Stegert's Psi functions

Stegert's \(\Psi\) functions are two ordinal collapsing functions created by Jan-Carl Stegert as part of his 2010 Ph.D. dissertation, alongside which he also introduced collapsing hierarchies \(\mathfrak{M}\), and developed a powerful ordinal notation based on the work of Rathjen.Stegert, Jan-Carl. "Ordinal proof theory of Kripke-Platek set theory augmented by strong reflection principles". He then used \(\Psi\) for an analysis of an extension of Kripke-Platek set theory called \(\mathsf{Stability}\). (p.141)
==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 (p.4). As part of a characterization of the functions provably recursive in \(\Pi_\omega\textrm{-ref}\) (p.96), 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 (p.75).
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 \(\Pi_\omega\textrm{-ref}\).
==Sources==
==See also==
Category:Ordinal collapsing functions
Category:Ordinal notations