User blog:P進大好きbot/Rathjen-type Ordinal Notation

This is an attempt to extend Rathjen's ordinal notaion based on the weakly Mahlo cardinal and my 非想非非想 notation explained here.

= Caution =

Before the definition, I note several important points, which might be asked by readers.

Ordinal Notation
Since many googologists misunderstands the definition of an ordinal notation, you might have caught a wrong description of an ordinal notation in blog posts written by them. If you do not know the precise definition of the notion of an ordinal notation, please read this blog post before asking me something non-sense like "why should you define standard forms and an ordering?" or "why don't you UNOCF, which is the greatest ordinal notation?"

Complexity
Although many googologists incorrectly talk as if any equivalent simplification of an OCF yielded an equivalent simpler ordinal notation, simplification causes the problem that we lose the computability of the \(\in\)-relation encoded into a binary relation on expressions in the notation. Although they incorrectly believe that complicated conditions in the definition of the OCF were used only in proofs, those complicated conditions are actually used in the construction of the algorithm to compute the encoded \(\in\)-relation.

If we want to work in the realm of computable googology, we should be careful about the computability of the whole notation. Otherwise, we can use uncomputable notations such as Kleene's \(\mathcal{O}\). This is an attempt to extend the ordinal notation associated to Rathjen's standard OCF based on the weakly Mahlo cardinal in a valid way, i.e. without losing the computability of the encoded \(\in\)-relation.

Strength
I do not know the strength, i.e. the ordinal type, of this notation. Since I intend that it is an extension of Rathjen's ordinal notation based on the weakly Mahlo cardinal, it goes beyond \(\lim_{\alpha \in M^{\Gamma}} \psi_{\Omega_1}(\alpha)\) with respect to Rathjen's standard OCF based on the weakly Mahlo cardinal if I am correct. Since it is difficult to compute fundamental sequences for Rathjen's ordinal notation, I am possibly writing an incorrect algorithm to compute fundamental sequences for my notation.

At least, this notation is much weaker than my strongest computable notation explained here. It diagonalises ordinal notations provably well-founded in \(\textrm{ZFC}\) set theory.

= Definition =

I will define a notation \((OT,<)\), which is expected to be an ordinal notation. Although I used an OCF defined in my own study in order to encode the \(\in\) relation in a recursive way purely in arithmetic, I will not write the definition of the OCF because it is also complicated and is not formalised in \(\textrm{ZFC}\) set theory augmented by specific large cardinal axioms. Namely, I assumed several (possiblly inconsistent) schema which play roles analogous to large cardinal axioms in the construction of the OCF. At least, the associated notation is formalised in \(\textrm{ZFC}\) set theory, as I precisely define in this section in a self-contained way.

Notation
I define a recursive set \(S\) of strings of letters \(0,+,\varphi,\chi,\psi\) and parentheses in the following recursive way: I note that one can avoid the use of subscripts and superscripts simply by replacing \(\varphi^{i}_{a}(b)\), \(\chi^{\kappa}_{a}(b)\), and \(\psi^{\kappa}(a)\) by \(\varphi(i,a,b)\), \(\chi(\kappa,a,b)\), and \(\psi(\kappa,a)\) respectively. In order to improve the readability, I employ comma-free expressions with subscripts and superscripts.
 * 1) \(0 \in T\).
 * 2) For any \((a,b) \in T^2\), \(a+b \in T\).
 * 3) For any \((i,a,b) \in T^3\), \(\varphi^{i}_{a}(b) \in T\).
 * 4) For any \((\kappa,a,b) \in T^3\), \(\chi^{\kappa}_{a}(b) \in T\).
 * 5) For any \((\kappa,a) \in T\), \(\psi^{\kappa}(a) \in T\).

In order to shorten expressions, I employ the following abbereviation:

Syntax
I define a recursive subset \(PT \subset T\) of principal terms in the following recursive way: Principal terms play roles analogous to additive principal numbers.
 * 1) \(0 \notin PT\).
 * 2) For any \((a,b) \in T^2\), \(a+b \notin PT\).
 * 3) For any \((i,a,b) \in T^3\), \(\varphi^{i}_{a}(b) \in PT\).
 * 4) For any \((\kappa,a,b) \in T^3\), \(\chi^{\kappa}_{a}(b) \in PT\).
 * 5) For any \((\kappa,a) \in T^2\), \(\psi^{\kappa}(a) \in PT\).

I define a recursive subset \(ST \subset T\) of successor terms in the following recursive way: Successor terms play roles analogous to successor ordinals.
 * 1) \(0 \notin ST\).
 * 2) For any \((a,b) \in PT \times T\), \(a+b \in ST\) if and only if \(b \in ST\).
 * 3) For any \((i,a,b) \in T^3\), \(\varphi^{i}_{a}(b) \in ST\) if and only if \((i,a,b) = (0,0,0)\).
 * 4) For any \((\kappa,a,b) \in T^3\), \(\chi^{\kappa}_{a}(b) \notin ST\).
 * 5) For any \((\kappa,a) \in T^3\), \(\psi^{\kappa}(a) \notin ST\).

