This is an attempt to extend Rathjen's ordinal notaion based on the weakly Mahlo cardinal^[1] and my 非想非非想 notation explained here.

How It Works[]

Before explaining the precise definition of the notation, I show how it works as expressions of ordinals. I use three functions \(\varphi^{i}_{a}(b)\), \(\chi^{\kappa}_{a}(b)\), and \(\psi^{\kappa}(a)\). In addition, I introduce \(\textrm{deg}\) function, which allows us to determine whether a given regular cardinal can be collapsed by \(\chi\)-function. Here, I often refer to the convention of Rathjen's standard OCF based on the least weakly Mahlo cardinal^[1].

φ function[]

If \(i = 0\), then \(\varphi^{i}_{a}(b)\) is just the same as Veblen function \(\varphi_{a}(b)\). If \(i = 1\), then \(\varphi^{i}_{a}(b)\) is just the same as Rathjen's \(\Phi\). For example, \(\varphi^{1}_{0}(b)\) is the \((1+b)\)-th uncountable cardinal \(\Omega_{1+b}\), \(\varphi^{1}_{1}(b)\) is the \((1+b)\)-th fixed point of \(x \mapsto \varphi^{1}_{0}(x)\), i.e. omega fixed point, \(\varphi^{1}_{2}(b)\) is the \((1+b)\)-th fixed point of \(x \mapsto \varphi^{1}_{1}(x)\), and so on. If \(i = 2\), then \(\varphi^{i}_{a}(b)\) is the \((1+b)\)-th cardinal in the set of weakly \(a\)-Mahlo cardinals and their limits. Then it is natural to ask "How about the case \(i = 3\)?" The question is related to how \(\textrm{deg}\) function is suppoed to work, which will be explained in the next subsection.

deg function[]

I use a function \(\textrm{deg}\), which scales the degree of the regularity. For example, a regular cardinal of \(\textrm{deg} = 1\) is just a regular cardinal, and a regular cardinal of \(\textrm{deg} = 2\) is a weakly Mahlo cardinal. When \(\kappa\) is a regular cardinal such that \(\textrm{deg}(\kappa)\) is a successor ordinal greater than \(1\), e.g. \(2,3,4,\ldots,\omega + 1,\ldots,\Omega_1+1,\ldots,M+1,\ldots\), then \(\chi^{\kappa}\) collapses \(\kappa\) into a smaller ordinal of the predecessor degree. Although I do not have any formalisation of higher degree in terms of large cardinal axioms, it actually works because at least the expansion rule is completely defined in this blog post. (At least, a regular cardinal of \(\textrm{deg} = 3\) should be much greater than the least weakly hyper-Mahlo cardinal. I do not know whether the weak compactness satisfies required properties or not.)

χ function[]

If \(\kappa = M\), i.e. the least weakly Mahlo cardinal \(\varphi^{2}_{0}(0)\), then \(\chi^{\kappa}_{a}(b)\) is quite similar to Rathjen's \(\chi\). For any \(b \in M\), \(\chi^{M}_{0}(b)\) is the \((1+b)\)-th cardinal \(\chi_{1}(b)\) in the set of weakly inaccessible cardinals and their limits, \(\chi^{M}_{1}(b)\) is the \((1+b)\)-th ordinal \(\chi_{2}(b)\) in the set of weakly 2-inaccessible cardinals and their limits, and \(\chi^{M}_{M}(0)\) is the least weakly \((1,0)\)-inaccessible cardinal \(\chi_{M}(0)\).

Roughly speaking, for any finite sequence \((a_0,\ldots,a_n)\) in \(M\), \(\chi^{M}_{M^n \times a_n + \cdots + M^0 \times a_0}\) "enumerates" weakly \((a_n,\ldots,a_0)\)-inaccessible cardinals and their limits below \(M\). Strictly speaking, this is not a precise description. For example, \(\chi^{M}_{\chi^{M}_{M}(0)}(b)\) is the \((2+b)\)-th cardinal \(\chi_{\chi_{M}(0)}(b)\) in the set of weakly \(\chi_{M}(0)\)-inaccessible cardinals and their limits, but not the \((1+b)\)-th one in that set. This difference occurs because \(x \mapsto \chi_x(0)\) does not admit a fixed point due to the important condition \(a \in B(a,\chi_{a}(b))\) in the definition of Rathjen's \(\chi\), which are often deleted in "simplifications". Thanks to the non-existence of the fixed point, we have an explicit algorithm to compute the comparison. Therefore I strongly doubt that such "simplifications" will make the notation "simpler". We will just lose an explicit algorithm to compute the comparison, and hence an explicit algorithm to compute fundamental sequences.

Anyway, \(\chi^{M}_{M^{\omega}} = \chi_{M^{\omega}}\) enumerates cardinals which are weakly \((1,\underbrace{0,\ldots,0}_n)\)-inaccessible for any \(n \in \mathbb{N}\) and their limits. I omit the description of \(\chi^{M}_{a}\) above \(M^{\omega}\) in order to avoid the complexity.

ψ function[]

Now, we should pay attention to the fact that \(\chi\) function itself does not express their infinite iteration. When we have ordinals "outside expressions", then it is time to introduce an OCF. When \(\kappa\) is a regular cardinal below \(M\), \(\psi^{\kappa}\) is quite similar to Rathjen's \(\psi_{\kappa}\) except the we use \(\varphi^{i}_{a}(b)\) instead of Rathjen's \(\Phi\). It collapses regular cardinals into smaller ordinals which are not "expressed" by other functions. Unlike Rathjen's \(\psi\), we allow expressions like \(\psi^{M}(0)\). This yields a little differences between Rathjen's \(\psi\) and our \(\psi\).

Finally, I demonstrate several characteristic countable values of \(\psi\) in the following:

