Not to be confused with Rathjen's psi function.

Rathjen's $$\Psi$$ function based on the least weakly compact cardinal is an ordinal collapsing function. 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:

1. $$C(\alpha,\pi)\cap\mathcal K = \pi$$
2. $$\forall(\xi\in C(\alpha,\pi) \cap \alpha)(M^{\xi} \text{ is stationary in }\pi)$$
3. $$\alpha\in C(\alpha,\pi)$$

$$C(\alpha,\beta)$$ is the closure of $$\beta\cup\{0,\mathcal K\}$$ under:

• $$(\xi,\eta)\mapsto\,$$$$\varphi$$$$(\xi,\eta)$$,
• $$\xi\mapsto\Omega_\xi$$ 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

1. Rathjen, Michael. "Proof Theory of Reflection", Annals of Pure and Applied Logic 68, 181--224 (1994).