I define a recursive subset \(RT \subset T\) of regular terms in the following recursive way: Regular terms play roles analogous to uncountable regular cardinals.
 * 1) \(0 \notin RT\).
 * 2) For any \((a,b) \in PT \times T\), \(a+b \notin RT\).
 * 3) For any \((i,a,b) \in T^3\), \(\varphi^{i}_{a}(b) \in RT\) if and only if \(i \neq 0\), \(a = 0\), and \(b \in \{0\} \cup ST\).
 * 4) For any \((\kappa,a,b) \in T^3\), \(\chi^{\kappa}_{a}(b) \in RT\) if and only if \(b \in \{0\} \cup ST\).
 * 5) For any \((\kappa,a) \in T^3\), \(\psi^{\kappa}(a) \notin RT\).

I define a recursive map \begin{eqnarray*} \textrm{pred} \colon T & \to & T \\ s & \mapsto & \textrm{pred}(s) \end{eqnarray*} in the following recursive way: The \(\textrm{pred}\) map plays a role analogous to the map assigning the predecessors to each successor ordinals.
 * 1) If \(s \in \{0\} \cup PT\), then \(\textrm{pred}(s) := 0\).
 * 2) Suppose \(s = a+b\) for a unique \((a,b) \in PT \times T\).
 * 3) If \(\textrm{pred}(b) = 0\), then \(\textrm{pred}(s) := a\).
 * 4) If \(\textrm{pred}(b) \neq 0\), then \(\textrm{pred}(s) := a + \textrm{pred}(b)\).

I define recursive map \begin{eqnarray*} \textrm{deg} \colon T & \to & T \\ s & \mapsto & \textrm{deg}(s) \end{eqnarray*} in the following recursive way: The \(\textrm{deg}\) map will be used to describe the ordering and the expansion rule.
 * 1) If \(s \notin PT\), then \(\textrm{deg}(s) := 0\).
 * 2) If \(s = \varphi^{i}_{a}(b)\) for a unique \((i,a,b) \in T^3\), then \(\textrm{deg}(s) := i\).
 * 3) If \(s = \chi^{\kappa}_{a}(b)\) for a unique \((\kappa,a,b) \in T^3\), then \(\textrm{deg}(s) := \textrm{pred}(\textrm{deg}(\kappa))\).
 * 4) If \(s = \psi^{\kappa}(a)\) for a unique \((\kappa,a) \in T^2\), then \(\textrm{deg}(s) := 0\).

For an \(s \in T\), I define a finite subset \(\textrm{AP}(s) \subset T\) in the following recursive way: The assignment \(\textrm{AP}\) will play a role analogous to Buchholz's \(\textrm{P}\) function.
 * 1) If \(s = 0\), then \(\textrm{AP}(s) := \emptyset\).
 * 2) If \(s = a+b\) for a unique \((a,b) \in PT \times T\), then \(\textrm{AP}(s) := \{a\} \cup \textrm{AP}(b)\).
 * 3) If \(s \in PT\), then \(\textrm{AP}(s) := \{s\}\).

For an \(s \in T\), I define a finite subset \(\textrm{SC}(s) \subset T\) in the following recursive way: The assignment \(\textrm{SC}\) will play a role analogous to Rathjen's \(\textrm{SC}\) function.
 * 1) If \(s = 0\), then \(\textrm{SC}(s) := \emptyset\).
 * 2) If \(s = a+b\) for a unique \((a,b) \in PT \times T\), then \(\textrm{SC}(s) := \textrm{SC}(a) \cup \textrm{SC}(b)\).
 * 3) If \(s = \varphi^{i}_{a}(b)\) for a unique \((i,a,b) \in T^3\), then \(\textrm{SC}(s) := \textrm{SC}(i) \cup \textrm{SC}(a) \cup \textrm{SC}(b)\).
 * 4) If \(s = \chi^{\kappa}_{a}(b)\) for a unique \((\kappa,a,b) \in T^3\), then \(\textrm{SC}(s) := \{s\}\).
 * 5) If \(s = \psi^{\kappa}_{a}(b)\) for a unique \((\kappa,a,b) \in T^3\), then \(\textrm{SC}(s) := \{s\}\).

For a \(\kappa \in T\), I define a finite subset \(\kappa^{-} \subset T\) in the following recursive way: The assignment \({-}\) will play a role analogous to Rathjen's \({-}\) function.
 * 1) Suppose \(s = \chi^{\kappa}_{a}(b)\) for a unique \((\kappa,a,b) \in T^3\).
 * 2) If \(b = 0\), then \(\kappa^{-} := \textrm{SC}(a)\).
 * 3) If \(b \neq 0\), then \(\kappa^{-} := \{\chi^{\kappa}_{a}(\textrm{Pred}(b))\}\).
 * 4) Otherwise, \(\kappa^{-} := \emptyset\).