\begin{eqnarray*} & & \psi^{\chi^{M}_{0}(0)}(0) = \psi^{\Omega_1}(0) = \varphi^{0}_{\cdot_{\cdot_{\cdot_{\varphi^{0}_{0}(0)}\cdot}\cdot}\cdot}(0) = \Gamma_0 \\ & < & \psi^{\Omega_1}(1) = \varphi^{0}_{\cdot_{\cdot_{\cdot_{\psi^{\Omega_1}(0)+1}\cdot}\cdot}\cdot}(0) = \Gamma_1 \\ & < & \psi^{\Omega_1}(\omega) = \psi^{\Omega_1}(1+1+\cdots) = \Gamma_{\omega} \\ & < & \psi^{\Omega_1}(\Omega_1) = \psi^{\Omega_1}(\cdots \psi^{\Omega}(0)\cdots) = \varphi(1,0,0) \\ & < & \psi^{\Omega_1}(\Omega_1+\Omega_1) = \psi^{\Omega_1}(\Omega_1+\psi^{\Omega_1}(\cdots \psi^{\Omega}(\Omega_1+\psi^{\Omega_1}(0))\cdots)) \\ & < & \psi^{\Omega_1}(\varphi^{0}_{\Omega_1}(1)) = \psi^{\Omega_1}(\varphi^{0}_{\psi^{\Omega_1}(\varphi^{0}_{\cdot_{\cdot_{\cdot_{\psi^{\Omega_1}(0)}\cdot}\cdot}\cdot}(\Omega_1+1))}(\Omega_1+1)) \\ & < & \psi^{\Omega_1}(\varphi^{0}_{\Omega_1+1}(0)) = \psi^{\Omega_1}(\varphi^{0}_{\Omega_1}(\cdots \varphi^{0}_{\Omega_1}(0)\cdots)) \\ & < & \psi^{\Omega_1}(\psi^{\chi^{M}_{0}(1)}(0)) = \psi^{\Omega_1}(\psi^{\Omega_2}(0)) = \psi^{\Omega_1}(\varphi^{0}_{\cdot_{\cdot_{\cdot_{\Omega_1+1}\cdot}\cdot}\cdot}(0)) = \psi^{\Omega_1}(\Gamma_{\Omega_1+1}) \\ & < & \psi^{\Omega_1}(\psi^{\Omega_2}(1)) = \psi^{\Omega_1}(\varphi^{0}_{\cdot_{\cdot_{\cdot_{\psi^{\Omega_2}(0)+1}\cdot}\cdot}\cdot}(0)) = \psi^{\Omega_1}(\Gamma_{\Omega_1+2}) \\ & < & \psi^{\Omega_1}(\psi^{\Omega_2}(\omega)) = \psi^{\Omega_1}(\psi^{\Omega_2}(1+1+\cdots)) = \psi^{\Omega_1}(\Gamma_{\Omega_1+\omega}) \\ & < & \psi^{\Omega_1}(\Omega_2) = \psi^{\Omega_1}(\psi^{\Omega_2}(\cdots \psi^{\Omega_1}(0)\cdots)) \\ & < & \psi^{\Omega_1}\chi^{M}_{0}(\omega)) = \psi^{\Omega_1}\Omega_{\omega}) = \psi^{\Omega_1}(\Omega_{1+1+\cdots}) \\ & < & \psi^{\Omega_1}(\varphi^{1}_{0}(0)) = \psi^{\Omega_1}(\Omega_{\Omega_{\cdot_{\cdot_{\cdot_{\Omega_1}\cdot}\cdot}\cdot}}) \\ & < & \psi^{\Omega_1}(\psi^{\chi^{M}_{1}(0)}(0)) = \psi^{\Omega_1}(\psi^{I}(0)) = \psi^{\Omega_1}(\varphi^{1}_{\cdot_{\cdot_{\cdot_{\varphi^{1}_{0}(0)}\cdot}\cdot}\cdot}(0)) = \psi^{\Omega_1}(\Phi(\cdots \Phi(0,0)\cdots,0)) \\ & < & \psi^{\Omega_1}(\psi^{I}(1)) = \psi^{\Omega_1}(\varphi^{1}_{\cdot_{\cdot_{\cdot_{\psi^{I}(0)}\cdot}\cdot}\cdot}(0)) = \psi^{\Omega_1}(\Phi(\cdots \Phi(\psi^{I}(0)+1,0)\cdots,0)) \\ & < & \psi^{\Omega_1}(\psi^{I}(\omega)) = \psi^{\Omega_1}(\psi^{I}(1+1+\cdots)) \\ & < & \psi^{\Omega_1}(I) = \psi^{\Omega_1}(\psi^{I}(\cdots \psi^{I}(0)\cdots)) \\ & < & \psi^{\Omega_1}(\psi^{\chi^{M}_{1}(1)}(0)) = \psi^{\Omega_1}(\psi^{I_0(1)}) = \psi^{\Omega_1}(\varphi^{1}_{\cdot_{\cdot_{\cdot_{\varphi^{1}_{I+1}(0)}\cdot}\cdot}\cdot}(0)) = \psi^{\Omega_1}(\Phi(\cdots \Phi(I+1,0)\cdots,0)) \\ & < & \psi^{\Omega_1}(\psi^{I_1(1)}(1)) = \psi^{\Omega_1}(\varphi^{1}_{\cdot_{\cdot_{\cdot_{\psi^{I_1(1)}(0)}\cdot}\cdot}\cdot}(0)) = \psi^{\Omega_1}(\Phi(\cdots \Phi(\psi^{I_1(1)}(0)+1,0)\cdots,0)) \\ & < & \psi^{\Omega_1}(\psi^{I_1(1)}(\omega)) = \psi^{\Omega_1}(\psi^{I_1(1)}(1+1+\cdots)) \\ & < & \psi^{\Omega_1}(I_1(1)) = \psi^{\Omega_1}(\psi^{I_1(1)}(\cdots \psi^{I_1(1)}(0)\cdots)) \\ & < & \psi^{\Omega_1}(\chi^{M}_{1}(\omega)) = \psi^{\Omega_1}(I_1(\omega)) = \psi^{\Omega_1}(I_1(1+1+\cdots)) \\ & < & \psi^{\Omega_1}(\psi^{\chi^{M}_{M}(0)}(0)) = \psi^{\Omega_1}(\psi_{I(1,0,0)}(0)) = \psi^{\Omega_1}(\Lambda_0) = \psi^{\Omega_1}(\chi^{M}_{\cdot_{\cdot_{\cdot_{\chi^{M}_{0}(0)}\cdot}\cdot}\cdot}(0)) \\ & < & \psi^{\Omega_1}(\psi^{I(1,0,0)}(1)) = \psi^{\Omega_1}(\chi^{M}_{\cdot_{\cdot_{\cdot_{\chi^{M}_{\psi^{I(1,0,0)}(0)}(0)}\cdot}\cdot}\cdot}(0)) \\ & < & \psi^{\Omega_1}(\psi^{I(1,0,0)}(2)) = \psi^{\Omega_1}(\chi^{M}_{\cdot_{\cdot_{\cdot_{\chi^{M}_{\psi^{I(1,0,0)}(1)}(0)}\cdot}\cdot}\cdot}(0)) \\ & < & \psi^{\Omega_1}(\psi^{I(1,0,0)}(\omega)) = \psi^{\Omega_1}(\psi^{I(1,0,0)}(1+1+\cdots)) \\ & < & \psi^{\Omega_1}(\psi^{I(1,0,0)}(I(1,0,0))) = \psi^{\Omega_1}(\psi^{I(1,0,0)}(\cdots \psi^{I(1,0,0)}(0)\cdots)) \\ & < & \psi^{\Omega_1}(I(1,0,0)) = \psi^{\Omega_1}(\psi^{I(1,0,0)}(\cdots \psi^{I(1,0,0)}(0)\cdots)) \\ & < & \psi^{\Omega_1}(\psi^{\chi^{M}_{I(1,0,0)}(0)}(0)) = \psi^{\Omega_1}(\psi^{I_{I(1,0,0)}(1)}(0)) = \psi(I_{\psi^{I(1,0,0)}(I_{\cdot_{\cdot_{\cdot_{\psi^{I(1,0,0)}(I(1,0,0)+1)}\cdot}\cdot}\cdot}(I(1,0,0)+1))}(I(1,0,0)+1)) \\ & < & \psi^{\Omega_1}(I_{I(1,0,0)}(1)) = \psi(\psi^{I_{I(1,0,0)}(1)}(\cdots \psi^{I_{I(1,0,0)}(1)}(0)\cdots)) \\ & < & \psi^{\Omega_1}(\psi^{\chi^{M}_{M}(1)}) = \psi^{\Omega_1}(\psi^{I(1,0,1)}(0)) = \psi^{\Omega_1}(\chi^{M}_{\cdot_{\cdot_{\cdot_{\chi^{M}_{I(1,0,0)}(0)}\cdot}\cdot}\cdot}(0)) = \psi^{\Omega_1}(I_{\cdot_{\cdot_{\cdot_{I_{I_{I(1,0,0)}(1)}(0)}\cdot}\cdot}\cdot}(0)) \\ & < & \psi^{\Omega_1}(I(1,0,1)) = \psi^{\Omega_1}(\psi^{I(1,0,1)}(\cdots \psi^{I(1,0,1)}(0)\cdots)) \\ & < & \psi^{\Omega_1}(\psi^{\chi^{M}_{\varphi^{0}_{M}(M)}(0)}(0)) = \psi^{\Omega_1}(\psi^{I_{\varphi_{M}(M)}(0)}(0)) = \psi^{\Omega_1}(\chi^{M}_{\varphi^{0}_{M}(\cdot_{\cdot_{\cdot_{\chi^{M}_{\varphi^{0}_{M}(\chi^{M}_{M}(0))}(0)}\cdot}\cdot}\cdot)}(0)) \\ & < & \psi^{\Omega_1}(\psi^{\chi^{M}_{\varphi^{0}_{M+1}(0)}(0)}(0)) = \psi^{\Omega_1}(\psi^{I_{\varphi_{M+1}(0)}(0)}(0)) = \psi^{\Omega_1}(\chi^{M}_{\varphi^{0}_{M}(\cdots \varphi^{0}_{M}(0)\cdots)}(0)) \\ & < & \psi^{\Omega_1}(I_{\varphi_{M+1}(0)}(0)) = \psi^{\Omega_1}(\psi^{I_{\varphi_{M+1}(0)}(0)}(\cdots \psi^{I_{\varphi_{M+1}(0)}(0)}(0)\cdots)) \\ & < & \psi^{\Omega_1}(\psi^{M}(0)) = \psi^{\Omega_1}(\chi^{M}_{\varphi^{0}_{\cdot_{\cdot_{\cdot_{\varphi^{0}_{M+1}(0)}\cdot}\cdot}\cdot}(0)}(0)) \\ & < & \psi^{\Omega_1}(\psi^{M}(1)) = \psi^{\Omega_1}(\chi^{M}_{\varphi^{0}_{\cdot_{\cdot_{\cdot_{\varphi^{0}_{M+1}(0)}\cdot}\cdot}\cdot}(0)}(\psi^{M}(0)+1)) \\ & < & \psi^{\Omega_1}(\psi^{M}(\omega)) = \psi^{\Omega_1}(\psi^{M}(1+1+\cdots)) \\ & < & \psi^{\Omega_1}(M) = \psi^{\Omega_1}(\psi^{M}(\cdots \psi^{M}(0)\cdots)) \\ & < & \psi^{\Omega_1}(\varphi^{2}_{0}(1)) = \psi^{\Omega_1}(M_0(1)) = \psi^{\Omega_1}(\psi^{M_0(1)}(\cdots \psi^{M_0(1)}(0)\cdots)) \\ & < & \psi^{\Omega_1}(\psi^{\chi^{\varphi^{3}_{0}(0)}_{0}(0)}(0)) = \psi^{\Omega_1}(\chi^{\chi^{\varphi^{3}_{0}(0)}_{0}(0)}_{\varphi^{0}_{\cdot_{\cdot_{\cdot_{\chi^{\varphi^{3}_{0}(0)}_{0}(0)+1}\cdot}\cdot}\cdot}(0)}(0)) \\ & < & \psi^{\Omega_1}(\chi^{\varphi^{3}_{0}(0)}_{0}(0)) = \psi^{\Omega_1}(\psi^{\chi^{\varphi^{3}_{0}(0)}_{0}(0)}(\cdots \psi^{\chi^{\varphi^{3}_{0}(0)}_{0}(0)}(0)\cdots)) \\ & < & \psi^{\Omega_1}(\varphi^{3}_{0}(0)) = \psi^{\Omega_1}(\chi^{\varphi^{3}_{0}(0)}_{\varphi^{0}_{\cdot_{\cdot_{\cdot_{\varphi^{0}_{\varphi^{3}_{0}(0)+1}(0)}\cdot}\cdot}\cdot}(0)}(0)) \\ & < & \psi^{\Omega_1}(\varphi^{\omega}_{0}(0)) = \psi^{\Omega_1}(\varphi^{1+1+\cdots}_{0}(\omega+1)) \\ & < & \psi^{\Omega_1}(\varphi^{\omega+1}_{0}(0)) = \psi^{\Omega_1}(\chi^{\varphi^{\omega+1}_{0}(0)}_{\varphi^{0}_{\cdot_{\cdot_{\cdot_{\varphi^{\omega+1}_{0}(0)+1}\cdot}\cdot}\cdot}(0)}(0)) \\ & < & \psi^{\Omega_1}(\varphi^{\varphi^{\omega}_{0}(0)}_{0}(0)) = \psi^{\Omega_1}(\varphi^{\varphi^{1+1+\cdots}_{0}(\omega+1)}_{0}(\varphi^{\omega}_{0}(0)+1)) \\ & < & \psi^{\Omega_1}(\varphi^{\varphi^{\omega}_{0}(0)+1}_{0}(0)) = \psi^{\Omega_1}(\chi^{\varphi^{\varphi^{\omega}_{0}(0)+1}_{0}(0)}_{\varphi^{0}_{\cdot_{\cdot_{\cdot_{\varphi^{\varphi^{\omega}_{0}(0)+1}_{0}(0)+1}\cdot}\cdot}\cdot}(0)}(0)) \\ & < & \psi^{\Omega_1}(\varphi^{\varphi^{\omega+1}_{0}(0)}_{0}(0)) = \psi^{\Omega_1}(\varphi^{\psi^{\varphi^{\omega+1}_{0}(0)}(\cdot^{\cdot^{\cdot^{\varphi^{\psi^{\varphi^{\omega+1}}_{0}(0)}(0)}_{0}(\varphi^{\varphi^{\omega+1}_{0}(0)+1)}\cdot}\cdot}\cdot)}_{0}(\varphi^{\varphi^{\omega+1}}_{0}(0)+1)) \end{eqnarray*} The limit of this notation is \(\psi^{\Omega_1}(\varphi^{\cdot^{\cdot^{\cdot^{\varphi^{0}_{0}(0)}\cdot}\cdot}\cdot}_{0}(0))\).

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 \(T\) of strings of letters \(0,+,\varphi,\chi,\psi\) and parentheses in the following recursive way:

\(0 \in T\).
For any \((a,b) \in T^2\), \(a+b \in T\).
For any \((i,a,b) \in T^3\), \(\varphi^{i}_{a}(b) \in T\).
For any \((\kappa,a,b) \in T^3\), \(\chi^{\kappa}_{a}(b) \in T\).
For any \((\kappa,a) \in T\), \(\psi^{\kappa}(a) \in T\).

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 instead of script-free expressions with commas.

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

term	abbreviation
\(\varphi^{0}_{0}(0)\)	\(1\)
\(\varphi^{0}_{0}(1)\)	\(\omega\)
\(\varphi^{1}_{0}(0)\)	\(\Omega_1\)

Syntax[]

I define a recursive subset \(PT \subset T\) of principal terms in the following recursive way:

\(0 \notin PT\).
For any \((a,b) \in T^2\), \(a+b \notin PT\).
For any \((i,a,b) \in T^3\), \(\varphi^{i}_{a}(b) \in PT\).
For any \((\kappa,a,b) \in T^3\), \(\chi^{\kappa}_{a}(b) \in PT\).
For any \((\kappa,a) \in T^2\), \(\psi^{\kappa}(a) \in PT\).

Principal terms play roles analogous to additive principal numbers.

I define a recursive subset \(ST \subset T\) of successor terms in the following recursive way:

\(0 \notin ST\).
For any \((a,b) \in PT \times T\), \(a+b \in ST\) if and only if \(b \in ST\).
For any \((i,a,b) \in T^3\), \(\varphi^{i}_{a}(b) \in ST\) if and only if \((i,a,b) = (0,0,0)\).
For any \((\kappa,a,b) \in T^3\), \(\chi^{\kappa}_{a}(b) \notin ST\).
For any \((\kappa,a) \in T^3\), \(\psi^{\kappa}(a) \notin ST\).

Successor terms play roles analogous to successor ordinals.

I define a recursive subset \(RT \subset T\) of regular terms in the following recursive way:

\(0 \notin RT\).
For any \((a,b) \in PT \times T\), \(a+b \notin RT\).
For any \((i,a,b) \in T^3\), \(\varphi^{i}_{a}(b) \in RT\) if and only if \(i \neq 0\), \(a = 0\), and \(b \in \{0\} \cup ST\).
For any \((\kappa,a,b) \in T^3\), \(\chi^{\kappa}_{a}(b) \in RT\) if and only if \(b \in \{0\} \cup ST\).
For any \((\kappa,a) \in T^3\), \(\psi^{\kappa}(a) \notin RT\).

Regular terms play roles analogous to uncountable regular cardinals.

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:

If \(s \in \{0\} \cup PT\), then \(\textrm{pred}(s) := 0\).
Suppose \(s = a+b\) for a unique \((a,b) \in PT \times T\).
1. If \(b = 1\), then \(\textrm{pred}(s) := a\).
2. If \(b \neq 1\) and \(\textrm{pred}(b) = 0\), then \(\textrm{pred}(s) := 0\).
3. If \(\textrm{pred}(b) \neq 0\), then \(\textrm{pred}(s) := a + \textrm{pred}(b)\).

The \(\textrm{pred}\) map plays a role analogous to the map assigning the predecessors to each successor ordinals.

I define recursive map \begin{eqnarray*} \textrm{deg} \colon T & \to & T \\ s & \mapsto & \textrm{deg}(s) \end{eqnarray*} in the following recursive way:

If \(s \notin PT\), then \(\textrm{deg}(s) := 0\).
If \(s = \varphi^{i}_{a}(b)\) for a unique \((i,a,b) \in T^3\), then \(\textrm{deg}(s) := i\).
If \(s = \chi^{\kappa}_{a}(b)\) for a unique \((\kappa,a,b) \in T^3\), then \(\textrm{deg}(s) := \textrm{pred}(\textrm{deg}(\kappa))\).
If \(s = \psi^{\kappa}(a)\) for a unique \((\kappa,a) \in T^2\), then \(\textrm{deg}(s) := 0\).

The \(\textrm{deg}\) map roughly indicates how many times \(s\) can be collapsible, will be used to describe the ordering and the expansion rule.

