A couple of years ago I defined my own extension to the inaccessible ordinal notation here.  It goes pretty far, but we can clean up the presentation a bit.

## Multiple weakly Mahlo cardinals

We can of course add more weakly Mahlo cardinals, defining $$M(\alpha)$$ as the $$\alpha$$th weakly Mahlo cardinal.  We can then define $$M(1,\alpha)$$ as the $$\alpha$$th weakly Mahlo cardinal that is a fixed point of $$\beta \mapsto M(\beta)$$, and more generally, $$M(\alpha, \beta)$$ is the $$\beta$$th weakly Mahlo cardinal that is a fixed point of $$\delta \mapsto M(\gamma, \delta)$$ for all $$\gamma < \alpha$$.

We can continue adding variables, even into the transfinite, and then use a large cardinal to index a Bachmann-Howard style hierarchy.  The natural choice is the smallest 1-weakly Mahlo cardinal, which we can call $$\Xi$$.  A 1-weakly Mahlo cardinal is a weakly Mahlo cardinal such that the set of weakly Mahlo cardinals less than it form a stationary subset.

Our notation is now:

$$C_0 (\alpha, \beta) = \beta \cup \lbrace 0, \Xi \rbrace$$

$$C_{n+1} (\alpha, \beta) = \lbrace \gamma + \delta, \varphi(\gamma, \delta), \Omega_{\gamma}, \chi(\gamma, \delta), \Xi (\gamma, \delta) \psi^0_\pi(\eta), \psi^1_\pi(\eta)$$

$$| \gamma, \delta, \eta, \pi \in C_n (\alpha, \beta); \eta < \alpha; \pi \text{ is a regular cardinal} \rbrace$$

$$C ( \alpha, \beta) = \bigcup_{n = 1}^{\infty} C_n ( \alpha, \beta)$$

$$\chi (\alpha, \beta) =$$ the $$beta$$th ordinal in the set $$\lbrace \gamma | C(\alpha, \gamma) \cap \Xi = \gamma, \gamma \text{ is a regular cardinal} \rbrace$$

$$\Xi (\alpha, \beta) =$$ the $$beta$$th ordinal in the set $$\lbrace \gamma | C(\alpha, \gamma) \cap \Xi = \gamma, \gamma \text{ is a weakly Mahlo cardinal} \rbrace$$

$$\psi_\pi^0 (\alpha) = \min (\lbrace \beta < \pi | C( \alpha, \beta) \cap \pi \subseteq \beta \wedge \pi \in C( \alpha, \beta) \rbrace \cup \lbrace \pi \rbrace)$$

$$\psi_\pi^1 (\alpha) = \min (\lbrace \beta < \pi | C( \alpha, \beta) \cap \pi \subseteq \beta \wedge \pi \in C( \alpha, \beta) \wedge \beta \text { is a regular cardinal} \rbrace \cup \lbrace \pi \rbrace)$$

## $$\alpha$$-weakly Mahlo cardinals and higher

Similar to the above, we can define $$\Xi(\alpha)$$ to be the $$\alpha$$th 1-weakly Mahlo cardinal, then define $$\Xi(\alpha, \beta)$$ as the $$\beta$$th 1-weakly Mahlo cardinal that is a fixed point of $$\delta \mapsto \Xi(\gamma, \delta)$$ for all $$\gamma < \alpha$$.  Again, we can add more and more variables, and finally index by a large cardinal, which will obviously be the smallest 2-weakly Mahlo cardinal, which we denote by $$\Xi$$.

We continue this process for higher and higher $$\alpha$$-weakly Mahlo cardinals.  In general, an $$\alpha$$-weakly Mahlo cardinal is a weakly Mahlo cardinal such that the set of $$\beta$$-weakly Mahlo cardinals form a stationary subset for all $$\beta < \alpha$$.  And, of course, we can define a (1,0)-weakly Mahlo cardinal as a cardinal $$\alpha$$ that is $$\alpha$$-weakly Mahlo.  An $$(\alpha, 0)$$-weakly Mahlo cardinal is a weakly Mahlo cardinal $$\beta$$ that is $$(\gamma, \beta)$$-weakly Mahlo for all $$\gamma < \alpha$$, and an $$(\alpha, \beta)$$-weakly Mahlo cardinal is a weakly Mahlo cardinal such that the $$(\alpha,\gamma)$$-weakly Mahlo cardinals below it form a stationary subset for all $$\gamma < \beta$$.  Again we continue for arbritrarily many places, and define Bachmann-Howard style hierarchy that diagonalizes over hyper-weakly Mahlo cardinals.  For this we need a really big cardinal; we will use the smallest weakly compact cardinal, which we will denote by K.  For a cardinal $$\alpha$$, consider the complete graph whose vertices are the ordinals less than $$\alpha$$; $$\alpha$$ is weakly compact if it is uncountable and, no matter how we color the edges with two colors, we can choose a subset of the vertices of cardinality $$\alpha$$ such that all edges adjoining vertices in the set are of the same color.  The important point is that a weakly compact cardinal $$\alpha$$ is weakly Mahlo, hyper-weakly Mahlo, $$(1 @ \alpha)$$-weakly Mahlo, and so on for as far as we care to diagonalize.  So K will suit are purposes perfectly.