Standard Form
For an \(n \in \mathbb{N}\), I define a finite subset \(OT_n \subset T\), recursive relations \(s \leq_{n} t\) and \(s <_{n} t\) on \((s,t) \in OT_n^2\), and \(s \triangleleft_{n} (\kappa,a)\) on \((s,\lambda,c) \in T^3\) simultaneously in the following recursive way: The relation \(\triangleleft_{n}\) is an analogue to Rathjen's \(K\) function.
 * Definition of \(OT_n\)
 * 1) If \(n = 0\), then \(OT_n := \{0\}\)
 * 2) Suppose \(n \neq 0\).
 * 3) \(0 \in OT_n\).
 * 4) For any \((a,b) \in PT \times T\), \(a+b \in OT_n\) is equivalent to \(a \in OT_{n-1}\), \(b \in OT_{n-1}\), \(b \neq 0\), \(\textrm{AP}(b) \subset OT_{n-1}\), and \(b' \leq_{n-1} a\) for any \(b' \in \textrm{AP}(b)\).
 * 5) For any \((i,a,b) \in T^3\), \(\varphi^{i}_{a}(b) \in OT_n\) is equivalent to \(i \in OT_{n-1}\), \(a \in OT_{n-1}\), \(b \in OT_{n-1}\), and that one of the following holds:
 * 6) \(b = 0\) and \(a \notin \textrm{PT}\);
 * 7) \(b = 0\), \(a = \varphi^{j}_{c}(d)\) for a unique \((j,c,d) \in T^3\), \(c <_{n-1} a\), \(d <_{n-1} a\), \(j \leq_{n-1} i\);
 * 8) \(b = 0\), \(a = \chi^{\lambda}_{c}(d)\) for a unique \((\lambda,c,d) \in T^3\), \(d <_{n-1} a\), \(\textrm{deg}(a) \leq_{n-1} i\);
 * 9) \(b = 0\), \(a = \chi^{\lambda}_{c}(d)\) for a unique \((\lambda,c,d) \in T^3\), \(d <_{n-1} a\), \(\lambda = \chi^{\mu}_{e}(f)\), \(\textrm{cof}(f) \in \{0,1\}\), \(\textrm{deg}(\lambda) = i\);
 * 10) \(b = 0\), \(a = \psi^{\lambda}(c)\) for a unique \((\lambda,c) \in T^2\), \(c \triangleleft_{n-1} (\lambda,c)\), \(\textrm{deg}(\lambda) <_{n-1} i\)
 * 11) \(b = 0\), \(a = \psi^{\lambda}(c)\) for a unique \((\lambda,c) \in T^2\), \(c \triangleleft_{n-1} (\lambda,c)\), \(\lambda = \varphi^{i}_{0}(f)\), \(f <_{n-1} \lambda\), \(\textrm{cof}(f) \in \{0,1\}\);
 * 12) \(b \neq 0\), \(b \leq_{n-1} a\);
 * 13) \(b = c+d\) for a unique \((c,d) \in PT \times T\);
 * 14) \(b = \varphi^{i}_{c}(d)\) for a unique \((c,d) \in T^2\), \(i <_{n-1} b\), \(c <_{n-1} b\), \(a <_{n-1} b\), \(c \leq_{n-1} a\);
 * 15) \(b = \varphi^{j}_{c}(d)\) for a unique \((j,c,d) \in T^3\), \(j <_{n-1} b\), \(c <_{n-1} b\), \(a <_{n-1} b\), \(j <_{n-1} i\);
 * 16) \(b = \chi^{\lambda}_{c}(d)\) for a unique \((\lambda,c,d) \in T^3\), \(d <_{n-1} b\), \(\textrm{deg}(\lambda) <_{n-1} i\); or
 * 17) \(b = \chi^{\lambda}_{c}(d)\) for a unique \((\lambda,c,d) \in T^3\), \(d <_{n-1} b\), \(\lambda = \varphi^{i}_{0}(f)\), \(\textrm{cof}(f) \in \{0,1\}\).
 * 18) For any \((\kappa,a,b) \in T^3\), \(\chi^{\kappa}_{a}(b) \in OT_n\) is equivalent to \(\kappa \in RT \cap OT_{n-1}\), \(a \in OT_{n-1}\), \(b \in OT_{n-1}\), \(\textrm{pred}(\textrm{deg}(\kappa)) \neq 0\), and that one of the following holds:
 * 19) \(b \notin PT\);
 * 20) \(b = \varphi^{j}_{c}(d)\) for a unique \((j,c,d) \in T^3\);
 * 21) \(b = \chi^{\kappa}_{c}(d)\) for a unique \((c,d) \in T^2\) and \(c \leq_{n-1} a\);
 * 22) \(b = \chi^{\kappa}_{c}(d)\) for a unique \((c,d) \in T^2\) and \(b \leq_{n-1} a'\) for some \(a' \in \textrm{SC}(a)\);
 * 23) \(b = \chi^{\lambda}_{c}(d)\) for a unique \((\lambda,c,d) \in T^3\) and \(\lambda \leq_{n-1} \chi^{\kappa}_{a}(b)\);