I define recursive map \begin{eqnarray*} \textrm{index} \colon T & \to & T^2 \\ s & \mapsto & \textrm{index}(s) \end{eqnarray*} in the following recursive way:

If \(s = \varphi^{i}_{0}(b)\) for a unique \((i,b) \in T^2\), then \(\textrm{index}(s) := (i,0)\).
Suppose \(s = \chi^{\kappa}_{a}(b)\) for a unique \((\kappa,a,b) \in T^3\).
1. If \(a = 0\), then \(\textrm{index}(s) := \textrm{index}(\kappa)\).
2. If \(a \neq 0\), then \(\textrm{index}(s) := (\kappa,a)\).
Otherwise, \(\textrm{index}(s) := (0,0)\).

The \(\textrm{index}\) map roughly indicates an \(i \in T\) such that \(s\) is not closed by \(\varphi^{i}\) or a \((\kappa,a)\) such that \(s\) is not closed under \(\chi^{\kappa}_{a}\) in some sense, and will be used to describe the expansion rule.

For an \(s \in T\), I define an \(s^{+} \in T\) in the following recursive way:

If \(s = 0\), then \(s^{+} := 1\).
If \(s = \varphi^{1}_{0}(b)\) for a unique \(b \in T \setminus \{0\}\), then \(s^{+} := \varphi^{1}_{0}(b+1)\).
Otherwise, \(s^{+} := \varphi^{1}_{0}(s+1)\).

The assignment \({+}\) will play a role analogous to the successor cardinal.

For an \((s,t) \in T^2\), I define an \(s^{t}_{\bullet} \in T\) in the following recursive way:

Suppose \(s = \chi^{\kappa}_{a}(b)\) for a unique \((\kappa,a,b) \in T^3\).
1. If \(\kappa = t\), then \(s^{t}_{\bullet} := a\).
2. If \(\kappa \neq t\), then \(s^{t}_{\bullet} := \kappa^{t}_{\bullet}\).
Otherwise, \(s^{t}_{\bullet} := 0\).

The value \(s^{t}_{\bullet}\) roughly means the subscript of itself or a superscript whose superscript is \(t\).

Standard Form[]

I define a recursive relation \(s \leq t\) on \((s,t) \in T^2\), a recursive relation \(s < t\) on \((s,t) \in T^2\), a recursive relation \(s \triangleleft (\kappa,a)\) on \((s,\lambda,c) \in T^3\), a recursive subset \(OT \subset T\), an \(s^{* \geq t} \in OT\) for an \((s,t) \in T^2\), and an \(s^{-} \in OT\) for an \(s \in T\) simultaneously in the following recursive way:

Definition of \(s \leq t\)

If \(s = t\), then \(s \leq t\) is true.
If \(s \neq t\), then \(s \leq t\) is equivalent to \(s < t\).

Definition of \(s < t\)

If \(t = 0\), then \(s < t\) is false.
If \(s = 0\) and \(t \neq 0\), then \(s < t\) is true.
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 \(s < t\) is equivalent to that one of the following holds:
1. \(a < c\); or
2. \(a = c\) and \(b < d\).
If \(s = a+b\) for a unique \((a,b) \in PT \times T\) and \(t \in PT\), then \(s < t\) is equivalent to \(a < t\).
If \(s \in PT\) and and \(t = c+d\) for a unique \((c,d) \in PT \times T\), then \(s < t\) is equivalent to the negation of \(t < s\).
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 \(s < t\) is equivalent to that one of the following hold:
1. \(i = j\), \(a = c\), and \(b < d\);
2. \(i = j\), \(a < c\), and \(b \leq t\);
3. \(i = j\), \(c < a\), and \(s < d\);
4. \(i < j\), \(a < t\), and \(b \leq t\);
5. \(i < j\), \(a = t\), and \(b = 0\);
6. \(j < i\), \(s \leq c\), and \(c \neq s\);
7. \(j < i\), \(s \leq c\), and \(d \neq 0\);
8. \(j < i\), \(s < d\), and \(c \neq s\);
9. \(j < i\), \(s < d\), and \(d \neq 0\);
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 \(s < t\) is equivalent to then \(s < t\) is equivalent to \(s < \lambda\) and one of the following holds:
1. \(i = t\), \(a = 0\), and \(b = 0\);
2. \(i < t\), \(a = t\), and \(b = 0\); or
3. \(i < t\), \(a < t\), and \(b \leq t\).
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 \(s < t\) is equivalent to \(s < \lambda\) and one of the following holds:
1. \(i = t\), \(a = 0\), and \(b = 0\);
2. \(i < t\), \(a = t\), and \(b = 0\); or
3. \(i < t\), \(a < t\), and \(b \leq t\).
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 \(s < t\) is equivalent to the negation of \(t < s\).
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 \(s < t\) is equivalent to that one of the following holds:
1. \(\kappa = \lambda\), \(a = c\), and \(b < d\);
2. \(\kappa = \lambda\), \(a < c\), \(a^{* \geq \lambda} < t\), and \(b < t\);
3. \(\kappa = \lambda\), \(c < a\), and \(s \leq c^{* \geq \kappa}\);
4. \(\kappa = \lambda\), \(c < a\), and \(s \leq d\);
5. \(\kappa \leq t\);
6. \(s \leq t^{-}\); or
7. \(s < \lambda\), \(s^{-} < t\), and \(a < \lambda^{\kappa}_{\bullet}\).
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 \(s < t\) is equivalent to \(s < \lambda\) and \(s \triangleleft (\lambda,c)\).
If \(s = \psi^{\kappa}(a)\) for a unique \((\kappa,a) \in T^2\) and \(t = \varphi^{j}_{c}(d)\) for a unique \((j,c,d) \in T^3\), then \(s < t\) is equivalent to the negation of \(t < s\).
If \(s = \psi^{\kappa}(a)\) for a unique \((\kappa,a) \in T^2\) and \(t = \chi^{\lambda}_{c}(d)\) for a unique \((\lambda,c,d) \in T^3\), then \(s < t\) is equivalent to the negation of \(t < s\).
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 \(s < t\) is equivalent to that one of the following holds:
1. \(\kappa = \lambda\) and \(a < c\);
2. \(\kappa < \lambda\) and \(\kappa^{* \geq \lambda} < t\); or
3. \(s < \lambda\) and \(\lambda < \kappa\).

I note that \(<\) is not a well-ordering unless it is restricted to \(OT\), and even is not compatible with the \(\in\)-relation for some natural correspondence to ordinals. It will be used to define \(OT\), under a careful avoidance of a common trap to use \(<\) to implement "\(\beta \in \chi_{\alpha}(\beta)\)" for Rathjen's \(\chi\) and "\(\alpha, \beta < \Phi_{\alpha}(\beta)\)" for Rathjen's \(\Phi\).

Definition of \(s \triangleleft (\lambda,c)\)

If \(s = 0\), then \(s \triangleleft (\lambda,c)\) is true;
If \(s = a+b\) for a unique \((a,b) \in PT \times T\), then \(s \triangleleft (\lambda,c)\) is equivalent to \(a \triangleleft (\lambda,c)\) and \(b \triangleleft (\lambda,c)\).
If \(s = \varphi^{i}_{a}(b)\) for a unique \((i,a,b) \in T^3\), then \(s \triangleleft (\lambda,c)\) is equivalent to \(i \triangleleft (\lambda,c)\), \(a \triangleleft (\lambda,c)\), and \(b \triangleleft (\lambda,c)\).
If \(s = \chi^{\kappa}_{a}(b)\) for a unique \((\kappa,a,b) \in T^3\), then \(s \triangleleft (\lambda,c)\) is equivalent to \(\kappa \triangleleft (\lambda,c)\), \(a \triangleleft (\lambda,c)\), and \(b \triangleleft (\lambda,c)\).
If \(s = \psi^{\kappa}(a)\) for a unique \((\kappa,a) \in T^2\), then \(s \triangleleft (\lambda,c)\) is equivalent to that one of the following hold:
1. \(s \leq \lambda^{-}\).
2. \(\lambda^{-} < s\), \(\kappa < \lambda\), and \(a \triangleleft (\lambda,c)\).
3. \(\lambda^{-} < s\), \(\lambda \leq \kappa\), \(a < c\), \(\kappa \triangleleft (\lambda,c)\), and \(a \triangleleft (\lambda,c)\).

The relation \(\triangleleft\) is an analogue to Rathjen's \(K\) function.

Definition of \(OT\)

\(0 \in OT\)
For any \((a,b) \in PT \times T\), \(a+b \in OT\) is equivalent to \(a \in OT\), \(b \in OT\), \(b \neq 0\), and \(b < \varphi^{0}_{0}(a+1)\).
For any \((i,a,b) \in T^3\), \(\varphi^{i}_{a}(b) \in OT\) is equivalent to \(i \in OT\), \(a \in OT\), \(b \in OT\), and that one of the following holds:
1. \(b = 0\) and \(a \notin \textrm{PT}\);
2. \(b = 0\), \(a = \varphi^{j}_{c}(d)\) for a unique \((j,c,d) \in T^3\), and \(j \leq i\);
3. \(b = 0\), \(a = \chi^{\lambda}_{c}(d)\) for a unique \((\lambda,c,d) \in T^3\), and \(\textrm{deg}(\lambda) \leq i\);
4. \(b = 0\), \(a = \psi^{\lambda}(c)\) for a unique \((\lambda,c) \in T^2\), and \(\textrm{deg}(\lambda) < i\);
5. \(b = 0\), \(a = \psi^{\lambda}(c)\) for a unique \((\lambda,c) \in T^2\), and \(\lambda = \varphi^{i}_{0}(f)\) for a unique \(f \in T\);
6. \(b \neq 0\), and \(b \leq a\);
7. \(b = c+d\) for a unique \((c,d) \in PT \times T\);
8. \(b = \varphi^{i}_{c}(d)\) for a unique \((c,d) \in T^2\), and \(c \leq a\);
9. \(b = \varphi^{j}_{c}(d)\) for a unique \((j,c,d) \in T^3\), and \(j < i\);
10. \(b = \chi^{\lambda}_{c}(d)\) for a unique \((\lambda,c,d) \in T^3\), and \(\textrm{deg}(\lambda) \leq i\);
11. \(b = \psi^{\lambda}(c)\) for a unique \((\lambda,c) \in T^2\), and \(\textrm{deg}(\lambda) < i\); or
12. \(b = \psi^{\lambda}(c)\) for a unique \((\lambda,c) \in T^2\), and \(\lambda = \varphi^{i}_{0}(f)\) for a unique \(f \in T\).
For any \((\kappa,a,b) \in T^3\), \(\chi^{\kappa}_{a}(b) \in OT\) is equivalent to \(\kappa \in RT \cap OT\), \(a \in OT\), \(b \in OT\), \(b < \kappa\), \(\textrm{pred}(\textrm{deg}(\kappa)) \neq 0\), \(a < \psi^{\kappa^{+}}(0)\), and that one of the following holds:
1. \(b \leq \chi^{\kappa}_{a}(0)^{-}\);
2. \(b \notin PT\);
3. \(b = \varphi^{j}_{c}(d)\) for a unique \((j,c,d) \in T^3\), and \(j < \textrm{deg}(\kappa)\);
4. \(b = \chi^{\lambda}_{c}(d)\) for a unique \((\lambda,c,d) \in T^3\), \(\textrm{deg}(\lambda) = \textrm{deg}(\kappa)\), and \(c \leq a\);
5. \(b = \chi^{\lambda}_{c}(d)\) for a unique \((\lambda,c,d) \in T^3\), and \(\textrm{deg}(\lambda) < \textrm{deg}(\kappa)\);
6. \(b = \psi^{\lambda}(c)\) for a unique \((\lambda,c) \in T^2\), and the negation of \(a \triangleleft (\lambda,c)\) holds;
7. \(b = \psi^{\lambda}(c)\) for a unique \((\lambda,c) \in T^2\), \(\lambda = \varphi^{j}_{0}(d)\) for a unique \((j,d) \in T^2\), and \(j < \textrm{deg}(\kappa)\).
8. \(b = \psi^{\lambda}(c)\) for a unique \((\lambda,c) \in T^2\), \(\lambda = \chi^{\mu}_{e}(f)\) for a unique \((\mu,e,f) \in T^3\), \(\textrm{deg}(\mu) = \textrm{deg}(\kappa)\), and \(e \leq a\); or
9. \(b = \psi^{\lambda}(c)\) for a unique \((\lambda,c) \in T^2\), \(\lambda = \chi^{\mu}_{e}(f)\) for a unique \((\mu,e,f) \in T^3\), \(\textrm{deg}(\mu) < \textrm{deg}(\kappa)\).
For any \((\kappa,a) \in T^2\), \(\psi^{\kappa}(a) \in OT\) is equivalent to \(\kappa \in RT \cap OT\), \(a \in OT\), and \(a \triangleleft (\kappa,a)\)

I note that "\(\chi^{\kappa}_{a}(b) \in OT\)" should not be determined by "\(\kappa \in OT\), \(a \in OT\), \(b \in OT\), and \(b < \chi^{\kappa}_{a}(b)\)" as I clarified in the definition of \(<\).

