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).