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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 \(\Xi[1]\).  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:

\(C_0 (\alpha, \beta) = \beta \cup \lbrace 0, \Xi[1] \rbrace\)

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

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

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

\(\chi (\alpha, \beta) = \) the \(beta\)th ordinal in the set \(\lbrace \gamma | C(\alpha, \gamma) \cap \Xi[1] = \gamma, \gamma \text{ is a regular cardinal} \rbrace\)

\(\Xi (\alpha, \beta) = \) the \(beta\)th ordinal in the set \(\lbrace \gamma | C(\alpha, \gamma) \cap \Xi[1] = \gamma, \gamma \text{ is a weakly Mahlo cardinal} \rbrace\)

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

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

\(\alpha\)-weakly Mahlo cardinals and higher

Similar to the above, we can define \(\Xi[1](\alpha)\) to be the \(\alpha\)th 1-weakly Mahlo cardinal, then define \(\Xi[1](\alpha, \beta)\) as the \(\beta\)th 1-weakly Mahlo cardinal that is a fixed point of \(\delta \mapsto \Xi[1](\gamma, \delta)\) 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 \(\Xi[2]\).

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:

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

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

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

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

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

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 the \(\beta\)th weakly Mahlo cardinal.

\(\Psi_{\Xi(3, 0)} (2, \Xi(3, 0)\alpha + \beta\) is \(M(\alpha, \beta)\) 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 \(\Psi_{\Omega_1}(0, \epsilon_{K+1})\).

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