Definition of \(s^{* \geq t}\)

If \(s \in \{0,t\}\) or \(s \notin OT\), then \(s^{* \geq t} := 0\).
Suppose \(s \notin \{0,t\}\) and \(s \in OT\).
1. Suppose \(s = a+b\) for a unique \((a,b) \in PT \times T\).
  1. If \(a^{* \geq t} \leq b^{* \geq t}\), then \(s^{* \geq t} := b^{* \geq t}\).
  2. If \(b^{* \geq t} < a^{* \geq t}\), then \(s^{* \geq t} := a^{* \geq t}\).
2. Suppose \(s = \varphi^{i}_{a}(b)\) for a unique \((i,a,b) \in T^3\).
  1. If \(i^{* \geq t} \leq b^{* \geq t}\) and \(a^{* \geq t} \leq b^{* \geq t}\), then \(s^{* \geq t} := b^{* \geq t}\).
  2. If \(i^{* \geq t} \leq a^{* \geq t}\) and \(b^{* \geq t} < a^{* \geq t}\), then \(s^{* \geq t} := a^{* \geq t}\).
  3. If \(a^{* \geq t} < i^{* \geq t}\) and \(b^{* \geq t} < i^{* \geq t}\), then \(s^{* \geq t} := i^{* \geq t}\).
3. Suppose \(s = \chi^{\kappa}_{a}(b)\) for a unique \((\kappa,a,b) \in T^3\).
  1. If \(t \leq \kappa\), then \(s^{* \geq t} := s\).
  2. Suppose \(\kappa < t\).
    1. If \(\kappa^{* \geq t} \leq b^{* \geq t}\) and \(a^{* \geq t} \leq b^{* \geq t}\), then \(s^{* \geq t} := b^{* \geq t}\).
    2. If \(\kappa^{* \geq t} \leq a^{* \geq t}\) and \(b^{* \geq t} < a^{* \geq t}\), then \(s^{* \geq t} := a^{* \geq t}\).
    3. If \(a^{* \geq t} < \kappa^{* \geq t}\) and \(b^{* \geq t} < \kappa^{* \geq t}\), then \(s^{* \geq t} := \kappa^{* \geq t}\).
4. If \(s = \psi^{\kappa}(a)\) for a unique \((\kappa,a) \in T^2\), then \(s^{* \geq t} := s\).

The assignment \({* \geq}\) will play a role analogous to Rathjen's \({*}\) function. Roughly speaking, \(s^{* \geq t}\) works as an upper bound of building blocks of \(s\) with respect to the closure of \(\{0,t\}\), \(+\), \(\varphi\), and \(\chi^{\lambda}\) with \(\lambda < t\). I note that the computation of \(s^{* \geq t}\) refers to the criterion of \(s \in OT\), the criterion of \(s \in OT\) refers to \({-}\) only when \(s\) is of the form \(\chi^{\kappa}_{a}(0)\) for a \((\kappa,a) \in T^2\), and the computation of \(\chi^{\kappa}_{a}(0)^{-}\) refers to \({*}\) only for the computation of \(a^{* \geq \kappa}\). Since \(a\) is a proper subexpression of \(s\), this mutual recursion of \({*}\), \(OT\), and \({-}\) does not cause a simple infinite loop. In addition, although the value of \(s^{* \geq t}\) is not necessarily smaller than \(t\) unlike Rathjen's \({*}\) function whose image is contained in the least weakly Mahlo cardinal, we have \(a^{* \geq \kappa} < \kappa\) for any \((\kappa,a) \in OT^2\) with \(\chi^{\kappa}_{a}(0) \in OT\) due to the restriction \(a < \psi^{\kappa^{+}}(0)\).

Definition of \(s^{-}\)

Suppose \(s = \varphi^{i}_{0}(b)\) for a unique \((i,b) \in OT^2\).
1. If \(b = 0\), then \(s^{-} := i\).
2. Suppose \(b \neq 0\).
  1. If \(\varphi^{i}_{0}(\textrm{pred}(b)) \in OT\), then \(s^{-} := \varphi^{i}_{0}(\textrm{pred}(b))\).
  2. If \(\varphi^{i}_{0}(\textrm{pred}(b)) \notin OT\), then \(s^{-} := \textrm{pred}(b)\).
Suppose \(s = \chi^{\kappa}_{a}(b)\) for a unique \((\kappa,a,b) \in OT^3\).
1. Suppose \(b = 0\).
  1. If \(\kappa^{-} \leq a^{* \geq \kappa}\), then \(s^{-} := a^{* \geq \kappa}\).
  2. If \(a^{* \geq \kappa} < \kappa^{-}\), then \(s^{-} := \kappa^{-}\).
2. Suppose \(b \neq 0\).
  1. If \(\chi^{\kappa}_{a}(\textrm{pred}(b)) \in OT\), then \(s^{-} := \chi^{\kappa}_{a}(\textrm{pred}(b))\).
  2. If \(\chi^{\kappa}_{a}(\textrm{pred}(b)) \notin OT\), then \(s^{-} := \textrm{pred}(b)\).
Otherwise, \(s^{-} := 0\).

The assignment \({-}\) will play a role analogous to Rathjen's \({-}\) function. Roughly speaking, \(\chi^{\kappa}_{a}(b)^{-}\) works as a lower bound of \(\chi^{\kappa}_{a}(b)\) and an upper bound of building blocks of \(\chi^{\kappa}_{a}(b)\) with respect to the closure of \(\{0,\kappa\}\), \(+\), \(\varphi\), and \(\chi^{\lambda}\) with \(\lambda < \kappa\).

I also denote by \(<\) the restriction of \(<\) to \(OT\). I expect that \((OT,<)\) forms an ordinal notation, i.e. the restricted \(<\) 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.

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:

If \(s = 0\), then \(\textrm{dom}(s) := 0\).
If \(s = a+b\) for a unique \((a,b) \in PT \times T\), then \(\textrm{dom}(s) := \textrm{dom}(b)\).
Suppose \(s = \varphi^{i}_{a}(b)\) for a unique \((i,a,b) \in T^3\).
1. If \(\textrm{dom}(a) = 0\), \(\textrm{dom}(i) = 0\), and \(\textrm{dom}(b) = 0\), then \(\textrm{dom}(s) := 1\).
2. If \(\textrm{dom}(a) = 0\), \(\textrm{dom}(i) = 0\), and \(\textrm{dom}(b) = 1\), then \(\textrm{dom}(s) := \omega\).
3. If \(\textrm{dom}(a) = 0\), \(\textrm{dom}(i) \neq 0\), and \(\textrm{dom}(b) \in \{0,1\}\), then \(\textrm{dom}(s) := s\).
4. If \(\textrm{dom}(a) = 1\) and \(\textrm{dom}(b) \in \{0,1\}\), then \(\textrm{dom}(s) := \omega\).
5. If \(\textrm{dom}(a) \notin \{0,1\}\) and \(\textrm{dom}(b) \in \{0,1\}\), then \(\textrm{dom}(s) := \textrm{dom}(a)\).
6. If \(\textrm{dom}(b) \notin \{0,1\}\), then \(\textrm{dom}(s) := \textrm{dom}(b)\).
Suppose \(s = \chi^{\kappa}_{a}(b)\) for a unique \((\kappa,a,b) \in T^3\).
1. If \(\textrm{dom}(b) \in \{0,1\}\), then \(\textrm{dom}(s) := s\).
2. If \(\textrm{dom}(b) \notin \{0,1\}\), then \(\textrm{dom}(s) := \textrm{dom}(b)\).
Suppose \(s = \psi^{\kappa}(a)\) for a unique \((\kappa,a) \in T^2\).
1. If \(\textrm{dom}(a) \in \{0,1\}\) and \(\chi^{\kappa}_{0}(0) \in OT\), then \(\textrm{dom}(s) := \omega\).
2. If \(\textrm{dom}(a) \in \{0,1\}\), \(\chi^{\kappa}_{0}(0) \notin OT\), and \(\textrm{index}(\kappa) = (j,0)\) for a unique \(j \in T\) with \(\textrm{dom}(j) = 1\), then \(\textrm{dom}(s) := \omega\).
3. If \(\textrm{dom}(a) \in \{0,1\}\), \(\chi^{\kappa}_{0}(0) \notin OT\), and \(\textrm{index}(\kappa) = (j,0)\) for a unique \(j \in T\) with \(\textrm{dom}(j) \neq 1\), then \(\textrm{dom}(s) := \textrm{dom}(j)\).
4. Suppose \(\textrm{dom}(a) \in \{0,1\}\), \(\chi^{\kappa}_{0}(0) \notin OT\), and \(\textrm{index}(\kappa) = (\lambda,c)\) for a unique \((\lambda,c) \in T^2\) with \(c \neq 0\).
  1. If \(\textrm{dom}(c) \neq 1\) and \(\textrm{dom}(c) < \lambda\), then \(\textrm{dom}(s) := \textrm{dom}(c)\).
  2. Otherwise, \(\textrm{dom}(s) := \omega\).
5. If \(\textrm{dom}(a) \notin \{0,1\}\) and \(\textrm{dom}(a) < \kappa\), then \(\textrm{dom}(s) := \textrm{dom}(a)\).
6. If \(\textrm{dom}(a) \notin \{0,1\}\) and \(\kappa \leq \textrm{dom}(a)\), then \(\textrm{dom}(s) := \omega\).

The \(\textrm{dom}\) map is an analogue of Buchholz's \(\textrm{dom}\) map.

Expansion[]

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:

If \(\textrm{dom}(n) = 1\), then \(\Gamma^{i}(s,n) := \varphi^{i}_{\Gamma^{i}(s,\textrm{pred}(n))}(0)\).
If \(\textrm{dom}(n) \neq 1\) and \(\varphi^{i}_{s}(0) \in OT\), then \(\Gamma^{i}(s,n) := s\).
If \(\textrm{dom}(n) \neq 1\) and \(\varphi^{i}_{s}(0) \notin OT\), then \(\Gamma^{i}(s,n) := s+1\).

The \(\Gamma\) map is an analogue of Gamma numbers.

I define a recursive map \begin{eqnarray*} \overline{\varphi} \colon OT^3 & \to & OT \\ (i,a,b) & \mapsto & \overline{\varphi}(i,a,b) \end{eqnarray*} in the following recursive way:

If \(\varphi^{i}_{a}(b) \in OT\), put \(\overline{\varphi}(i,a,b) := \varphi^{i}_{a}(b)\).
If \(\varphi^{i}_{a}(b) \notin OT\) and \(b = 0\), put \(\overline{\varphi}(i,a,b) := a\).
If \(\varphi^{i}_{a}(b) \notin OT\) and \(b \neq 0\), put \(\overline{\varphi}(i,a,b) := b\).

The \(\overline{\varphi}\) map returns an element of \(OT\) which is a "standardisation" of \(\varphi^{i}_{a}(b)\).

I define a recursive map \begin{eqnarray*} \psi^{-} \colon OT^2 & \to & OT \\ (\kappa,a) & \mapsto & \psi^{-}(\kappa,a) \end{eqnarray*} in the following recursive way:

If \(\textrm{dom}(a) = 0\) and \(\kappa^{-} \neq 0\), then \(\psi^{-}(\kappa,a) := \kappa^{-} + 1\).
If \(\textrm{dom}(a) = 1\) and \(\psi^{\kappa}(a[0]) \in OT\), then \(\psi^{-}(\kappa,a) := \psi^{\kappa}(a[0])+1\).
Otherwise, \(\psi^{-}(a,\kappa) := \kappa^{-}\).

The \(\psi^{-}\) map returns an element of \(OT\) which is smaller than \(\psi^{\kappa}(a)\).

