User blog:P進大好きbot/Cheatsheet on Properties of OCFs

I list significant properties, e.g. typical expansions of standard expressions of ordinals, of several OCFs which are helpful in analysis. I am not good at explicit computation, and hence tables below might contain many errors.

= Buchholz's OCF =

References:
 * 1) W. Buchholz, A new system of proof-theoretic ordinal functions, Annals of Pure and Applied Logic, Volume 32 (1986), pp. 195--207.
 * 2) W. Buchholz, Relating ordinals to proofs in a prespicious way, unpublished article.
 * 3) Buchholz's function, wiki article.

I list specific values of Buchholz's \(\psi\).

I list non-standard expressions of ordinals with respect to Buchholz's \(\psi\).

I list nest-free expansions of standard expressions of ordinals with respect to Buchholz's \(\psi\).

I list nesting expansions of standard expressions of countable ordinals with respect to Buchholz's \(\psi\).

= Extended Buchholz's OCF =

References:
 * 1) D. Maksudov, The extension of Buchholz's function, Traveling To The Infinity.
 * 2) Buchholz's function#Extension, wiki article.

Since extended Buchholz's OCF restricted to \(\varepsilon_{\Omega_{\omega}+1}\) coincides with Buchholz's OCF, I only list expansions of ordinals above \(\varepsilon_{\Omega_{\omega}+1}\) in extended Buchholz's \(\psi_0\).

WIP

= Rathjen's OCF Based on a Weakly Mahlo Cardinal =

References:
 * 1) M. Rathjen, Ordinal Notations Based on a Weakly Mahlo Cardinal, Archive for Mathematical Logic, Volume 29, Issue 4 (1990), pp. 249--263.
 * 2) Ordinal notation#Rathjen's ψ, wiki article.

I note that it does not coincide with the OCF introduced in The Realm of Ordinal Analysis, and is not a formal symbol in the ordinal notation system introduced in Proof-Theoretic Analysis of KPM.

I list expansions of ordinals in Rathjen's \(\psi_{\Omega_1}\) based on the least weakly Mahlo cardinal \(M\) introduced in the first reference above.

WIP

= Rathjen's OCF Based on a Weakly Compact Cardinal =

References:
 * 1) M. Rathjen, Proof Theory of Reflection, Annals of Pure and Applied Logic, Volume 68, Issue 2 (1994), pp. 181--224.

I list expansions of ordinals in Rathjen's \(\psi^{0}_{\Omega_1}\) based on the least weakly compact cardinal \(K\).

WIP

= Arai's OCF =

References:
 * 1) M. Rathjen, Proof Theory of Reflection, Annals of Pure and Applied Logic, Volume 68, Issue 2 (1994), pp. 181--224.

I list expansions of ordinals in Arai's \(\psi_{\Omega_1}\) based on the least \(\Pi^1_{N-2}\)-indescribable cardinal \(\mathbb{N}\) for a fixed \(N \in \mathbb{N}\) greater than \(2\).

WIP