Buchholz's psi-functions: Difference between revisions

==History==

<nowiki>In 1950, H. Bachmann defined the first ordinal collapsing function, Bachmann's \(\varphi\). While able to succinctly describe the Bachmann-Howard ordinal as \(\varphi_{\varepsilon_{\Omega+1}}(0)\)</nowiki><ref>W. Buchholz, [https://www.mathematik.uni-muenchen.de/~buchholz/articles/jaegerfestschr_buchholz3.pdf A survey on ordinal notations around the Bachmann-Howard ordinal]</ref>, Bachmann's \(\varphi\) had a complicated definition. Subsequently, Feferman made a simultaneous simplification and extension of Bachmann's \(\varphi\) up to the level of the [[Takeuti-Feferman-Buchholz ordinal]]<ref>W. Buchholz, Relating ordinals to proofs in a perspicuous way</ref><ref>S. Feferman, [https://math.stanford.edu/~feferman/papers/id-saga.pdf The proof theory of classical and constructive inductive definitions. A 40 year saga, 1968-2008.]</ref>, and then Buchholz further simplified Feferman's \( \theta \) to an ordinal collapsing function with behaviour more similar to the original by Bachmann.<ref>M. Rathjen, [https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf The Art of Ordinal Analysis]</ref>

 

* \(C_\nu^0(\alpha) = \Omega_\nu\)

* \(C_\nu(\alpha) = \bigcup\{C_\nu^n(\alpha): n < \omega\}\)

* \(\psi_\nu(\alpha) = \operatorname{mex}(C_\nu(\alpha))\), where \(\operatorname{mex}\) denotes minimal excludant.

 

* \(C_\nu^0(\alpha) = \Omega_\nu\)

* \(C_\nu(\alpha) = \bigcup\{C_\nu^n(\alpha): n < \omega\}\)

* \(\psi_\nu(\alpha) = \operatorname{mex}(C_\nu(\alpha))\), where \(\operatorname{mex}\) denotes minimal excludant.

 

This admits an ordinal notation too, as well as a canonical set of fundamental sequences.