I define a recursive map \begin{eqnarray*} [ \ ] \colon \{(s,n) \in OT^2 \mid n < \textrm{dom}(s) \lor \textrm{dom}(s) < \omega\} & \to & OT \\ (s,n) & \mapsto & s[n] \end{eqnarray*} in the following recursive way:

If \(s = 0\), then \(s[n] := 0\).
Suppose \(s = a+b\) for a unique \((a,b) \in PT \times T\),
1. If \(b[n] = 0\), then \(s[n] := a\).
2. If \(b[n] \neq 0\), then \(s[n] := a+b[n]\).
Suppose \(s = \varphi^{i}_{a}(b)\) for a unique \((i,a,b) \in T^3\).
1. If \(\textrm{dom}(a) = 0\), \(\textrm{dom}(i) = 0\), and \(\textrm{dom}(b) = 0\), then \(s[n] := 0\).
2. Suppose \(\textrm{dom}(a) = 0\), \(\textrm{dom}(i) = 0\), and \(\textrm{dom}(b) = 1\).
  1. If \(\textrm{dom}(n) = 1\), then \(s[n] := s[n[0]]+\overline{\varphi}(i,a,b[n])\).
  2. If \(\textrm{dom}(n) \neq 1\), then \(s[n] := \overline{\varphi}(i,a,b[n])\).
3. If \(\textrm{dom}(a) = 0\), \(\textrm{dom}(i) \neq 0\), and \(\textrm{dom}(b) \in \{0,1\}\), then \(s[n] := n\).
4. Suppose \(\textrm{dom}(a) = 1\) and \(\textrm{dom}(b) \in \{0,1\}\).
  1. If \(\textrm{dom}(n) = 1\), then \(s[n] := \varphi^{i}_{a[0]}(s[n[0]])\).
  2. If \(\textrm{dom}(n) \neq 1\), \(\textrm{dom}(b) = 0\), and \(\varphi^{i}_{a[0]}(0) \in OT\), then \(s[n] := \varphi^{i}_{a[0]}(0)\).
  3. If \(\textrm{dom}(n) \neq 1\), \(\textrm{dom}(b) = 0\), and \(\varphi^{i}_{a[0]}(0) \notin OT\), then \(s[n] := a[0]+1\).
  4. If \(\textrm{dom}(n) \neq 1\) and \(\textrm{dom}(b) = 1\), then \(s[n] := \varphi^{i}_{a[0]}(\overline{\varphi}(i,a,b[0])+1)\).
5. If \(\textrm{dom}(a) \notin \{0,1\}\), \(\textrm{dom}(b) = 0\), and \(\varphi^{i}_{a[n]}(0) \in OT\), then \(s[n] := \varphi^{i}_{a[n]}(0)\).
6. If \(\textrm{dom}(a) \notin \{0,1\}\), \(\textrm{dom}(b) = 0\), and \(\varphi^{i}_{a[n]}(0) \notin OT\), then \(s[n] := a[n]+1\).
7. If \(\textrm{dom}(a) \notin \{0,1\}\) and \(\textrm{dom}(b) = 1\), then \(s[n] := \varphi^{i}_{a[n]}(\overline{\varphi}(i,a,b[0])+1)\).
8. If \(\textrm{dom}(b) \notin \{0,1\}\), then \(s[n] := \overline{\varphi}(i,a,b[n])\).
Suppose \(s = \chi^{\kappa}_{a}(b)\) for a unique \((\kappa,a,b) \in T^3\).
1. If \(\textrm{dom}(b) \in \{0,1\}\), then \(s[n] := n\).
2. If \(\textrm{dom}(b) \notin \{0,1\}\), then \(s[n] := \chi^{\kappa}_{a}(b[n])\).
Suppose \(s = \psi^{\kappa}(a)\) for a unique \((\kappa,a) \in T^2\).
1. If \(\textrm{dom}(a) = 0\) and \(\chi^{\kappa}_{0}(0) \in OT\), then \(s[n] := \chi^{\kappa}_{\Gamma^{0}(\kappa,n)}(0)\).
2. If \(\textrm{dom}(a) = 1\) and \(\chi^{\kappa}_{0}(0) \in OT\), then \(s[n] := \chi^{\kappa}_{\Gamma^{0}(\kappa,n)}(\psi^{\kappa}(a[0])+1)\).
3. If \(\textrm{dom}(a) \in \{0,1\}\), \(\chi^{\kappa}_{0}(0) \notin OT\), and \(\textrm{index}(\kappa) = (j,0)\) for a unique \(j \in T\) with \(\textrm{dom}(j) = 1\), then \(s[n] := \Gamma^{j[0]}(\psi^{-}(\kappa,a),n)\).
4. If \(\textrm{dom}(a) \in \{0,1\}\), \(\chi^{\kappa}_{0}(0) \notin OT\), and \(\textrm{index}(\kappa) = (j,0)\) for a unique \(j \in T\) with \(\textrm{dom}(j) \neq 1\), then \(s[n] := \varphi^{j[n]}_{\psi^{-}(\kappa,a)}(0)\).
5. Suppose \(\textrm{dom}(a) \in \{0,1\}\), \(\chi^{\kappa}_{0}(0) \notin OT\), and \(\textrm{index}(\kappa) = (\lambda,c)\) for a unique \((\lambda,c) \in T^2\) with \(c \neq 0\).
  1. Suppose \(\textrm{dom}(c) = 1\).
    1. If \(\textrm{dom}(n) = 1\), then \(s[n] := \chi^{\lambda}_{c[0]}(s[n[0]])\).
    2. If \(\textrm{dom}(n) \neq 1\), then \(s[n] := \chi^{\lambda}_{c[0]}(\psi^{-}(\kappa,a))\).
  2. If \(1 < \textrm{dom}(c)\) and \(\textrm{dom}(c) < \lambda\), then \(s[n] := \chi^{\lambda}_{c[n]}(\psi^{-}(\kappa,a))\).
  3. Suppose \(\lambda \leq \textrm{dom}(c)\).
    1. If \(\textrm{dom}(n) = 1\), then \(s[n] := \chi^{\lambda}_{c[s[n[0]]}(0)\).
    2. If \(\textrm{dom}(n) \neq 1\), then \(s[n] := \chi^{\lambda}_{c[\psi^{-}(\kappa,a)]}(0)\).
6. If \(\textrm{dom}(a) \notin \{0,1\}\) and \(\textrm{dom}(a) < \kappa\), then \(s[n] := \psi^{\kappa}(a[n])\).
7. Suppose \(\textrm{dom}(a) \notin \{0,1\}\) and \(\kappa \leq \textrm{dom}(a)\).
  1. If \(\textrm{dom}(n) = 1\) and \(s[n[0]] = \psi^{\kappa}(a')\) for a unique \(a' \in OT\), then \(s[n] := \psi^{\kappa}(a[\psi^{\textrm{dom}(a)}(a')])\).
  2. Otherwise, \(s[n] := \psi^{\kappa}(a[\psi^{\textrm{dom}(a)}(0)])\).

I denote by \(CT \subset OT\) the recursive subset \(\{s \in OT \mid s < \Omega_1\}\). The \([ \ ]\) operator restricted to \(CT\) plays a role of a recursive system of fundamental sequences.

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:

If \(n = 0\), then \(\lfloor n \rfloor := 0\).
If \(n = 1\), then \(\lfloor n \rfloor := 1\).
If \(n > 1\), then \(\lfloor n \rfloor := \lfloor n-1 \rfloor + 1\).

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 \(A\) is the empty array, then \(f(A,n) := n\).
Suppose that \(A\) is not the empty array.
1. Denote by \(t \in CT\) the rightmost entry of \(A\).
2. Denote by \(G \in CT^{<\omega}\) the array obtained by deleting the rightmost entry from \(A\).
3. If \(t = 0\), then \(f(A,n) := f(G,n+1)\).
4. Suppose \(t \neq 0\).
  1. Denote by \(A' \in CT^{<\omega}\) the concatenation of \(G\) and \((t[\lfloor n \rfloor])_{i=0}^{n-1}\).
  2. Then \(f(A,n) := f(A',n)\).

I define a recursive map \begin{eqnarray*} \perp \colon OT & \to & OT \\ n & \mapsto & \perp(n) \end{eqnarray*} in the following recursive way:

If \(\textrm{dom}(n) = 1\), then \(\perp(n) := \varphi^{\perp(\textrm{pred}(n))}_{0}(0)\).
If \(\textrm{dom}(n) \neq 1\), then \(\perp(n) := \varphi^{0}_{0}(0)\).

If \((OT,<)\) is actually 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 \(f((\psi^{\Omega_1}(\perp(1+1+1+1+1+1+1+1+1+1))),10)\) is a computable large number.

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 \(\psi\)^[1] in the ordinal column. This is the least strongly inaccessible WIP.

Abbreviation[]

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

term	abbreviation
\(\varphi^{0}_{0}(0)\)	\(1\)
\(\varphi^{0}_{0}(1)\)	\(\omega\)
\(\varphi^{1}_{0}(0)\)	\(\Omega_1\)
\(\varphi^{1}_{0}(1)\)	\(\Omega_2\)
\(\varphi^{1}_{0}(\omega)\)	\(\Omega_{\omega}\)
\(\chi^{\varphi^{2}_{0}(0)}_{0}(t)\)	\(I_t\)
\(\varphi^{2}_{0}(t)\)	\(M_t\)
\(\varphi^{3}_{0}(t)\)	\(K_t\)

Up to Γ_0[]

term	ordinal	expansion
\(0\)	\(0\)	\(0\)
\(1\)	\(1\)	\(0\)
\(1+1\)	\(2\)	\(1\)
\(1+1+1\)	\(3\)	\(1+1\)
\(\omega\)	\(\omega\)	\(1+1+\cdots\)
\(\omega+1\)	\(\omega+1\)	\(\omega\)
\(\omega+1+1\)	\(\omega+2\)	\(\omega+1\)
\(\omega+\omega\)	\(\omega \times 2\)	\(\omega+1+1+\cdots\)
\(\varphi^{0}_{0}(1+1)\)	\(\omega^2\)	\(\omega+\omega+\cdots\)
\(\varphi^{0}_{0}(\omega)\)	\(\omega^{\omega}\)	\(\varphi^{0}_{0}(1+1+\cdots)\)
\(\varphi^{0}_{0}(\varphi^{0}_{0}(\omega))\)	\(\omega^{\omega^{\omega}}\)	\(\varphi^{0}_{0}(\varphi^{0}_{0}(1+1+\cdots))\)
\(\varphi^{0}_{1}(0)\)	\(\varepsilon_0\)	\(\varphi^{0}_{0}(\cdots \varphi^{0}_{0}(0)\cdots)\)
\(\varphi^{0}_{0}(\varphi^{0}_{1}(0)+1)\)	\(\omega^{\varepsilon_0+1}\)	\(\varphi^{0}_{1}(0)+\varphi^{0}_{1}(0)+\cdots\)
\(\varphi^{0}_{1}(1)\)	\(\varepsilon_1\)	\(\varphi^{0}_{0}(\cdots \varphi^{0}_{0}(\varphi^{0}_{0}(0)+1)\cdots)\)
\(\varphi^{0}_{1}(1+1)\)	\(\varepsilon_2\)	\(\varphi^{0}_{0}(\cdots \varphi^{0}_{0}(\varphi^{0}_{0}(1)+1)\cdots)\)
\(\varphi^{0}_{1}(\omega)\)	\(\varepsilon_{\omega}\)	\(\varphi^{0}_{1}(1+1+\cdots)\)
\(\varphi^{0}_{1}(\varphi^{0}_{0}(\omega))\)	\(\varepsilon_{\omega^{\omega}}\)	\(\varphi^{0}_{1}(\varphi^{0}_{0}(1+1+\cdots))\)
\(\varphi^{0}_{1}(\varphi^{0}_{1}(0))\)	\(\varepsilon_{\varepsilon_0}\)	\(\varphi^{0}_{1}(\varphi^{0}_{0}(\cdots \varphi^{0}_{0}(0)\cdots))\)
\(\varphi^{0}_{2}(0)\)	\(\zeta_0\)	\(\varphi^{0}_{1}(\cdots \varphi^{0}_{1}(0)\cdots)\)
\(\varphi^{0}_{\omega}(0)\)	\(\varphi_{\omega}(0)\)	\(\varphi^{0}_{1+1+\cdots}(0)\)
\(\Gamma^{0}(0,1+1)\)	\(\varphi_{\varepsilon_0}(0)\)	\(\varphi^{0}_{\varphi^{0}_{0}(\cdots \varphi^{0}_{0}(0)\cdots)}(0)\)
\(\Gamma^{0}(\omega,1+1)\)	\(\varphi_{\varphi_{\omega}(0)}(0)\)	\(\Gamma^{0}(1+1+\cdots,1+1)\)
\(\psi^{\Omega_1}(0)\)	\(\Gamma_0\)	\(\Gamma^{0}(0,1+1+\cdots)\)

As you see above, \(\varphi^{0}_{a}(b)\) works in the same way as \(\varphi_{a}(b)\), and \(\psi^{\Omega_1}\) starts enumerating fixed points of \(a \mapsto \varphi_{a}(0)\), i.e. Gamma numbers.

