\).
3. For any \((a,b) \in PT_{\psi} \times (T_{\psi} \setminus \{0\})\), \(a+b \in T_{\psi} \cap T\).
4. For any \((a,b) \in T^2\), \(\theta(a,b) \in PT_{\theta} \cap T_{\theta} \cap PT \cap T\).
5. For any \((a,b) \in T_{\theta} \times T_{\psi}\), \(\psi(a,b) \in PT_{\psi} \cap T_{\psi} \cap PT \cap T\).

For each \((a,b) \in T_{\theta} \times T_{\psi}\), \(\psi(a,b)\) is abbreviated as \(\psi_a(b)\).

We use \(\omega\) as a formal string in order to keep the whole system computable, and we define \(\mathbb{N}^+ := \mathbb{N} \cup \{\omega\}\).

A computable total map \begin{eqnarray*} $ \colon \mathbb{N}^+ & \to & T \\ n & \mapsto & $n \end{eqnarray*} is defined in the following recursive way:

1. If \(n = 0\), then \($n := 0\).
2. If \(n = 1\), then \($n := \psi_0(0)\).
3. If \(n = \omega\), then \($n := \psi_0($1)\).
4. If \(n \notin \{0,1,\omega\}\), then \($n := $(n-1)+$1\).

Ordering

\(2\)-ary relations \(s \le t\) and \(s \lt t\) on \(T\) are simultaneously defined in the following recursive way:

Definition of \(s \le t\)

1. If \(s = t\), then \(s \le t\) is true.
2. If \(s \neq t\), then \(s \le t\) is equivalent to \(s \lt t\).

Definition of \(s \lt t\)

1. If \(t = 0\), then \(s \lt t\) is false.
2. If \(t \neq 0\) and \(s = 0\), then \(s \lt t\) is true.
3. Suppose that \(t \neq 0\) and \(s = a+b\) for some \((a,b) \in PT \times (T \setminus \{0\})\).
   1. If \(t = c+d\) for some \((c,d) \in PT \times (T \setminus \{0\})\), then \(s \lt t\) is equivalent to that either one of the following holds:
      1. \(a \lt c\).
      2. \(a = c\) and \(b \lt d\).
   2. If \(t = \theta(c,d)\) for some \((c,d) \in T^2\), then \(s \lt t\) is equivalent to \(a \lt t\).
   3. If \(t = \psi_c(d)\) for some \((c,d) \in T_{\theta} \times T_{\psi}\), then \(s \lt t\) is equivalent to \(a \lt t\).
4. Suppose that \(t \neq 0\) and \(s = \theta(a,b)\) for some \((a,b) \in T^2\).
   1. If \(t = c+d\) for some \((c,d) \in PT \times (T \setminus \{0\})\), then \(s \lt t\) is equivalent to \(s \le c\).
   2. If \(t = \theta(c,d)\) for some \((c,d) \in T^2\), then \(s \lt t\) is equivalent to that either one of the following holds:
      1. \(a \lt c\).
      2. \(a = c\) and \(b \lt d\).
   3. If \(t = \psi_c(d)\) for some \((c,d) \in T_{\theta} \times T_{\psi}\), then \(s \lt t\) is false.
5. Suppose that \(t \neq 0\) and \(s = \psi_a(b)\) for some \((a,b) \in T_{\theta} \times T_{\psi}\).
   1. If \(t = c+d\) for some \((c,d) \in PT \times (T \setminus \{0\})\), then \(s \lt t\) is equivalent to \(s \le c\).
   2. If \(t = \theta(c,d)\) for some \((c,d) \in T^2\), then \(s \lt t\) is true.
   3. If \(t = \psi_c(d)\) for some \((c,d) \in T_{\theta} \times T_{\psi}\), then \(s \lt t\) is equivalent to that either one of the following holds:
      1. \(a \lt c\).
      2. \(a = c\) and \(b \lt d\).

Cofinality

A computable total map \begin{eqnarray*} \textrm{dom} \colon T & \to & T \\ s & \mapsto & \textrm{dom}(s) \end{eqnarray*} is defined in the following recursive way:

1. If \(s = 0\), then \(\textrm{dom}(s) := 0\).
2. If \(s = a+b\) for some \((a,b) \in PT \times (T \setminus \{0\})\), then \(\textrm{dom}(s) := \textrm{dom}(b)\).
3. Suppose that \(s = \theta(a,b)\) for some \((a,b) \in T^2\).
   1. Suppose that \(\textrm{dom}(b) = 0\).
      1. If \(\textrm{dom}(a) \in \{0,$1\}\), then \(\textrm{dom}(s) := s\).
      2. If \(\textrm{dom}(a) = \theta(0,0)\), then \(\textrm{dom}(s) := \theta(0,\theta(0,0))\).
      3. If \(\textrm{dom}(a) \notin \{0,$1,\theta(0,0)\}\), then \(\textrm{dom}(s) := \textrm{dom}(a)\).
   2. If \(\textrm{dom}(b) = $1\), then \(\textrm{dom}(s) := $\omega\).
   3. If \(\textrm{dom}(b) = \theta(0,0)\), then \(\textrm{dom}(s) := \theta(0,\theta(0,0))\).
   4. Suppose that \(\textrm{dom}(b) \notin \{0,$1,\theta(0,0)\}\).
      1. If \(\textrm{dom}(b) \lt s\), then \(\textrm{dom}(s) := \textrm{dom}(b)\).
      2. If \(s \le \textrm{dom}(b)\), then \(\textrm{dom}(s) := $\omega\).
4. Suppose that \(s = \psi_a(b)\) for some \((a,b) \in T_{\theta} \times T_{\psi}\).
   1. Suppose that \(\textrm{dom}(b) = 0\).
      1. If \(\textrm{dom}(a) \in \{0,\theta(0,0),\theta(0,\theta(0,0))\}\), then \(\textrm{dom}(s) := s\).
      2. Suppose that \(\textrm{dom}(a) \notin \{0,\theta(0,0),\theta(0,\theta(0,0))\}\).
         1. If \(\textrm{dom}(a) \lt s\), then \(\textrm{dom}(s) := \textrm{dom}(a)\).
         2. If \(s \le \textrm{dom}(a)\), then \(\textrm{dom}(s) := $\omega\).
   2. If \(\textrm{dom}(b) = $1\), then \(\textrm{dom}(s) := $\omega\).
   3. Suppose that \(\textrm{dom}(b) \notin \{0,$1\}\).
      1. If \(\textrm{dom}(b) \lt s\), then \(\textrm{dom}(s) := \textrm{dom}(b)\).
      2. If \(s \le \textrm{dom}(b)\), then \(\textrm{dom}(s) := $\omega\).

Search function

A computable total map \begin{eqnarray*} \textrm{BP} \colon T & \to & T \\ s & \mapsto & \textrm{BP}(s) \end{eqnarray*} is defined in the following recursive way:

1. If \(s = 0\), then \(\textrm{BP}(s) := 0\).
2. Suppose that \(s = a+b\) for some \((a,b) \in PT \times (T \setminus \{0\})\).
   1. If \(a \in PT_{\theta}\), then \(\textrm{BP}(s) := \textrm{BP}(b)\).
   2. If \(a \notin PT_{\theta}\), then \(\textrm{BP}(s) := s\).
3. Suppose that \(s = \theta(a,b)\) for some \((a,b) \in T^2\).
   1. If \(b = 0\), then \(\textrm{BP}(s) := \textrm{BP}(a)\).
   2. If \(b \neq 0\), then \(\textrm{BP}(s) := \textrm{BP}(b)\).
4. If \(s = \psi_a(b)\) for some \((a,b) \in T_{\theta} \times T_{\psi}\), then \(\textrm{BP}(s) := s\).

Fundamental sequence

A computable total map \begin{eqnarray*} [ \ ] \colon T^2 & \to & T \\ (s,t) & \mapsto & s[t] \end{eqnarray*} is defined in the following recursive way:

1. If \(s = 0\), then \(s[t] := 0\).
2. If \(s = a+b\) for some \((a,b) \in PT \times (T \setminus \{0\})\), then put \(b' := b[t]\).
   1. If \(b' = 0\), then \(s[t] := a\).
   2. If \(b' \neq 0\), then \(s[t] := a+b'\).
3. Suppose that \(s = \theta(a,b)\) for some \((a,b) \in T^2\).
   1. Suppose that \(\textrm{dom}(b) = 0\).
      1. If \(\textrm{dom}(a) \in \{0,$1\}\), then \(s[t] := t\).
      2. If \(\textrm{dom}(a) \notin \{0,$1\}\), then \(s[t] := \theta(a[t],b)\).
   2. Suppose that \(\textrm{dom}(b) = $1\).
      1. If \(t = t[0]+$1\), then \(s[t] := s[t[0]]+s[$1]\).
      2. If \(t \neq t[0]+$1\), then \(s[t] := \theta(a,b[0])\).
   3. Suppose that \(\textrm{dom}(b) \notin \{0,$1\}\).
      1. If \(\textrm{dom}(b) \lt s\), then \(s[t] := \theta(a,b[t])\).
      2. If \(s \le \textrm{dom}(b)\), then there is a \(c \in T\) such that \(\textrm{dom}(b) = \theta(c,0)\).
         1. If \(t = $i\) for some \(i \in \mathbb{N} \setminus \{0\}\) and \(s[t[0]] = \theta(a,\Gamma)\) for some \(\Gamma \in T\), then \(s[t] := \theta(a,b[\theta(c[0],\Gamma)])\).
         2. Otherwise, then \(s[t] := \theta(a,b[\theta(c[0],0)])\).
4. Suppose that \(s = \psi_a(b)\) for some \((a,b) \in T_{\theta} \times T_{\psi}\).
   1. Suppose that \(\textrm{dom}(b) = 0\).
      1. If \(\textrm{dom}(a) \in \{0,\theta(0,0)\}\), then \(s[t] := t\).
      2. If \(\textrm{dom}(a) = \theta(0,\theta(0,0))\), then \(s[t] := \psi_{a[t]}(b)\).
      3. Suppose that \(\textrm{dom}(a) \notin \{0,\theta(0,0),\theta(0,\theta(0,0))\}\).
         1. If \(\textrm{dom}(a) \lt s\), then \(s[t] := \psi_{a[t]}(b)\).
         2. If \(s \le \textrm{dom}(a)\), then there is a \(c \in T_{\theta}\) such that \(\textrm{dom}(a) = \psi_c(0)\).
            1. Suppose that \(\textrm{dom}(c) = \theta(0,0)\).
               1. If \(t = $i\) for some \(i \in \mathbb{N} \setminus \{0\}\) and \(s[t[0]] = \psi_{\Gamma}(b)\) for some \(\Gamma \in T\), then \(s[t] := \psi_{a[\psi_{c[0]}(\textrm{BP}(\Gamma))]}(b)\).
               2. Otherwise, then \(s[t] := \psi_a(b[\psi_{c[0]}(0)])\).
            2. Suppose that \(\textrm{dom}(c) \neq \theta(0,0)\).
               1. If \(t = $i\) for some \(i \in \mathbb{N} \setminus \{0\}\) and \(s[t[0]] = \psi_{\Gamma}(b)\) for some \(\Gamma \in T\), then \(s[t] := \psi_{a[\textrm{BP}(\Gamma)]}(b)\).
               2. Otherwise, then \(s[t] := \psi_{a[0]}(b)\).
   2. Suppose that \(\textrm{dom}(b) = $1\).
      1. If \(t = t[0]+$1\), then \(s[t] := s[t[0]]+s[$1]\).
      2. If \(t \neq t[0]+$1\), then \(s[t] := \psi_a(b[0])\).
   3. Suppose that \(\textrm{dom}(b) \notin \{0,$1\}\).
      1. If \(\textrm{dom}(b) \lt s\), then \(s[t] := \psi_a(b[t])\).
      2. If \(s \le \textrm{dom}(b)\), then there is a \(c \in T_{\theta}\) such that \(\textrm{dom}(b) = \psi_c(0)\).
         1. Suppose that \(\textrm{dom}(c) = \theta(0,0)\).
            1. If \(t = $i\) for some \(i \in \mathbb{N} \setminus \{0\}\) and \(s[t[0]] = \psi_a(\Gamma)\) for some \(\Gamma \in T\), then \(s[t] := \psi_a(b[\psi_{c[0]}(\Gamma)])\).
            2. Otherwise, then \(s[t] := \psi_a(b[\psi_{c[0]}(0)])\).
         2. Suppose that \(\textrm{dom}(c) \neq \theta(0,0)\).
            1. If \(t = $i\) for some \(i \in \mathbb{N} \setminus \{0\}\) and \(s[t[0]] = \psi_a(\Gamma)\) for some \(\Gamma \in T\), then \(s[t] := \psi_a(b[\Gamma])\).
            2. Otherwise, then \(s[t] := \psi_a(b[0])\).

FGH

A computable partial map \begin{eqnarray*} f \colon T \times \mathbb{N}^2 & \to & \mathbb{N} \\ (s,m,n) & \mapsto & f_s^m(n) \end{eqnarray*} is defined in the following recursive way:

1. If \(m = 0\), then \(f_s^m(n) := n\).
2. Suppose that \(m = 1\).
   1. If \(\textrm{dom}(s) = 0\), then \(f_s^m(n) := n+1\).
   2. If \(\textrm{dom}(s) = $1\), then \(f_s^m(n) := f_{s[0]}^n(n)\).
   3. If \(\textrm{dom}(s) \notin \{0,$1\}\), then \(f_s^m(n) := f_{s[$n]}^1(n)\).
3. If \(m \notin \{0,1\}\), then \(f_s^m(n) := f_s^1(f_s^{m-1}(n))\).

Limit function

A computable total map \begin{eqnarray*} \Lambda \colon \mathbb{N} & \to & PT_{\theta} \\ n & \mapsto & \Lambda(n) \end{eqnarray*} is defined in the following recursive way:

1. If \(n = 0\), then \(\Lambda(n) := \theta(0,0)\).
2. If \(n \neq 0\), then \(\Lambda(n) := \theta(\Lambda(n-1),0)\).

Standard form

A subset \(OT \subset T\) is defined in the following recursive way:

1. For any \(n \in \mathbb{N}\), \(\psi_0(\psi_{\theta(0,\Lambda(n))}(0)) \in OT\).
2. For any \((s,n) \in OT \times \mathbb{N}\), \(s[$n] \in OT\).

My expectation is that \((OT,\lt)\) is an ordinal notation, or \(OT\) is recursive and the restriction of \(\lt\) to \(OT\) is well-ordered.

Naming

A computable partial map \begin{eqnarray*} F \colon \mathbb{N} & \to & \mathbb{N} \\ n & \mapsto & F(n) \end{eqnarray*} is defined as \(F(n) := f_{\psi_0(\psi_{\theta(0,\Lambda(n))}(0))}^1(n)\).

I name \(F^{10^{100}}(10^{100})\) "2-var theta psi number".

I name an ordinal which correspond to the limit of 2-var θψ function "2-var theta psi ordinal" (2-TPO).