Rathjen's Psi function

Rathjen's \(\Psi\) function based on the least weakly compact cardinal is an ordinal collapsing function.Rathjen, Michael. "Proof Theory of Reflection", Annals of Pure and Applied Logic 68, 181--224 (1994). A weakly compact cardinal can be defined as a cardinal \(\mathcal{K}\) such that it is \(\Pi_1^1\)-indescribable. He uses this to diagonalise over the weakly Mahlo hierarchy.
== Definition ==
The functions \(M^{\alpha}\), \(C(\alpha,\pi)\), \(\Xi(\alpha)\), and \(\Psi^{\xi}_{\pi}(\alpha)\) are defined using simultaneous recursion in the following way:
\(M^0=\mathcal K\cap\text{Lim}\), where \(\textrm{Lim}\) denotes the class of limit ordinals.
For \(\alpha>0\), \(M^\alpha\) is the set of \(\pi<\mathcal K\) such that \(\pi\) satisfies these 3 conditions:
#\(C(\alpha,\pi)\cap\mathcal K = \pi\)
#\(\forall(\xi\in C(\alpha,\pi) \cap \alpha)(M^{\xi} \text{ is stationary in }\pi)\)
#\(\alpha\in C(\alpha,\pi)\)
\(C(\alpha,\beta)\) is the closure of \(\beta\cup\{0,\mathcal K\}\) under:
*addition,
*\((\xi,\eta)\mapsto\,\)\(\varphi\)\((\xi,\eta)\),
*\(\xi\mapsto\Omega_\xi\) given \(\xi<\mathcal K\),
*\(\xi\mapsto\Xi(\xi)\) given \(\xi<\alpha\),
*\((\xi,\pi,\delta)\mapsto\Psi^\xi_\pi(\delta)\) given \(\xi\le\delta<\alpha\).
\(\Xi(\alpha)=\min(M^\alpha \cup\{\mathcal K\})\).
For \(\xi\le\alpha\), \(\Psi^\xi_\pi(\alpha)=\min(\{\rho\in M^\xi\cap\pi\,\colon C(\alpha,\rho) \cap \pi = \rho \land (\pi,\alpha)\in C(\alpha,\rho)^2\} \cup \{\pi\})\).
== Ordinal notation ==
Rathjen created an ordinal notation associated to \(\Psi\). Readers should be careful that googologists tend to "simplify" Rathjen's OCFs being unaware of the issue that such a "simplification" might not admit an ordinal notation associated to them, because the notion of an ordinal notation is quite difficult for the majority of them.
== Sources ==
Category:Ordinal collapsing functions
Category:Ordinal notations
Category:Set theory