|
== Other similar OCFs ==
=== Arai's ''ψ'' ===
'''Arai's ''ψ'' function''' is an ordinal collapsing function introduced by Toshiyasu Arai (husband of [[Noriko H. Arai]]) in his paper: ''A simplified ordinal analysis of first-order reflection''. <math>\psi_\Omega(\alpha)</math> is a collapsing function such that <math>\psi_\Omega(\alpha) < \Omega</math>, where <math>\Omega</math> represents the [[first uncountable ordinal]] (it can be replaced by the [[Nonrecursive ordinal|Church–Kleene ordinal]] at the cost of extra technical difficulty). Throughout the course of this article, <math>\mathsf{KP\Pi_N}</math> represents [[Kripke–Platek set theory]] for a <math>\mathsf{\Pi_N}</math>-reflecting universe, <math>\mathbb{K}_N</math> is the smallestleast <math>\mathsf{\Pi_NPi}^1_{N-2}</math>-indescribable cardinal (it may be replaced with the least <math>\mathsf{\Pi}_N</math>-reflecting ordinal at the cost of extra technical difficulty), <math>N</math> is a fixed natural number <math>>\ge 23</math>, and <math>\Omega_0 = 0</math>.
Suppose <math>\mathsf{KP\Pi_N} \vdash \theta}</math> for a <math>\mathsf{\Sigma_1}</math> (<math>\Omega</math>)-sentence <math>\mathsf{\theta}</math>. Then, [[Existential quantification|there exists]] a finite <math>n</math> such that for <math>\alpha = \psi_\Omega(\Omega_nomega_n(\mathbb{K}_N + 1))</math>, <math>L_\alpha \models \theta</math>. It can also be proven that <math>\mathsf{KP\Pi_N}</math> proves that each initial segment <math>\{\alpha \in OT: \alpha < \psi_\Omega(\Omega_nomega_n(\mathbb{K}_N + 1))\}; n = 1, 2, \ldots</math> is [[Well-founded relation|well-founded]], and therefore, the [[Ordinal analysis|proof-theoretic ordinal]] of <math>\psi_\Omega(\varepsilon_{\mathbb{K}_N+1})</math> is the [[Ordinal analysis|proof-theoretic ordinal]] of <math>\mathsf{KP\Pi_N}</math>. Using this, <math>\psi_\Omega(\varepsilon_{\mathbb{K}_N+1}) = \min(\{\alpha \leq \Omega \mid \forall \theta \in \Sigma_1(\mathsf{KP\Pi_N} \vdash \theta^{L_\Omega} \rightarrow L_\alpha \models \theta)\})</math>. One can then make the following conversions:
* <math>\psi_{\Omega(A) = \psi_0}(\Omega) = |\mathsf{PA}| = \varphi(1, 0)</math>, where <math>A\Omega</math> is either the least admissiblerecursively regular ordinal or the least uncountable cardinal, <math>\mathsf{PA}</math> is [[Peano axioms|Peano arithmetic]] and <math>\varphi</math> is the [[Veblen function|Veblen hierarchy]].
* <math>\psi_{\Omega(\varepsilon_{A +1}) = \psi_0(\varepsilon_{\Omega + 1}) = |\mathsf{KP\omega}| = \mathsf{BHO}</math>, where <math>A\Omega</math> is either the least admissiblerecursively regular ordinal or the least uncountable cardinal, <math>\mathsf{KP\omega}</math> is [[Kripke–Platek set theory]] with infinity and <math>\mathsf{BHO}</math> is the [[Bachmann–Howard ordinal]].
* <math>\psi_{\Omega(I) = \psi_0}(\Omega_{\omega}) = |\mathsf{\Pi^1_1-CA_0}| = \mathsf{BO}</math>, where <math>I\Omega_{\omega}</math> is either the least recursivelylimit of admissible ordinals or the least limit of inaccessibleinfinite ordinalcardinals and <math>\mathsf{BO}</math> is [[Buchholz's ordinal]].
* <math>\psi_{\Omega(\varepsilon_{I +1}) = \psi_0(\varepsilon_{\Omega_{\omega} + 1}) = |\mathsf{KPIKPl}| = \mathsf{TFBO}</math>, where <math>I\Omega_{\omega}</math> is either the least recursivelylimit inaccessibleof ordinaladmissible ordinals or the least limit of infinite cardinals, <math>\mathsf{KPIKPl}</math> is Kripke–PlatekKPi setwithout theory with a recursivelythe inaccessiblecollection universescheme and <math>\mathsf{TFBO}</math> is the [[Takeuti–Feferman–Buchholz ordinal]].
* <math>\psi_{\Omega}(\varepsilon_{I + 1}) = |\mathsf{KPi}|</math>, where <math>I</math> is either the least recursively inaccessible ordinal or the least weakly inaccessible cardinal and <math>\mathsf{KPi}</math> is Kripke–Platek set theory with a recursively inaccessible universe.
=== Bachmann's ''ψ'' ===
| 