Up to φ(1,0,0,0)[]

term	ordinal	expansion
\(\psi^{\Omega_1}(1)\)	\(\Gamma_1\)	\(\Gamma^{0}(\varphi^{0}_{0}(\psi^{\Omega_1}(0)+1),1+1+\cdots)\)
\(\psi^{\Omega_1}(2)\)	\(\Gamma_2\)	\(\Gamma^{0}(\varphi^{0}_{0}(\psi^{\Omega_1}(1)+1),1+1+\cdots)\)
\(\psi^{\Omega_1}(\omega)\)	\(\Gamma_{\omega}\)	\(\psi^{\Omega_1}(1+1+\cdots)\)
\(\psi^{\Omega_1}(\psi^{\Omega_1}(0))\)	\(\Gamma_{\Gamma_0}\)	\(\psi^{\Omega_1}(\Gamma^{0}(0,1+1+\cdots))\)
\(\psi^{\Omega_1}(\Omega_1)\)	\(\varphi(1,1,0)\)	\(\psi^{\Omega_1}(\cdots \psi^{\Omega_1}(0)\cdots)\)
\(\psi^{\Omega_1}(\Omega_1+1)\)	\(\Gamma_{\varphi(1,1,0)+1}\)	\(\Gamma^{0}(\psi^{\Omega_1}(\Omega_1)+1,1+1+\cdots)\)
\(\psi^{\Omega_1}(\Omega_1+\omega)\)	\(\Gamma_{\varphi(1,1,0)+\omega}\)	\(\psi^{\Omega_1}(\Omega_1+1+1+\cdots)\)
\(\psi^{\Omega_1}(\Omega_1+\psi^{\Omega_1}(\Omega_1))\)	\(\Gamma_{\varphi(1,1,0) \times 2}\)	\(\psi^{\Omega_1}(\Omega_1+\psi^{\Omega_1}(\cdots \psi^{\Omega_1}(0)\cdots))\)
\(\psi^{\Omega_1}(\Omega_1+\psi^{\Omega_1}(\Omega_1+\psi^{\Omega_1}(\Omega_1)))\)	\(\Gamma_{\Gamma_{\varphi(1,1,0) \times 2}}\)	\(\psi^{\Omega_1}(\Omega_1+\psi^{\Omega_1}(\Omega_1+\psi^{\Omega_1}(\cdots \psi^{\Omega_1}(0)\cdots)))\)
\(\psi^{\Omega_1}(\Omega_1+\Omega_1)\)	\(\varphi(1,1,1)\)	\(\psi^{\Omega_1}(\Omega_1+\cdots \psi^{\Omega_1}(\Omega_1+\psi^{\Omega_1}(0))\cdots)\)
\(\psi^{\Omega_1}(\Omega_1+\Omega_1+\Omega_1)\)	\(\varphi(1,1,2)\)	\(\psi^{\Omega_1}(\Omega_1+\Omega_1+\cdots \psi^{\Omega_1}(\Omega_1+\Omega_1+\psi^{\Omega_1}(0))\cdots)\)
\(\psi^{\Omega_1}(\varphi^{0}_{0}(\Omega_1+1))\)	\(\varphi(1,1,\omega)\)	\(\psi^{\Omega_1}(\Omega_1+\Omega_1+\cdots)\)
\(\psi^{\Omega_1}(\varphi^{0}_{0}(\Omega_1+\omega))\)	\(\varphi(1,1,\omega^{\omega})\)	\(\psi^{\Omega_1}(\varphi^{0}_{0}(\Omega_1+1+1+\cdots))\)
\(\psi^{\Omega_1}(\varphi^{0}_{0}(\Omega_1+\psi^{\Omega_1}(0)))\)	\(\varphi(1,1,\Gamma_0)\)	\(\psi^{\Omega_1}(\varphi^{0}_{0}(\Omega_1+\Gamma^{0}(0,1+1+\cdots)))\)
\(\psi^{\Omega_1}(\varphi^{0}_{0}(\Omega_1+\psi^{\Omega_1}(\Omega_1)))\)	\(\varphi(1,1,\varphi(1,1,0))\)	\(\psi^{\Omega_1}(\varphi^{0}_{0}(\Omega_1+\psi^{\Omega_1}(\cdots \psi^{\Omega_1}(0)\cdots)))\)
\(\psi^{\Omega_1}(\varphi^{0}_{0}(\Omega_1+\psi^{\Omega_1}(\Omega_1+\Omega_1)))\)	\(\varphi(1,1,\varphi(1,1,1))\)	\(\psi^{\Omega_1}(\varphi^{0}_{0}(\Omega_1+\psi^{\Omega_1}(\Omega_1+\cdots \psi^{\Omega_1}(\Omega_1+\psi^{\Omega_1}(0))\cdots)))\)
\(\psi^{\Omega_1}(\varphi^{0}_{0}(\Omega_1+\psi^{\Omega_1}(\varphi^{0}_{0}(\Omega_1+1))))\)	\(\varphi(1,1,\varphi(1,1,\omega))\)	\(\psi^{\Omega_1}(\varphi^{0}_{0}(\Omega_1+\psi^{\Omega_1}(\Omega_1+\Omega_1+\cdots)))\)
\(\psi^{\Omega_1}(\varphi^{0}_{0}(\Omega_1+\Omega_1))\)	\(\varphi(1,2,0)\)	\(\psi^{\Omega_1}(\varphi^{0}_{0}(\Omega_1+ \cdots \psi^{\Omega_1}(\varphi^{0}_{0}(\Omega_1+\psi^{\Omega_1}(0)))\cdots))\)
\(\psi^{\Omega_1}(\varphi^{0}_{0}(\varphi^{0}_{0}(\Omega_1+1)))\)	\(\varphi(1,\omega,0)\)	\(\psi^{\Omega_1}(\varphi^{0}_{0}(\Omega_1+\Omega_1+\cdots))\)
\(\psi^{\Omega_1}(\varphi^{0}_{0}(\varphi^{0}_{0}(\Omega_1+\omega)))\)	\(\varphi(1,\omega^{\omega},0)\)	\(\psi^{\Omega_1}(\varphi^{0}_{0}(\varphi^{0}_{0}(\Omega_1+1+1+\cdots)))\)
\(\psi^{\Omega_1}(\varphi^{0}_{0}(\varphi^{0}_{0}(\Omega_1+\psi^{\Omega_1}(0))))\)	\(\varphi(1,\Gamma_0,0)\)	\(\psi^{\Omega_1}(\varphi^{0}_{0}(\varphi^{0}_{0}(\Omega_1+\Gamma^{0}(0,1+1+\cdots))))\)
\(\psi^{\Omega_1}(\varphi^{0}_{0}(\varphi^{0}_{0}(\Omega_1+\psi^{\Omega_1}(\Omega_1))))\)	\(\varphi(1,\varphi(1,1,0),0)\)	\(\psi^{\Omega_1}(\varphi^{0}_{0}(\varphi^{0}_{0}(\Omega_1+\psi^{\Omega_1}(\cdots \psi^{\Omega_1}(0) \cdots))))\)
\(\psi^{\Omega_1}(\varphi^{0}_{0}(\varphi^{0}_{0}(\Omega_1+\Omega_1)))\)	\(\varphi(2,0,0)\)	\(\psi^{\Omega_1}(\varphi^{0}_{0}(\varphi^{0}_{0}(\Omega_1+\cdots \psi^{\Omega_1}(\varphi^{0}_{0}(\varphi^{0}_{0}(\Omega_1+\psi^{\Omega_1}(0))))\cdots)))\)
\(\psi^{\Omega_1}(\varphi^{0}_{0}(\varphi^{0}_{0}(\varphi^{0}_{0}(\Omega_1+1))))\)	\(\varphi(\omega,0,0)\)	\(\psi^{\Omega_1}(\varphi^{0}_{0}(\varphi^{0}_{0}(\Omega_1+\Omega_1+\cdots)))\)
\(\psi^{\Omega_1}(\varphi^{0}_{0}(\varphi^{0}_{0}(\varphi^{0}_{0}(\Omega_1+\omega))))\)	\(\varphi(\omega^{\omega},0,0)\)	\(\psi^{\Omega_1}(\varphi^{0}_{0}(\varphi^{0}_{0}(\varphi^{0}_{0})(\Omega_1+1+1+\cdots))))\)
\(\psi^{\Omega_1}(\varphi^{0}_{0}(\varphi^{0}_{0}(\varphi^{0}_{0}(\Omega_1+\psi^{\Omega_1}(0)))))\)	\(\varphi(\Gamma_0,0,0)\)	\(\psi^{\Omega_1}(\varphi^{0}_{0}(\varphi^{0}_{0}(\varphi^{0}_{0})(\Omega_1+\Gamma^{0}(0,1+1+\cdots)))))\)
\(\psi^{\Omega_1}(\varphi^{0}_{0}(\varphi^{0}_{0}(\varphi^{0}_{0}(\Omega_1+\Omega_1))))\)	\(\varphi(1,0,0,0)\)	\(\psi^{\Omega_1}(\varphi^{0}_{0}(\varphi^{0}_{0}(\varphi^{0}_{0})(\Omega_1+\cdots \psi^{\Omega_1}(\varphi^{0}_{0}(\varphi^{0}_{0}(\varphi^{0}_{0})(\Omega_1+\psi^{\Omega_1}(0)))))\cdots))))\)

As you see above, the correspondence between ordinals expressed by multivariable Veblen function and terms in my notation is a little complicated. Since the correspondence between ordinals expressed by Rathjen's \(\psi\) and terms in my notation is much simpler, I stop dealing with multivariable Veblen function here.

Up to ψ_{Ω_1}(Φ_1(0))[]