 * 24) \(b = \chi^{\lambda}_{c}(d)\) for a unique \((\lambda,c,d) \in T^3\), \(b <_{n-1} \kappa\), and \(\kappa <_{n-1} \lambda\);
 * 25) \(b = \psi^{\lambda}(c)\) for a unique \((\lambda,c) \in T^2\) and the negation of \(a \triangleleft_{n-1} (\lambda,c)\);
 * 26) \(b = \psi^{\lambda}(c)\) for a unique \((\lambda,c) \in T^2\) and \(\lambda \leq_{n-1} a'\) for some \(a' \in \textrm{SC}(a)\);
 * 27) \(b = \psi^{\lambda}(c)\) for a unique \((\lambda,c) \in T^2\), \(\lambda = \chi^{\kappa}_{e}(f)\) for a unique \((e,f) \in T^2\), and \(e \leq_{n-1} a\);
 * 28) \(b = \psi^{\lambda}(c)\) for a unique \((\lambda,c) \in T^2\), \(\lambda = \chi^{\mu}_{e}(f)\) for a unique \((\mu,e,f) \in T^3\), \(\kappa \leq_{n-1} \lambda\); or
 * 29) \(b = \psi^{\lambda}(c)\) for a unique \((\lambda,c) \in T^2\), \(\lambda = \chi^{\mu}_{e}(f)\) for a unique \((\mu,e,f) \in T^3\), \(\mu <_{n-1} \kappa\), \(\chi^{\kappa}_{a}(b) \leq_{n-1} \mu\).
 * 30) For any \((\kappa,a) \in T^2\), \(\psi^{\kappa}(a) \in OT_n\) is equivalent to \(\kappa \in RT \cap OT_{n-1}\), \(a \in OT_{n-1}\), and \(a \triangleleft_{n-1} (\kappa,a)\)
 * Definition of \(s \leq_{n-1} t\)
 * 1) \(s \leq_{n} t\) is equivalent to \(s <_{n} t\) or \(s = t\).
 * Definition of \(s <_{n} t\)
 * 1) If \(n = 0\), then \(s < t\) is false.
 * 2) If \(n \neq 0\) and \((s,t) \in OT_{n-1}^2\), then \(s <_{n} t\) is equivalent to \(s <_{n-1} t\).
 * 3) Suppose \(n \neq 0\) and \((s,t) \notin OT_{n-1}^2\).
 * 4) If \(t = 0\), then \(s <_{n} t\) is false.
 * 5) If \(s = 0\) and \(t \neq 0\), then \(s <_{n} t\) is true.
 * 6) If \(s = a+b\) for a unique \((a,b) \in PT \times T\) and \(t = c+d\) for a unique \((c,d) \in PT \times T\), then \(s <_{n} t\) is equivalent to that one of the following holds:
 * 7) \(a <_{n} c\); or
 * 8) \(a = c\) and \(b <_{n} d\).
 * 9) If \(s = a+b\) for a unique \((a,b) \in PT \times T\) and \(t \in PT\), then \(s <_{n} t\) is equivalent to \(a <_{n} t\).
 * 10) If \(s \in PT\) and  and \(t = c+d\) for a unique \((c,d) \in PT \times T\), then \(s <_{n} t\) is equivalent to the negation of \(t <_{n} s\).
 * 11) If \(s = \varphi^{i}_{a}(b)\) for a unique \((i,a,b) \in T^3\) and \(t = \varphi^{j}_{c}(d)\) for a unique \((j,c,d) \in T^3\), then \(s <_{n} t\) is equivalent to that one of the following hold:
 * 12) \(i = j\), \(a = c\), and \(b <_{n} d\);
 * 13) \(i = j\), \(a <_{n} c\), and \(b \leq_{n} t\);
 * 14) \(i = j\), \(c <_{n} a\), and \(s <_{n} d\);
 * 15) \(i <_{n} j\), \(a <_{n} t\), and \(b \leq_{n} t\);
 * 16) \(i <_{n} j\), \(a = t\), and \(b = 0\);
 * 17) \(j <_{n} i\) and \(t <_{n} a\);
 * 18) \(j <_{n} i\), \(a = t\), and \(b \neq 0\);
 * 19) \(j <_{n} i\), \(t <_{n} b\), and \(a \neq t\); or
 * 20) \(j <_{n} i\), \(t <_{n} b\), and \(b \neq 0\).
 * 21) If \(s = \varphi^{i}_{a}(b)\) for a unique \((i,a,b) \in T^3\) and \(t = \chi^{\lambda}_{c}(d)\) for a unique \((\lambda,c,d) \in T^3\), then \(s <_{n} t\) is equivalent to \(s <_{n} \lambda\) and \(s' <_{n} t\) for any \(s' \in \textrm{SC}(s)\).
