Multiple weakly inaccessibles[]
The obvious next step is to add another weakly inaccessible cardinal, \(I_2\). We can let \(\psi_{I_2} (\alpha)\) be the \(\alpha\)th fixed point of \(\beta \mapsto \Omega_{I + \beta}\), then \(\psi_{I_2} (I_2 \alpha + \beta)\) is the \(\beta\)th fixed point of \(\gamma \mapsto \psi_{I_2} (\gamma)\), and so on.
Next, we can add \(I_\alpha\) for any \(\alpha\) in our notation:
\(C_0 (\alpha, \beta) = \beta \cup \lbrace 0 \rbrace\)
\(C_{n+1} (\alpha, \beta) = \lbrace \gamma + \delta, \varphi(\gamma, \delta), \Omega_{\gamma}, I_{\gamma}, \psi_\pi(\eta) | \gamma, \delta, \eta, \pi \in C_n (\alpha, \beta); \eta < \alpha; \pi \text{ is a regular cardinal} \rbrace \)
\(C ( \alpha, \beta) = \bigcup_{n = 1}^{\infty} C_n ( \alpha, \beta) \)
\(\psi_\pi (\alpha) = \min (\lbrace \beta < \pi | C( \alpha, \beta) \cap \pi \subseteq \beta \wedge \pi \in C( \alpha, \beta) \rbrace \cup \lbrace \pi \rbrace) \)
\(\alpha\)-weakly inaccessibles[]
We can go farther by defining \(\alpha\)-weakly inaccessibles. A cardinal is 0-weakly inaccessible if it is weakly inaccessible. A cardinal is 1-weakly inaccessible if it weakly inaccessible and a limit of weakly inaccessibles. More generally, a cardinal is \(\alpha\)-weakly inaccessible if it is weakly inaccessible and a limit of \(\beta\)-weakly inaccessibles for all \(\beta < \alpha\).
So, we can define \(I(\alpha, \beta)\) to be the \(\beta\)th \(\alpha\)-weakly inaccessible. We then have \(\psi_{I(1,0)}(\alpha)\) equal to the \(\alpha\)th fixed point of \(\beta \mapsto I(\beta)\), \(\psi_{I(1,0)}(I(1, 0) \alpha + \beta)\) is the \(\beta\)th fixed point of \(\gamma \mapsto \psi_{I(1, 0)} (\gamma)\), and so on. We have:
\(C_0 (\alpha, \beta) = \beta \cup \lbrace 0 \rbrace\)
\(C_{n+1} (\alpha, \beta) = \lbrace \gamma + \delta, \varphi(\gamma, \delta), \Omega_{\gamma}, I(\gamma, \delta), \psi_\pi(\eta) | \gamma, \delta, \eta, \pi \in C_n (\alpha, \beta); \eta < \alpha; \pi \text{ is a regular cardinal} \rbrace \)
\(C ( \alpha, \beta) = \bigcup_{n = 1}^{\infty} C_n ( \alpha, \beta) \)
\(\psi_\pi (\alpha) = \min (\lbrace \beta < \pi | C( \alpha, \beta) \cap \pi \subseteq \beta \wedge \pi \in C( \alpha, \beta) \rbrace \cup \lbrace \pi \rbrace) \)
hyper-weakly inaccessibles and beyond[]
Next, we define a hyper-weakly inaccessible, or (1,0)-weakly inaccessible, to be a cardinal \(\alpha\) that is \(\alpha\)-weakly inaccessible. More generally, a cardinal \(\gamma\) is \((\alpha, 0)\)-weakly inaccessible if it is weakly inaccessible and \((\delta, \gamma)\)-weakly inaccessible for all \(\delta < \alpha\), and it is \((\alpha, \beta)\)-weakly inaccessible if it is weakly inaccessible and a limit of \((\alpha, \delta)\)-weakly inaccessibles for all \(\delta < \beta\). We can then define \(I(\alpha, \beta, \gamma)\) as the \(\gamma\)th cardinal that is \((\alpha, \beta)\)-weakly inaccessible.