term	ordinal	expansion
\(\psi^{\Omega_1}(\varphi^{0}_{1}(\Omega_1+1))\)	\(\psi_{\Omega_1}(\varepsilon_{\Omega_1+1})\)	\(\psi^{\Omega_1}(\varphi^{0}_{0}(\cdots \varphi^{0}_{0}(\Omega_1+1)\cdots))\)
\(\psi^{\Omega_1}(\varphi^{0}_{1}(\Omega_1+1)+1)\)	\(\psi_{\Omega_1}(\varepsilon_{\Omega_1+1}+1)\)	\(\Gamma^{0}(\psi^{\Omega_1}(\varphi^{0}_{1}(\Omega_1+1)),1+1+\cdots)\)
\(\psi^{\Omega_1}(\varphi^{0}_{0}(\varphi^{0}_{1}(\Omega_1+1)+1))\)	\(\psi_{\Omega_1}(\omega^{\varepsilon_{\Omega_1+1}+1})\)	\(\psi^{\Omega_1}(\varphi^{0}_{1}(\Omega_1+1)+\varphi^{0}_{1}(\Omega_1+1)+\cdots)\)
\(\psi^{\Omega_1}(\varphi^{0}_{0}(\varphi^{0}_{0}(\varphi^{0}_{1}(\Omega_1+1)+1)))\)	\(\psi_{\Omega_1}(\omega^{\omega^{\varepsilon_{\Omega_1+1}+1}})\)	\(\psi^{\Omega_1}(\varphi^{0}_{0}(\varphi^{0}_{1}(\Omega_1+1)+\varphi^{0}_{1}(\Omega_1+1)+\cdots))\)
\(\psi^{\Omega_1}(\varphi^{0}_{1}(\Omega_1+1+1))\)	\(\psi_{\Omega_1}(\varepsilon_{\Omega_1+2})\)	\(\psi^{\Omega_1}(\varphi^{0}_{0}(\cdots \varphi^{0}_{0}(\varphi^{0}_{1}(\Omega_1+1)+1)\cdots))\)
\(\psi^{\Omega_1}(\varphi^{0}_{1}(\Omega_1+\omega))\)	\(\psi_{\Omega_1}(\varepsilon_{\Omega_1+\omega})\)	\(\psi^{\Omega_1}(\varphi^{0}_{1}(\Omega_1+1+1+\cdots))\)
\(\psi^{\Omega_1}(\varphi^{0}_{1}(\Omega_1+\psi^{\Omega_1}(0)))\)	\(\psi_{\Omega_1}(\varepsilon_{\Omega_1+\Gamma_0})\)	\(\psi^{\Omega_1}(\varphi^{0}_{1}(\Omega_1+\Gamma^{0}(0,1+1+\cdots)))\)
\(\psi^{\Omega_1}(\varphi^{0}_{1}(\Omega_1+\psi^{\Omega_1}(\Omega_1)))\)	\(\psi_{\Omega_1}(\varepsilon_{\Omega_1+\psi_{\Omega_1}(\Omega_1)})\)	\(\psi^{\Omega_1}(\varphi^{0}_{1}(\Omega_1+\psi^{\Omega_1}(\cdots \psi^{\Omega_1}(0)\cdots)))\)
\(\psi^{\Omega_1}(\varphi^{0}_{1}(\Omega_1+\psi^{\Omega_1}(\varphi^{0}_{0}(\Omega_1+1))))\)	\(\psi_{\Omega_1}(\varepsilon_{\Omega_1+\psi_{\Omega_1}(\omega^{\Omega_1+1})})\)	\(\psi^{\Omega_1}(\varphi^{0}_{1}(\Omega_1+\psi^{\Omega_1}(\Omega_1+\Omega_1+\cdots)))\)
\(\psi^{\Omega_1}(\varphi^{0}_{1}(\Omega_1+\psi^{\Omega_1}(\varphi^{0}_{1}(\Omega_1+1))))\)	\(\psi_{\Omega_1}(\varepsilon_{\Omega_1+\psi_{\Omega_1}(\varepsilon_{\Omega_1+1})})\)	\(\psi^{\Omega_1}(\varphi^{0}_{1}(\Omega_1+\psi^{\Omega_1}(\varphi^{0}_{0}(\cdots \varphi^{0}_{0}(\Omega_1+1)\cdots))))\)
\(\psi^{\Omega_1}(\varphi^{0}_{1}(\Omega_1+\psi^{\Omega_1}(\varphi^{0}_{1}(\Omega_1+\psi^{\Omega_1}(0)))))\)	\(\psi_{\Omega_1}(\varepsilon_{\Omega_1+\psi_{\Omega_1}(\varepsilon_{\Omega_1+\Gamma_0})})\)	\(\psi^{\Omega_1}(\varphi^{0}_{1}(\Omega_1+\psi^{\Omega_1}(\varphi^{0}_{1}(\Omega_1+\Gamma^{0}(0,1+1+\cdots)))))\)
\(\psi^{\Omega_1}(\varphi^{0}_{1}(\Omega_1+\Omega_1))\)	\(\psi_{\Omega_1}(\varepsilon_{\Omega_1+\Omega_1})\)	\(\psi^{\Omega_1}(\varphi^{0}_{1}(\Omega_1+\cdots \psi^{\Omega_1}(\varphi^{0}_{1}(\Omega_1+\psi^{\Omega_1}(0)))\cdots))\)
\(\psi^{\Omega_1}(\varphi^{0}_{1}(\varphi^{0}_{0}(\Omega_1+1)))\)	\(\psi_{\Omega_1}(\varepsilon_{\omega^{\Omega_1+1}})\)	\(\psi^{\Omega_1}(\varphi^{0}_{1}(\Omega_1+\Omega_1+\cdots))\)
\(\psi^{\Omega_1}(\varphi^{0}_{1}(\varphi^{0}_{0}(\varphi^{0}_{0}(\Omega_1+1))))\)	\(\psi_{\Omega_1}(\varepsilon_{\omega^{\omega^{\Omega_1+1}}})\)	\(\psi^{\Omega_1}(\varphi^{0}_{1}(\varphi^{0}_{0}(\Omega_1+\Omega_1+\cdots)))\)
\(\psi^{\Omega_1}(\varphi^{0}_{1}(\varphi^{0}_{1}(\Omega_1+1)))\)	\(\psi_{\Omega_1}(\varepsilon_{\varepsilon_{\Omega_1+1}})\)	\(\psi^{\Omega_1}(\varphi^{0}_{1}(\varphi^{0}_{0}(\cdots \varphi^{0}_{0}(\Omega_1+1)\cdots)))\)
\(\psi^{\Omega_1}(\varphi^{0}_{2}(\Omega_1+1))\)	\(\psi_{\Omega_1}(\zeta_{\Omega_1+1})\)	\(\psi^{\Omega_1}(\varphi^{0}_{1}(\cdots \varphi^{0}_{1}(\Omega_1+1)\cdots))\)
\(\psi^{\Omega_1}(\varphi^{0}_{\omega}(\Omega_1+1))\)	\(\psi_{\Omega_1}(\varphi_{\omega}(\Omega_1+1))\)	\(\psi^{\Omega_1}(\varphi^{0}_{1+1+\cdots}(\Omega_1+1))\)
\(\psi^{\Omega_1}(\varphi^{0}_{\psi^{\Omega_1}(0)}(\Omega_1+1))\)	\(\psi_{\Omega_1}(\varphi_{\Gamma_0}(\Omega_1+1))\)	\(\psi^{\Omega_1}(\varphi^{0}_{\Gamma^{0}(0,1+1+\cdots)}(\Omega_1+1))\)
\(\psi^{\Omega_1}(\varphi^{0}_{\Omega_1}(1))\)	\(\psi_{\Omega_1}(\varphi_{\Omega_1}(1))\)	\(\psi^{\Omega_1}(\varphi^{0}_{\cdot_{\cdot_{\cdot_{\psi^{\Omega_1}(\varphi^{0}_{\psi^{\Omega_1}(0)}(\Omega_1+1))}\cdot}\cdot}\cdot}(\Omega_1+1))\)
\(\psi^{\Omega_1}(\varphi^{0}_{\Omega_1}(\omega))\)	\(\psi_{\Omega_1}(\varphi_{\Omega_1}(\omega))\)	\(\psi^{\Omega_1}(\varphi^{0}_{\Omega_1}(1+1+\cdots))\)
\(\psi^{\Omega_1}(\varphi^{0}_{\Omega_1}(\Omega))\)	\(\psi_{\Omega_1}(\varphi_{\Omega_1}(\Omega))\)	\(\psi^{\Omega_1}(\varphi^{0}_{\Omega_1}(\cdots \psi^{\Omega_1}(\varphi^{0}_{\Omega_1}(0))\cdots))\)
\(\psi^{\Omega_1}(\varphi^{0}_{\Omega_1+1}(0))\)	\(\psi_{\Omega_1}(\varphi_{\Omega_1+1}(0))\)	\(\psi^{\Omega_1}(\varphi^{0}_{\Omega_1}(\cdots \varphi^{0}_{\Omega_1}(\Omega_1)\cdots)))\)
\(\psi^{\Omega_1}(\varphi^{0}_{\varphi^{0}_{\Omega_1+1}(0)}(0))\)	\(\psi_{\Omega_1}(\varphi_{\varphi_{\Omega_1+1}(0)}(0))\)	\(\psi^{\Omega_1}(\varphi^{0}_{\varphi^{0}_{\Omega_1}(\cdots \varphi^{0}_{\Omega_1}(\Omega_1)\cdots))}(0))\)
\(\psi^{\Omega_1}(\psi^{\Omega_2}(0))\)	\(\psi_{\Omega_1}(\psi_{\Omega_2}(0))\)	\(\psi^{\Omega_1}(\varphi^{0}_{\cdot_{\cdot_{\cdot_{\varphi^{0}_{\Omega_1+1}(0)}\cdot}\cdot}\cdot}(0))\)
\(\psi^{\Omega_1}(\psi^{\Omega_2}(\psi^{\Omega_2}(0)))\)	\(\psi_{\Omega_1}(\psi_{\Omega_2}(\psi_{\Omega_2}(0)))\)	\(\psi^{\Omega_1}(\psi^{\Omega_2}(\varphi^{0}_{\cdot_{\cdot_{\cdot_{\varphi^{0}_{\Omega_1+1}(0)}\cdot}\cdot}\cdot}(0)))\)
\(\psi^{\Omega_1}(\Omega_2)\)	\(\psi_{\Omega_1}(\Omega_2)\)	\(\psi^{\Omega_1}(\psi^{\Omega_2}(\cdots \psi^{\Omega_2}(0)\cdots)\)
\(\psi^{\Omega_1}(\Omega_2+\Omega_2)\)	\(\psi_{\Omega_1}(\Omega_2+\Omega_2)\)	\(\psi^{\Omega_1}(\Omega_2+\psi^{\Omega_2}(\cdots \psi^{\Omega_2}(\Omega_2+\psi^{\Omega_2}(0))\cdots)\)
\(\psi^{\Omega_1}(\varphi^{0}_{0}(\Omega_2+1))\)	\(\psi_{\Omega_1}(\omega^{\Omega_2+1})\)	\(\psi^{\Omega_1}(\Omega_2+\Omega_2+\cdots)\)
\(\psi^{\Omega_1}(\varphi^{0}_{0}(\varphi^{0}{0}(\Omega_2+1)))\)	\(\psi_{\Omega_1}(\omega^{\omega^{\Omega_2+1}})\)	\(\psi^{\Omega_1}(\varphi^{0}_{0}(\Omega_2+\Omega_2+\cdots))\)
\(\psi^{\Omega_1}(\varphi^{0}_{1}(\Omega_2+1))\)	\(\psi_{\Omega_1}(\varepsilon_{\Omega_2+1})\)	\(\psi^{\Omega_1}(\varphi^{0}_{0}(\cdots \varphi^{0}_{0}(\Omega_2+1)\cdots))\)
\(\psi^{\Omega_1}(\varphi^{0}_{\omega}(\Omega_2+1))\)	\(\psi_{\Omega_1}(\varphi_{\omega}(\Omega_2+1))\)	\(\psi^{\Omega_1}(\varphi^{0}_{1+1+\cdots}(\Omega_2+1))\)
\(\psi^{\Omega_1}(\varphi^{0}_{\Omega_1}(\Omega_2+1))\)	\(\psi_{\Omega_1}(\varphi_{\Omega_1}(\Omega_2+1))\)	\(\psi^{\Omega_1}(\varphi^{0}_{\cdot_{\cdot_{\cdot_{\psi^{\Omega_1}(\varphi^{0}_{\psi^{\Omega_1}(0)}(\Omega_2+1))}\cdot}\cdot}\cdot}(\Omega_2+1))\)
\(\psi^{\Omega_1}(\varphi^{0}_{\psi^{\Omega_2}(0)}(\Omega_2+1))\)	\(\psi_{\Omega_1}(\varphi_{\Omega_1}(\Omega_2+1))\)	\(\psi^{\Omega_1}(\varphi^{0}_{\varphi^{0}_{\cdot_{\cdot_{\cdot_{\varphi^{0}_{\Omega_1+1}(0)}\cdot}\cdot}\cdot}(0)}(\Omega_2+1))\)
\(\psi^{\Omega_1}(\varphi^{0}_{\Omega_2}(1))\)	\(\psi_{\Omega_1}(\varphi_{\Omega_2}(1))\)	\(\psi^{\Omega_1}(\varphi^{0}_{\psi^{\Omega_2}(\varphi^{0}_{\cdot_{\cdot_{\cdot_{\psi^{\Omega_2}(\varphi^{0}_{\psi^{\Omega_2}(0)}(\Omega_2+1))}\cdot}\cdot}\cdot}(\Omega_2+1))}(\Omega_2+1))\)
\(\psi^{\Omega_1}(\varphi^{0}_{\Omega_2+1}(0))\)	\(\psi_{\Omega_1}(\varphi_{\Omega_2+1}(0))\)	\(\psi^{\Omega_1}(\varphi^{0}_{\Omega_2}(\cdots \varphi^{0}_{\Omega_2}(\Omega_2)\cdots))\)
\(\psi^{\Omega_1}(\varphi^{0}_{\varphi^{0}_{\Omega_2+1}(0)}(0))\)	\(\psi_{\Omega_1}(\varphi_{\varphi_{\Omega_2+1}(0)}(0)\)	\(\psi^{\Omega_1}(\varphi^{0}_{\varphi^{0}_{\Omega_2}(\cdots \varphi^{0}_{\Omega_2}(\Omega_2)\cdots)}(0))\)
\(\psi^{\Omega_1}(\psi^{\varphi^{1}_{0}(1+1)}(0))\)	\(\psi_{\Omega_1}(\psi_{\Omega_3}(0))\)	\(\psi^{\Omega_1}(\varphi^{0}_{\cdot_{\cdot_{\cdot_{\varphi^{0}_{\Omega_2+1}(0)}\cdot}\cdot}\cdot}(0))\)
\(\psi^{\Omega_1}(\Omega_{\omega})\)	\(\psi_{\Omega_1}(\Omega_{\omega})\)	\(\psi^{\Omega_1}(\varphi^{1}_{0}(1+1+\cdots))\)
\(\psi^{\Omega_1}(\Omega_{\omega+\omega})\)	\(\psi_{\Omega_1}(\Omega_{\omega+\omega})\)	\(\psi^{\Omega_1}(\varphi^{1}_{0}(\omega+1+1+\cdots))\)
\(\psi^{\Omega_1}(\Omega_{\psi^{\Omega_1}(0)})\)	\(\psi_{\Omega_1}(\Omega_{\Gamma_0})\)	\(\psi^{\Omega_1}(\varphi^{1}_{0}(\varphi^{0}_{\cdot_{\cdot_{\cdot_{\varphi^{0}_{0}(0)}\cdot}\cdot}\cdot}(0)))\)
\(\psi^{\Omega_1}(\Omega_{\Omega_1})\)	\(\psi_{\Omega_1}(\Omega_{\Omega_1})\)	\(\psi^{\Omega_1}(\varphi^{1}_{0}(\cdots \psi^{\Omega_1}(\varphi^{1}_{0}(\psi^{\Omega_1}(0))\cdots)\)
\(\psi^{\Omega_1}(\Omega_{\Omega_2})\)	\(\psi_{\Omega_1}(\Omega_{\Omega_2})\)	\(\psi^{\Omega_1}(\varphi^{1}_{0}(\psi^{\Omega_2}(\cdots \psi^{\Omega_2}(\varphi^{1}_{0}(\psi^{\Omega_2}(0))\cdots))\)
\(\psi^{\Omega_1}(\Omega_{\Omega_{\omega}})\)	\(\psi_{\Omega_1}(\Omega_{\Omega_{\omega}})\)	\(\psi^{\Omega_1}(\varphi^{1}_{0}(\varphi^{1}_{0}(1+1+\cdots)))\)
\(\psi^{\Omega_1}(\Omega_{\Omega_{\Omega_1}})\)	\(\psi_{\Omega_1}(\Omega_{\Omega_{\Omega_1}})\)	\(\psi^{\Omega_1}(\varphi^{1}_{0}(\varphi^{1}_{0}(\cdots \psi^{\Omega_1}(\varphi^{1}_{0}(\varphi^{1}_{0}(\psi^{\Omega_1}(0))))\cdots)))\)
\(\psi^{\Omega_1}(\varphi^{1}_{1}(0))\)	\(\psi_{\Omega_1}(\Phi_1(0))\)	\(\psi^{\Omega_1}(\varphi^{1}_{0}(\cdots \varphi^{1}_{0}(0)\cdots))\)