 * 22) If \(s = \varphi^{i}_{a}(b)\) for a unique \((i,a,b) \in T^3\) and \(t = \psi^{\lambda}(c)\) for a unique \((\lambda,c) \in T^2\), then \(s <_{n} t\) is equivalent to \(s <_{n} \lambda\), \(i \in t\), \(a \in t\), and \(b \in t\).
 * 23) If \(s = \chi^{\kappa}_{a}(b)\) for a unique \((\kappa,a,b) \in T^3\) and \(t = \varphi^{j}_{c}(d)\) for a unique \((j,c,d) \in T^3\), then \(s <_{n} t\) is equivalent to the negation of \(t <_{n} s\).
 * 24) If \(s = \chi^{\kappa}_{a}(b)\) for a unique \((\kappa,a,b) \in T^3\) and \(t = \chi^{\lambda}_{c}(d)\) for a unique \((\lambda,c,d) \in T^3\), then \(s <_{n} t\) is equivalent to that one of the following holds:
 * 25) \(\kappa = \lambda\), \(a = c\), and \(b <_{n} d\);
 * 26) \(\kappa = \lambda\), (a <_{n} c\), \(a' <_{n} \chi^{\lambda}_{c}(d)\) for all \(a' \in \textrm{SC}(a)\), and \(b <_{n} \chi^{\lambda}_{c}(d)\);
 * 27) \(\kappa = \lambda\), \(c <_{n} a\), and \(\chi^{\kappa}_{a}(b) \leq_{n} c'\) for some \(c' \in \textrm{SC}(c)\);
 * 28) \(\kappa = \lambda\), \(c <_{n} a\), and \(\chi^{\kappa}_{a}(b) \leq_{n} d\);
 * 29) \(\kappa \leq_{n} \chi^{\lambda}_{c}(d)\); or
 * 30) \(\chi^{\kappa}_{a}(b) <_{n} \lambda\) and \(\lambda <_{n} \kappa\).
 * 31) If \(s = \chi^{\kappa}_{a}(b)\) for a unique \((\kappa,a,b) \in T^3\) and \(t = \psi^{\lambda}(c)\) for a unique \((\lambda,c) \in T^2\), then \(s <_{n} t\) is equivalent to \(s <_{n} \lambda\) and \(s \triangleleft_{n} (\lambda,c)\).
 * 32) If \(s = \psi^{\kappa}(a)\) for a unique \((\kappa,a) \in T^2\) and \(t = \psi^{\lambda}(c)\) for a unique \((\lambda,c) \in T^2\), then \(s <_{n} t\) is equivalent to that one of the following holds:
 * 33) \(\kappa = \lambda\) and \(a <_{n} c\);
 * 34) \(\kappa <_{n} \psi^{\lambda}(c)\); or
 * 35) \(\lambda <_{n} \kappa\) and \(\psi^{\kappa}(a) <_{n} \lambda\).
 * Definition of \(s \triangleleft_{n} (\lambda,c)\):
 * 1) If \(s = 0\), then \(s \triangleleft_{n} (\lambda,c)\) is true;
 * 2) If \(s = a+b\) for a unique \((a,b) \in PT \times T\), then \(s \triangleleft_{n} (\lambda,c)\) is equivalent to \(a \triangleleft_{n} (\lambda,c)\) and \(b \triangleleft_{n} (\lambda,c)\).
 * 3) If \(s = \varphi^{i}_{a}(b)\) for a unique \((i,a,b) \in T^3\), then \(s \triangleleft_{n} (\lambda,c)\) is equivalent to \(i \triangleleft_{n} (\lambda,c)\), \(a \triangleleft_{n} (\lambda,c)\), and \(b \triangleleft_{n} (\lambda,c)\).
 * 4) If \(s = \chi^{\kappa}_{a}(b)\) for a unique \((\kappa,a,b) \in T^3\), then \(s \triangleleft_{n} (\lambda,c)\) is equivalent to \(\kappa \triangleleft_{n} (\lambda,c)\), \(a \triangleleft_{n} (\lambda,c)\), and \(b \triangleleft_{n} (\lambda,c)\).
 * 5) If \(s = \psi^{\kappa}(a)\) for a unique \((\kappa,a) \in T^2\), then \(s \triangleleft_{n} (\lambda,c)\) is equivalent to that one of the following hold:
 * 6) \(s \leq_{n} t\) for some \(t \in \lambda^{-}\).
 * 7) \(t <_{n} s\) for any \(t \in \lambda^{-}\), \(\kappa <_{n} \lambda\), and \(a \triangleleft_{n} (\lambda,c)\).
 * 8) \(t <_{n} s\) for any \(t \in \lambda^{-}\), \(\lambda \leq_{n} \kappa\), \(a <_{n} c\), \(\kappa \triangleleft_{n} (\lambda,c)\), and \(a \triangleleft_{n} (\lambda,c)\).