So, we finally reach the following notation:

$$C_0 (\alpha, \beta) = \beta \cup \lbrace 0, K \rbrace$$

$$C_{n+1} (\alpha, \beta) = \lbrace \gamma + \delta, \varphi(\gamma, \delta), \Omega_{\gamma}, \Xi(\eta, \gamma), \Psi_\pi(\epsilon, \eta)$$

$$| \gamma, \delta, \epsilon, \eta, \pi \in C_n (\alpha, \beta); \epsilon \le \eta < \alpha; \pi \text{ is a regular cardinal} \rbrace$$

$$C ( \alpha, \beta) = \bigcup_{n = 1}^{\infty} C_n ( \alpha, \beta)$$

$$M(0) = \lbrace \beta < K : C(0, \beta) \cap K = \beta \rbrace$$

For $$\alpha > 0$$,
$$M(\alpha) = \lbrace \beta < K : C(\alpha, \beta) \cap K = \beta \wedge \beta \text { is regular } \wedge (\forall \gamma \in C(\alpha, \beta) \cap \alpha )$$

$$(M(\gamma) \text { is stationary in } \beta ) \rbrace$$

$$\Xi(\alpha, \beta) =$$ the $$\beta$$th ordinal in $$M(\alpha)$$

$$\Psi_\pi (\alpha, \beta) = \min (\lbrace \gamma : \gamma \in M(\alpha) \cap \pi \wedge C( \beta, \gamma) \cap \pi \subseteq \gamma \wedge \pi \in C( \beta, \gamma) \rbrace \cup \lbrace \pi \rbrace)$$

We have that

M(0) = the strongly critical ordinals (the ordinals of the form $$\Gamma_\alpha$$ for some $$\alpha$$)

M(1) = the regular cardinals.

M(2) = the weakly Mahlo cardinals.

M(3) = the 1-weakly Mahlo cardinals.

M(2 + $$\alpha$$) = the $$\alpha$$-weakly Mahlo cardinals.

M(K) = the (1,0)-weakly Mahlo cardinals.

M($$K \alpha + \beta$$) = the $$(\alpha, \beta)$$-weakly Mahlo cardinals.

M($$K^2 \alpha + K \beta + \gamma$$) = the $$(\alpha, \beta, \gamma)$$-weakly Mahlo cardinals.

$$\Psi_\pi (0, \beta)$$ is our regular collapsing function.

$$\Psi_\pi (1, \beta)$$ is restricted to regular cardinals, so for example

$$\Psi_{\Xi(2, 0)} (1, \beta)$$ is the $$\beta$$th weakly inaccessible cardinal.

$$\Psi_{\Xi(2, 0)} (1, \Xi(2, 0)\alpha + \beta)$$ is the $$\beta$$th $$\alpha$$-weakly inaccessible cardinal.

$$\Psi_{\Xi(2, 0)} (1, \Xi(2,0)^2 \alpha + \Xi(2, 0)\beta + \gamma)$$ is the $$\gamma$$th $$(\alpha, \beta)$$-weakly inaccessible cardinal.

$$\Psi_\pi (2, \beta)$$ is generates weakly Mahlo cardinals, so for example

$$\Psi_{\Xi(3, 0)} (2, \beta)$$ is $$M(1,\beta)$$ from the top of the article.

$$\Psi_{\Xi(3, 0)} (2, \Xi(3, 0)\alpha + \beta$$ is $$M(1+\alpha, \beta)$$ from the top of the article.

$$\Psi_{\Xi(3, 0)} (2, \Xi(3,0)^2 \alpha + \Xi(3, 0)\beta + \gamma)$$ is $$M(\alpha, \beta, \gamma)$$.

The proof theoretic ordinal of KP + $$\Pi$$-3 reflection is $$\Psi_{\Omega_1}(0, \epsilon_{K+1})$$.