As part of his 2010 Ph.D. dissertation, Jan-Carl Stegert introduced collapsing hierarchies $$\mathfrak{M}$$ and two collapsing functions $$\Psi$$, and developed a powerful ordinal notation based on the work of Rathjen.. $$\Psi$$ was then used for an analysis of an extension of $$\textrm{KP}$$ set theory called $$\mathsf{Stability}$$.

Overview

The original definition of the first OCF is long and difficult, however here is a simplified overview of it.

Reflection instances are defined using ordered quintuples, and $$\mathsf M\textrm -\mathsf P$$ expressions are characterized in terms of reflection configurations, ordinals, and a number $$n\in\omega$$ or $$n=-1$$. For an ordinal $$\kappa$$ and an $$\mathsf M\textrm -\mathsf P$$-expression $$R$$, a relation $$\kappa\vDash R$$ is defined (not to be confused with satifaction for models), which plays the role of translating the properties represented by the $$\mathsf M\textrm -\mathsf P$$-expression into indescribability. Then, for ordinal $$\alpha$$ and reflection instance $$\mathbb X$$, collapsing hierarchies $$\mathfrak M^\alpha_{\mathbb X}$$ are defined that play a role superficially similar to an extension of $$M^\alpha$$ in Rathjen's OCF using a weakly compact cardinal (note that there are some differences, e.g. the behavior of the 1st entry $$\pi$$ of $$\mathbb X$$). Finally, an ordinal collapsing function $$\Psi$$ is defined.

Fast-growing function

Let $$\Pi_\omega\textrm{-ref}$$ denote the theory $$\textrm{KP}$$ augmented by a certain first-order reflection scheme. As part of a characterization of the functions provably recursive in $$\Pi_\omega\textrm{-ref}$$, Stegert defines a function superficially similar to the middle-growing hierarchy (composed with tetration) along $$\mathsf T(\Xi)$$, using the concept of a "norm" in place of an explicit definition of fundamental sequences.

Since $$\varphi_1(\Xi+1)\subseteq\varphi_1(\Xi+1)+2\subseteq\varphi_2(\Xi+1)\subseteq\mathsf T(\Xi)$$, where $$\mathsf T(\Xi)$$ denotes the hull-set we use, then all of the following functions are well-defined, and the order of eventual domination is as follows:

• $$f^{\mathsf T(\Xi)}_{\Psi^{\varphi_2(\Xi+1)}_{(\omega^+;\mathsf P_0;\epsilon;\epsilon;0)}}$$ will eventually dominate $$f^{\mathsf T(\Xi)}_{\Psi^{\varphi_1(\Xi+1)+2}_{(\omega^+;\mathsf P_0;\epsilon;\epsilon;0)}}$$,
• Which eventually dominates $$f^{\mathsf T(\Xi)}_{\Psi^{\varphi_1(\Xi+1)}_{(\omega^+;\mathsf P_0;\epsilon;\epsilon;0)}}$$.

Therefore, the function $$f^{\mathsf T(\Xi)}_{\Psi^{\varphi_2(\Xi+1)}_{(\omega^+;\mathsf P_0;\epsilon;\epsilon;0)}}$$ will dominate all functions provably recursive in the theory $$\Pi_\omega\textrm{-ref}$$.