I denote by \(OT \subset T\) the union \(\bigcup_{n \in \mathbb{N}} OT_n \subset T\). Although it is not trivial from the definition using union, \(OT\) is a recursive subset of \(T\). Indeed, \(t \in OT\) is equivalent to \(t \in OT_n\), where \(n\) is the length of the script-free expression of \(t\).

Well-Ordering
By the definition, \((<_n)_{n \in \mathbb{N}}\) is a compatible system of recursive binary relations, and hence extends to a unique recursive binary relation \(<\) on \(OT\). Then \(s < t\) is equivalent to \(s <_{n} t\), where \(n\) is the least natural number satisfying \((s,t) \in OT_n^2\). I expect that \((OT,<)\) forms an ordinal notation, i.e. \(<\) is a well-ordering.

As I explained here, every ordinal notation admits an explicit canonical algorithm to compute fundamental sequences. However, the canonical algorithm is hard to execute, and hence I will define a simpler algorithm to compute fundamental sequences. Since creating a simple algorithm to compute fundamental sequences for an ordinal notation is not so easy, my algorithm is possibly wrong. Any correction is appreciated.

I denote by \(CT \subset OT\) the recursive subset \(\{s \in OT \mid s < \Omega_1\}\).

Cofinality
I define a recursive map \begin{eqnarray*} \textrm{dom} \colon OT & \to & OT \\ s & \mapsto & \textrm{dom}(s) \end{eqnarray*} in the following recursive way: The \(\textrm{dom}\) map is an analogue of Buchholz's \(\textrm{dom}\) map.
 * 1) If \(s = 0\), then \(\textrm{dom}(s) := 0\).
 * 2) If \(s = a+b\) for a unique \((a,b) \in PT \times T\), then \(\textrm{dom}(s) := \textrm{dom}(b)\).
 * 3) Suppose \(s = \varphi^{i}_{a}(b)\) for a unique \((i,a,b) \in T^3\).
 * 4) If \(\textrm{dom}(a) = 0\), \(\textrm{dom}(i) = 0\), and \(\textrm{dom}(b) = 0\), then \(\textrm{dom}(s) := 1\).
 * 5) If \(\textrm{dom}(a) = 0\), \(\textrm{dom}(i) = 0\), and \(\textrm{dom}(b) = 1\), then \(\textrm{dom}(s) := \omega\).
 * 6) If \(\textrm{dom}(a) = 0\), \(\textrm{dom}(i) \neq 0\), and \(\textrm{dom}(b) \in \{0,1\}\), then \(\textrm{dom}(s) := s\).
 * 7) If \(\textrm{dom}(a) = 1\) and \(\textrm{dom}(b) \in \{0,1\}\), then \(\textrm{dom}(s) := \omega\).
 * 8) If \(\textrm{dom}(a) \notin \{0,1\}\) and \(\textrm{dom}(b) \in \{0,1\}\), then \(\textrm{dom}(s) := \textrm{dom}(a)\).
 * 9) If \(\textrm{dom}(b) \notin \{0,1\}\), then \(\textrm{dom}(s) := \textrm{dom}(b)\).
 * 10) Suppose \(s = \chi^{\kappa}_{a}(b)\) for a unique \((\kappa,a,b) \in T^3\).
 * 11) If \(\textrm{dom}(b) \in \{0,1\}\), then \(\textrm{dom}(s) := s\).
 * 12) If \(\textrm{dom}(b) \notin \{0,1\}\), then \(\textrm{dom}(s) := \textrm{dom}(b)\).
 * 13) Suppose \(s = \psi^{\kappa}(a)\) for a unique \((\kappa,a) \in T^2\).
 * 14) If \(\textrm{dom}(a) \in \{0,1\}\) and \(\chi^{\kappa}_{0}(0) \in OT\), then \(\textrm{dom}(s) := \omega\).
 * 15) If \(\textrm{dom}(a) \in \{0,1\}\) and \(\kappa = \varphi^{1}_{0}(d)\) for a unique \(d \in T\), then \(\textrm{dom}(s) := \omega\).
 * 16) If \(\textrm{dom}(a) \in \{0,1\}\), \(\kappa = \varphi^{j}_{0}(d)\) for a unique \((j,d) \in T^2\), and \(\textrm{dom}(j) \neq 1\), then \(\textrm{dom}(s) := \textrm{dom}(j)\).
 * 17) If \(\textrm{dom}(a) \in \{0,1\}\), \(\chi^{\kappa}_{0}(0) \notin OT\), \(\kappa = \chi^{\lambda}_{c}(d)\) for a unique \((\lambda,c,d) \in T^3\), and \(\textrm{dom}(c) \in \{0,1\}\), then \(\textrm{dom}(s) := \omega\).
 * 18) If \(\textrm{dom}(a) \in \{0,1\}\), \(\chi^{\kappa}_{0}(0) \notin OT\), \(\kappa = \chi^{\lambda}_{c}(d)\) for a unique \((\lambda,c,d) \in T^3\), \(\textrm{dom}(c) \notin \{0,1\}\), and \(\textrm{dom}(c) < \lambda\), then \(\textrm{dom}(s) := \textrm{dom}(\lambda)\).
 * 19) If \(\textrm{dom}(a) \in \{0,1\}\), \(\chi^{\kappa}_{0}(0) \notin OT\), \(\kappa = \chi^{\lambda}_{c}(d)\) for a unique \((\lambda,c,d) \in T^3\), \(\textrm{dom}(c) \notin \{0,1\}\), and \(\lambda \leq \textrm{dom}(c)\), then \(\textrm{dom}(s) := \omega\).
 * 20) If \(\textrm{dom}(a) \notin \{0,1\}\) and \(\textrm{dom}(a) < \kappa\), then \(\textrm{dom}(s) := \textrm{dom}(a)\).
 * 21) If \(\textrm{dom}(a) \notin \{0,1\}\) and \(\kappa \leq \textrm{dom}(a)\), then \(\textrm{dom}(s) := \omega\).

Expansion
I define a recursive map \begin{eqnarray*} \Phi \colon OT & \to & OT \\ n & \mapsto & \Phi(n) \end{eqnarray*} in the following recursive way: I define a recursive map \begin{eqnarray*} \Gamma \colon OT^3 & \to & OT \\ (i,s,n) & \mapsto & \Gamma^{i}(s,n) \end{eqnarray*} in the following recursive way: The \(\Gamma\) map is an analogue of Gamma numbers.
 * 1) If \(\textrm{dom}(n) = 1\), then \(\Phi(n) := \varphi^{\Phi(\textrm{pred}(n))}_{0}(0)\).
 * 2) If \(\textrm{dom}(n) \neq 1\), then \(\Phi(n) := \varphi^{0}_{0}(0)\).
 * 1) If \(\textrm{dom}(n) = 1\), then \(\Gamma^{i}(s,n) := \varphi^{i}_{\Gamma^{i}(s,\textrm{pred}(n))}(0)\).
 * 2) If \(\textrm{dom}(n) \neq 1\) and \(\varphi^{i}_{s}(0) \in OT\), then \(\Gamma^{i}(s,n) := \varphi^{i}_{s}(0)\).
 * 3) If \(\textrm{dom}(n) \neq 1\) and \(\varphi^{i}_{s}(0) \notin OT\), then \(\Gamma^{i}(s,n) := s+1\).