We can clearly continue this to arbitrarily many variables, and even variables of transfinite index; For this, we can use a notation similar to the Schutte Klammersymbolen. For example, a cardinal \(\alpha\) is \((1 @ \omega)\)-weakly inaccessible if it is \((\alpha @ n)\)-weakly inaccessible for all \(n < \omega\). We then have:
\(C_0 (\alpha, \beta) = \beta \cup \lbrace 0 \rbrace\)
\(C_{n+1} (\alpha, \beta) = \lbrace \gamma + \delta, \varphi(\gamma, \delta), \Omega_{\gamma}, I(\gamma_1 @ \delta_1, \gamma_2 @ \delta_2, \ldots, \gamma_m @ \delta_m), \psi_\pi(\eta) | \gamma, \delta, \gamma_i, \delta_i, \eta, \pi \in C_n (\alpha, \beta); \eta < \alpha; \pi \text{ is a regular cardinal} \rbrace \)
\(C ( \alpha, \beta) = \bigcup_{n = 1}^{\infty} C_n ( \alpha, \beta) \)
\(\psi_\pi (\alpha) = \min (\lbrace \beta < \pi | C( \alpha, \beta) \cap \pi \subseteq \beta \wedge \pi \in C( \alpha, \beta) \rbrace \cup \lbrace \pi \rbrace) \)
Up to a weakly Mahlo[]
Of course, the next step after the Schutte Klammersymbolen is the Bachmann-Howard notation, and we can do something similar here. However, we need a very large ordinal to act as a diagonalization operator, like \(\Omega\) in the regular Bachmann-Howard notation. This is where the notion of weakly Mahlo cardinal comes in.
A set S of ordinals is closed if, for any ordinal \(\alpha\) such that \(\sup \lbrace \beta | \beta < \alpha \wedge \beta \in S \rbrace = \alpha\), \(\alpha\) is in S. In other words, if S contains a subset whose supremum is \(\alpha\), then S contains \(\alpha\). (In other words, S contains all limit points, which is the normal topological definiton of closed.)
A subset S of T is unbounded in T if, for any element \(\alpha\) in T, there exists an element \(\beta\) in S such that \(\beta \ge \alpha\). If S and T are sets of ordinals, we can also say S is unbounded in T if sup S = sup T.
A subset S of a set T of ordinals is stationary in T if every closed and unbounded subset of T contains an element of S.
An ordinal \(\alpha\) is weakly Mahlo if it is a limit ordinal and the set of regular cardinals less than \(\alpha\) is a stationary subset of \(\alpha\). Put another way, an ordinal \(\alpha\) is weakly Mahlo if it is a limit ordinal and every closed and unbounded subset of \(\alpha\) contains a regular cardinal.
It turns out that a weakly Mahlo cardinal \(\alpha\) is weakly inaccessible, hyper-weakly inaccessible, hyper-hyper-weakly inaccessible, \((1 @ \alpha)\)-weakly inaccessible, and so on as far as one cares to diagonalize. So a weakly Mahlo cardinal is perfectly suited for our needs.
We define:
\(C_0 (\alpha, \beta) = \beta \cup \lbrace 0, M \rbrace\)
\(C_{n+1} (\alpha, \beta) = \lbrace \gamma + \delta, \varphi(\gamma, \delta), \Omega_{\gamma}, \chi(\eta, \delta), \psi_\pi(\eta) | \gamma, \delta, \eta, \pi \in C_n (\alpha, \beta); \eta < \alpha; \pi \text{ is a regular cardinal} \rbrace \)
\(C ( \alpha, \beta) = \bigcup_{n = 1}^{\infty} C_n ( \alpha, \beta) \)
\(\chi (\alpha, \beta) = \) the \(\beta\)th ordinal in the set \(\lbrace \gamma | C(\alpha, \gamma) \cap M = \gamma, \gamma \text{ is a regular cardinal} \rbrace\)
\(\psi_\pi (\alpha) = \min (\lbrace \beta < \pi | C( \alpha, \beta) \cap \pi \subseteq \beta \wedge \pi \in C( \alpha, \beta) \rbrace \cup \lbrace \pi \rbrace) \)
We have
\(\chi(M^{\alpha_1} \beta_1 + M^{\alpha_2} \beta_2 + \ldots + M^{\alpha_n} \beta_n, \gamma) = I(\beta_1 @ 1 + \alpha_1, \beta_2 @ 1 + \alpha_2, \ldots, \beta_n @ 1 + \alpha_n, \gamma @ 0)\)
but of course we can continue to \(\chi(M^M), \chi(M^{M^M}), \chi(\epsilon_{M+1})\) and beyond.
\(\psi_{\Omega_1}(\epsilon_{M+1})\) is the proof theoretic ordinal of KP + "There exists a recursively Mahlo ordinal".