As you see above, \(\psi^{\Omega_1}\) in this realm works in the same way as Rathjen's \(\psi\).

Up to ψ_{Ω_1}(M^Γ)[]

term	ordinal	expansion
\(\psi^{\Omega_1}(\psi^{\varphi^{1}_{0}(\varphi^{1}_{1}(0)+1)}(0))\)	\(\psi_{\Omega_1}(\psi_{\Omega_{\Phi_1(0)+1}}(0))\)	\(\psi^{\Omega_1}(\varphi^{0}_{\cdot_{\cdot_{\cdot_{\varphi^{0}_{\Phi_1+1}(0)}\cdot}\cdot}\cdot}(0))\)
\(\psi^{\Omega_1}(\varphi^{1}_{0}(\varphi^{1}_{1}(0)+1))\)	\(\psi_{\Omega_1}(\Omega_{\Phi_1(0)+1})\)	\(\psi^{\Omega_1}(\psi^{\varphi^{1}_{0}(\varphi^{1}_{1}(0)+1)}(\cdots \psi^{\varphi^{1}_{0}(\varphi^{1}_{1}(0)+1)}(0)\cdots))\)
\(\psi^{\Omega_1}(\varphi^{1}_{0}(\varphi^{1}_{0}(\varphi^{1}_{1}(0)+1)))\)	\(\psi_{\Omega_1}(\Omega_{\Omega_{\Phi_1(0)+1}})\)	\(\psi^{\Omega_1}(\varphi^{1}_{0}(\psi^{\varphi^{1}_{0}(\varphi^{1}_{1}(0)+1)}(\cdots \varphi^{1}_{0}(\psi^{\varphi^{1}_{0}(\varphi^{1}_{1}(0)+1)}(0)\cdots)))\)
\(\psi^{\Omega_1}(\varphi^{1}_{1}(1))\)	\(\psi_{\Omega_1}(\Phi_1(1))\)	\(\psi^{\Omega_1}(\varphi^{1}_{0}(\cdots \varphi^{1}_{0}(\varphi^{1}_{1}(0)+1)\cdots))\)
\(\psi^{\Omega_1}(\varphi^{1}_{1}(1+1))\)	\(\psi_{\Omega_1}(\Phi_1(2))\)	\(\psi^{\Omega_1}(\varphi^{1}_{0}(\cdots \varphi^{1}_{0}(\varphi^{1}_{1}(1)+1)\cdots))\)
\(\psi^{\Omega_1}(\varphi^{1}_{1}(\omega))\)	\(\psi_{\Omega_1}(\Phi_1(\omega))\)	\(\psi^{\Omega_1}(\varphi^{1}_{1}(1+1+\cdots))\)
\(\psi^{\Omega_1}(\varphi^{1}_{1}(\Omega_1))\)	\(\psi_{\Omega_1}(\Phi_1(\Omega_1))\)	\(\psi^{\Omega_1}(\varphi^{1}_{1}(\cdots \psi^{\Omega_1}(\varphi^{1}_{1}(\psi^{\Omega_1}(0)))\cdots))\)
\(\psi^{\Omega_1}(\varphi^{1}_{1}(\varphi^{1}_{0}(\Omega_1)))\)	\(\psi_{\Omega_1}(\Phi_1(\Omega_{\Omega_1}))\)	\(\psi^{\Omega_1}(\varphi^{1}_{1}(\varphi^{1}_{0}(\cdots \psi^{\Omega_1}(\varphi^{1}_{1}(\varphi^{1}_{0}(\psi^{\Omega_1}(0))))\cdots)))\)
\(\psi^{\Omega_1}(\varphi^{1}_{1}(\varphi^{1}_{1}(0)))\)	\(\psi_{\Omega_1}(\Phi_1(\Phi_1(0)))\)	\(\psi^{\Omega_1}(\varphi^{1}_{1}(\varphi^{1}_{0}(\cdots \varphi^{1}_{0}(0)\cdots)))\)
\(\psi^{\Omega_1}(\varphi^{1}_{1+1}(0))\)	\(\psi_{\Omega_1}(\Phi_2(0))\)	\(\psi^{\Omega_1}(\varphi^{1}_{1}(\cdots \varphi^{1}_{1}(0)\cdots))\)
\(\psi^{\Omega_1}(\varphi^{1}_{\omega}(0))\)	\(\psi_{\Omega_1}(\Phi_{\omega}(0))\)	\(\psi^{\Omega_1}(\varphi^{1}_{1+1+\cdots}(0))\)
\(\psi^{\Omega_1}(\varphi^{1}_{\Omega_1}(0))\)	\(\psi_{\Omega_1}(\Phi_{\Omega_1}(0))\)	\(\psi^{\Omega_1}(\varphi^{1}_{\cdot_{\cdot_{\cdot_{\psi^{\Omega_1}(\varphi^{1}_{\psi^{\Omega_1}(0)}(0))}\cdots}\cdots}\cdots}(0))\)
\(\psi^{\Omega_1}(\varphi^{1}_{\varphi^{1}_{0}(\omega)}(0))\)	\(\psi_{\Omega_1}(\Phi_{\Omega_{\omega}}(0))\)	\(\psi^{\Omega_1}(\varphi^{1}_{\varphi^{1}_{0}(1+1+\cdots)}(0))\)
\(\psi^{\Omega_1}(\varphi^{1}_{\varphi^{1}_{1}(0)}(0))\)	\(\psi_{\Omega_1}(\Phi_{\Phi_1(0)}(0))\)	\(\psi^{\Omega_1}(\varphi^{1}_{\varphi^{1}_{0}(\cdots \varphi^{1}_{0}(0)\cdots)}(0))\)
\(\psi^{\Omega_1}(\psi^{I_0}(0))\)	\(\psi_{\Omega_1}(\psi_{\chi_1(0)}(0))\)	\(\psi^{\Omega_1}(\varphi^{1}_{\cdot_{\cdot_{\cdot_{\varphi^{1}_{0}(0)}\cdot}\cdot}\cdot}(0))\)
\(\psi^{\Omega_1}(\varphi^{1}_{\psi^{I_0}(0)}(1))\)	\(\psi_{\Omega_1}(\Phi_{\psi_{\chi_1(0)}(0)}(1))\)	\(\psi^{\Omega_1}(\varphi^{1}_{\varphi^{1}_{\cdot_{\cdot_{\cdot_{\varphi^{1}_{0}(0)}\cdot}\cdot}\cdot}(0)}(\psi^{I_0}(0)+1))\)
\(\psi^{\Omega_1}(\varphi^{1}_{\psi^{I_0}(0)+1}(0))\)	\(\psi_{\Omega_1}(\Phi_{\psi_{\chi_1(0)}(0)+1}(0))\)	\(\psi^{\Omega_1}(\varphi^{1}_{\psi^{I_0}(0)}(\cdots \varphi^{1}_{\psi^{I_0}(0)}(\psi^{I_0}(0))\cdots))\)
\(\psi^{\Omega_1}(\varphi^{1}_{\varphi^{1}_{\psi^{I_0}(0)+1}(0)}(0))\)	\(\psi_{\Omega_1}(\Phi_{\Phi_{\psi_{\chi_1(0)}(0)+1}(0)}(0))\)	\(\psi^{\Omega_1}(\varphi^{1}_{\varphi^{1}_{\psi^{I_0}(0)}(\cdots \varphi^{1}_{\psi^{I_0}(0)}(\psi^{I_0}(0))\cdots)}(0))\)
\(\psi^{\Omega_1}(\psi^{I_0}(1))\)	\(\psi_{\Omega_1}(\psi_{\chi_1(0)}(1))\)	\(\psi^{\Omega_1}(\varphi^{1}_{\cdot_{\cdot_{\cdot_{\varphi^{1}_{\psi^{I_0}(0)+1}(0)}\cdot}\cdot}\cdot}(0))\)
\(\psi^{\Omega_1}(\psi^{I_0}(\omega))\)	\(\psi_{\Omega_1}(\psi_{\chi_1(0)}(\omega))\)	\(\psi^{\Omega_1}(\psi^{I_0}(1+1+\cdots))\)
\(\psi^{\Omega_1}(\psi^{I_0}(\Omega_1))\)	\(\psi_{\Omega_1}(\psi_{\chi_1(0)}(\Omega_1))\)	\(\psi^{\Omega_1}(\psi^{I_0}(\cdots \psi^{\Omega_1}(\psi^{I_0}(0))\cdots))\)
\(\psi^{\Omega_1}(\psi^{I_0}(\psi^{I_0}(0)))\)	\(\psi_{\Omega_1}(\psi_{\chi_1(0)}(\psi_{\chi_1(0)}(0)))\)	\(\psi^{\Omega_1}(\psi^{I_0}(\varphi^{1}_{\cdot_{\cdot_{\cdot_{\varphi^{1}_{0}(0)}\cdot}\cdot}\cdot}(0)))\)
\(\psi^{\Omega_1}(I_0)\)	\(\psi_{\Omega_1}(\chi_1(0))\)	\(\psi^{\Omega_1}(\psi^{I_0}(\cdots \psi^{I_0}(0)\cdots))\)
WIP	WIP	WIP

Since I am not good at analysing, the table above might be wrong.

Up to the limit[]

term	expansion
\(\psi^{\Omega_1}(\varphi^{1+1}_{0}(0))\)
\(\psi^{\Omega_1}(\varphi^{\omega}_{0}(0))\)
\(\psi^{\Omega_1}(\varphi^{\Omega_1}_{0}(0))\)
\(\psi^{\Omega_1}(\varphi^{\varphi^{\omega}_{0}(0)}_{0}(0))\)
\(\psi^{\Omega_1}(\varphi^{\varphi^{\Omega_1}_{0}(0)}_{0}(0))\)
\(\psi^{\Omega_1}(\varphi^{\varphi^{\varphi^{\omega}_{0}(0)}_{0}(0)}_{0}(0))\)

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

Computation Programme[]

Naruyoko, p進大好きbot's Rathjen-type Ordinal Notation Implementation, Github page.

References[]

↑ ^1.0 ^1.1 ^1.2 M. Rathjen, Ordinal notations based on a weakly Mahlo cardinal, Archive for Mathematical Logic, Volume 29, Issue 4, pp. 249--263, 1990.

[Rathjen-1] 1.0 ^1.1 ^1.2 M. Rathjen, Ordinal notations based on a weakly Mahlo cardinal, Archive for Mathematical Logic, Volume 29, Issue 4, pp. 249--263, 1990.

[1]

Rathjen-type Ordinal Notation