I define a recursive map \begin{eqnarray*} [ \ ] \colon OT^2 & \to & OT \\ (s,n) & \mapsto & s[n] \end{eqnarray*} in the following recursive way: The \([ \ ]\) operator restricted to \(CT\) plays a role of a recursive system of fundamental sequences.
 * 1) If \(s = 0\), then \(s[n] := 0\).
 * 2) Suppose \(s = a+b\) for a unique \((a,b) \in PT \times T\),
 * 3) Put \(b' := b[n]\).
 * 4) If \(a+b' \in OT\),  then \(s[n] := a+b'\).
 * 5) If \(a+b' \notin OT\),  then \(s[n] := a\).
 * 6) Suppose \(s = \varphi^{i}_{a}(b)\) for a unique \((i,a,b) \in T^3\).
 * 7) Define \(b' \in OT\) on the following way:
 * 8) If \(\varphi^{i}_{a}(b[n]) \in OT\), put \(b' := \varphi^{i}_{a}(b[n])\).
 * 9) If \(\varphi^{i}_{a}(b[n]) \notin OT\), put \(b' := b[n]\).
 * 10) If \(\textrm{dom}(a) = 0\), \(\textrm{dom}(i) = 0\), and \(\textrm{dom}(b) = 0\), then \(s[n] := 0\).
 * 11) Suppose \(\textrm{dom}(a) = 0\), \(\textrm{dom}(i) = 0\), and \(\textrm{dom}(b) = 1\).
 * 12) If \(\textrm{dom}(n) = 1\), then \(s[n] := s[n[0]]+b'\).
 * 13) If \(\textrm{dom}(n) \neq 1\), then \(s[n] := b'\).
 * 14) If \(\textrm{dom}(a) = 0\), \(\textrm{dom}(i) \neq 0\), and \(\textrm{dom}(b) \in \{0,1\}\), then \(s[n] := n\).
 * 15) Suppose \(\textrm{dom}(a) = 1\) and \(\textrm{dom}(b) \in \{0,1\}\).
 * 16) If \(\textrm{dom}(n) = 1\), then \(s[n] := \varphi^{i}_{a[0]}(s[n[0]])\).
 * 17) If \(\textrm{dom}(n) \neq 1\) and \(\textrm{dom}(b) = 0\), then \(s[n] := \varphi^{i}_{a[0]}(0)\).
 * 18) If \(\textrm{dom}(n) \neq 1\) and \(\textrm{dom}(b) = 1\), then \(s[n] := \varphi^{i}_{a[0]}(b'+1)\).
 * 19) If \(\textrm{dom}(a) \notin \{0,1\}\) and \(\textrm{dom}(b) = 0\), then \(s[n] := \varphi^{i}_{a[n]}(0)\).
 * 20) If \(\textrm{dom}(a) \notin \{0,1\}\) and \(\textrm{dom}(b) = 1\), then \(s[n] := \varphi^{i}_{a[n]}(b'+1)\).
 * 21) If \(\textrm{dom}(b) \notin \{0,1\}\), then \(s[n] := \varphi^{i}_{a}(b')\).
 * 22) Suppose \(s = \chi^{\kappa}_{a}(b)\) for a unique \((\kappa,a,b) \in T^3\).
 * 23) If \(\textrm{dom}(b) \in \{0,1\}\), then \(s[n] := n\).
 * 24) If \(\textrm{dom}(b) \notin \{0,1\}\), then \(s[n] := \chi^{\kappa}_{a}(b[n])\).
 * 25) Suppose \(s = \psi^{\kappa}(a)\) for a unique \((\kappa,a) \in T^2\).
 * 26) If \(\textrm{dom}(a) = 0\) and \(\chi^{\kappa}_{0}(0) \in OT\), then \(s[n] := \chi^{\kappa}_{\Phi(n)}(0)\).
 * 27) If \(\textrm{dom}(a) = 1\) and \(\chi^{\kappa}_{0}(0) \in OT\), then \(s[n] := \chi^{\kappa}_{\Phi(n)}(\psi^{\kappa}(a[0])+1)\).
 * 28) Suppose \(\textrm{dom}(a) \in \{0,1\}\), \(\chi^{\kappa}_{0}(0) \notin OT\), and \(\kappa = \varphi^{j}_{0}(d)\) for a unique \((j,d) \in T^2\).
 * 29) Define \(d' \in OT\) in the following way:
 * 30) If \(\textrm{dom}(a) = 0\), \(\textrm{dom}(d) = 1\), and \(\varphi^{j}_{0}(d[0]) \in OT\), then put \(d' := \varphi^{j}_{0}(d[0])+1\).
 * 31) If \(\textrm{dom}(a) = 0\), \(\textrm{dom}(d) = 1\), and \(\varphi^{j}_{0}(d[0]) \notin OT\), then put \(d' := d[0]+1\).
 * 32) If \(\textrm{dom}(a) = 0\) and \(\textrm{dom}(d) \neq 1\), then put \(d' := 0\).
 * 33) If \(\textrm{dom}(a) = 1\), then put \(d' := \psi^{\kappa}(a[0])\).
 * 34) If \(j = 1\), then \(s[n] := \Gamma^{0}(d',n)\).
 * 35) If \(j \neq 1\), then \(s[n] := \varphi^{j[n]}_{0}(d')\).
 * 36) Suppose \(\textrm{dom}(a) \in \{0,1\}\), \(\chi^{\kappa}_{0}(0) \notin OT\), and \(\kappa = \chi^{\lambda}_{c}(d)\) for a unique \((\lambda,c,d) \in T^3\).
 * 37) Define \(d' \in OT\) in the following way:
 * 38) If \(\textrm{dom}(a) = 0\), \(\textrm{dom}(d) = 1\), and \(\chi^{\lambda}_{c}(d[0]) \in OT\), then put \(d' := \chi^{\lambda}_{c}(d[0])+1\).
 * 39) If \(\textrm{dom}(a) = 0\), \(\textrm{dom}(d) = 1\), and \(\chi^{\lambda}_{c}(d[0]) \notin OT\), then put \(d' := d[0]+1\).
 * 40) If \(\textrm{dom}(a) = 0\) and \(\textrm{dom}(d) \neq 1\), then put \(d' := 0\).
 * 41) If \(\textrm{dom}(a) = 1\), then put \(d' := \psi^{\kappa}(a[0])+1\).
 * 42) If \(\textrm{dom}(c) = 0\), then \(s[n] := \Gamma^{\textrm{deg}(\kappa)}(d',n)\).
 * 43) Suppose \(\textrm{dom}(c) = 1\).
 * 44) If \(\textrm{dom}(n) = 1\), then \(s[n] := \chi^{\lambda}_{c[0]}(s[n[0]])\).
 * 45) If \(\textrm{dom}(n) \neq 1\), then \(s[n] := \chi^{\lambda}_{c[0]}(d')\).
 * 46) If \(\textrm{dom}(c) \notin \{0,1\}\) and \(\textrm{dom}(c) < \lambda\), then \(s[n] := \chi^{\lambda}_{c[n]}(d')\).
 * 47) Suppose \(\textrm{dom}(c) \notin \{0,1\}\) and \(\lambda \leq \textrm{dom}(c)\).
 * 48) If \(\textrm{dom}(n) = 1\), then \(s[n] := \chi^{\lambda}_{c[s[n[0]]}(0)\).
 * 49) If \(\textrm{dom}(n) \neq 1\), then \(s[n] := \chi^{\lambda}_{c[d']}(0)\).
 * 50) If \(\textrm{dom}(a) \notin \{0,1\}\) and \(\textrm{dom}(a) < \kappa\), then \(s[n] := \psi^{\kappa}(a[n])\).
 * 51) Suppose \(\textrm{dom}(a) \notin \{0,1\}\) and \(\kappa \leq \textrm{dom}(a)\).
 * 52) If \(\textrm{dom}(n) = 1\) and \(s[n[0]] = \psi^{\kappa}(a')\) for a unique \(a' \in OT\), then \(s[n] := \psi^{\kappa}(a[\psi^{\textrm{dom}(a)}(a')])\).
 * 53) Otherwise, \(s[n] := \psi^{\kappa}(a[\psi^{\textrm{dom}(a)}(0)])\).

