Buchholz's psi-functions

Buchholz's \(\psi\)-functions are a family of functions \(\psi_\nu:(\omega+1)\times\textrm{Ord}\to\textrm{Ord},\;\alpha\mapsto\psi_\nu(\alpha)\) defined by Wilfried Buchholz in 1984.

Historical background

In 1950, H. Bachmann defined the first ordinal collapsing function, Bachmann's \(\varphi\). While able to succinctly describe the Bachmann-Howard ordinal as \(\varphi_{\varepsilon_{\Omega+1}}(0)\)^{[Citation needed]}, Bachmann's \(\varphi\) had a complicated definition

Possible sources for this section:

• M. Rathjen, The Art of Ordinal Analysis
• W. Buchholz, A survey on ordinal notations around the Bachmann-Howard ordinal
• About Bridge's work on Feferman theta: W. Buchholz, Relating ordinals to proofs in a perspicuous way
• About the conception of Feferman theta, and some more historical detail on pre-Feferman OCFs: S. Feferman, The proof theory of classical and constructive inductive definitions. A 40 year saga, 1968-2008.