## 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".