Computable Large Number
As stylistic beauty, I define the FGH associated to my notation in the computable realm, i.e. by a term rewriting.

I define a recursive map \begin{eqnarray*} \lfloor \ \rfloor \colon \mathbb{N} & \to & OT \\ n & \mapsto & \lfloor n \rfloor \end{eqnarray*} in the following recursive way: I denote by \(CT^{< \omega}\) the set of finite arrays in \(CT\), and define a recursive map \begin{eqnarray*} f \colon CT^{< \omega} \times \mathbb{N} & \to & \mathbb{N} \\ (A,n) & \mapsto & f(A,n) \end{eqnarray*} in the following recursive way: If \((OT,<)\) is an ordinal notation and the \([ \ ]\) operator restricted to \(CT\) actually gives an algorithm to compute fundamental sequences, then the well-foundedness of \(<\) ensure the termination of \(f\), and \(f((\psi^{\Omega_1}(\Phi(1+1+1+1+1+1+1+1+1+1))),10)\) is a computable large number.
 * 1) If \(n = 0\), then \(\lfloor n \rfloor := 0\).
 * 2) If \(n = 1\), then \(\lfloor n \rfloor := 1\).
 * 3) If \(n > 1\), then \(\lfloor n \rfloor := \lfloor n-1 \rfloor + 1\).
 * 1) If \(A\) is the empty array, then \(f(A,m,n) := n\).
 * 2) Suppose that \(A\) is not the empty array.
 * 3) Denote by \(t \in CT\) the rightmost entry of \(A\).
 * 4) Denote by \(G \in CT^{<\omega}\) the array obtained by deleting the rightmost entry from \(A\).
 * 5) If \(t = 0\), then \(f(A,m,n) := f(G,n+1)\).
 * 6) Suppose \(t \neq 0\).
 * 7) Denote by \(A' \in CT^{<\omega}\) the concatenation of \(G\) and \((t[\lfloor n \rfloor])_{i=0}^{n-1}\).
 * 8) Then \(f(A,n) := f(A',n)\).

= Analysis =

I informally list the ordinals corresponding to expressions in my notation. I use Veblen function \(\varphi\), Rathjen's \(\Phi\), Rathjen's \(\chi\), and Rathjen's \(\psi\) in the ordinal column. This is the least strongly inaccessible WIP.

Up to ψ_{Ω_1}(Φ_1(0))
WIP

Up to the limit
The limit of this notation below \(\Omega_1\) is given by the sequence \(\psi^{\Omega_1}(\Phi(1+1+\cdots))\).

= References =