Bachmann's function

Heinz Bachmann's function was the first true ordinal collapsing function, described by Rathjen as a "novel idea" for 1950.M. Rathjen, "A history of ordinal representations" (notes) (p.9) Archived 2007-06-12. An English translation of Bachmann's original paper was submitted to ArXiv in 2019 by Martin Dowd.M. Dowd, A Translation of "Die Normalfunktionen und das Problem der ausgezeichneten Folgen von Ordnungszahlen" by Heinz Bachmann
==Original==
Bachmann's original function, a binary function \(\varphi_\alpha(\beta)\), is somewhat cumbersome as it depends on fundamental sequences for a lot of limit ordinalsM. Rathjen, Proof theory: From arithmetic to set theory (p.13). Accessed 2021-06-19. (specifically those in a set \(\mathfrak B\) defined by Bachmann). The definition is omitted from this article due to its complexity.
Up to about two decades after the publication of Bachmann's original \(\varphi\), the study of OCFs developed, but the OCFs themselves were extremely complicated and required meticulous computation to keep track of fundamental sequences. Then, Feferman's theta function became the first OCF to base its methods of operation off of other concepts entirely, at a stroke simplifying the study of OCFs and removing the need for complicated fundamental sequence considerations going forward.
==Rathjen's recast==
Rathjen suggests a "recast" of the system as follows. Let \(\Omega\) be an uncountable ordinal such as \(\aleph_1\). Then define \(C^\Omega(\alpha, \beta)\) as the closure of \(\beta \cup \{0, \Omega\}\) under \(+, (\xi \mapsto \omega^\xi), (\xi \mapsto \psi_\Omega(\xi))_{\xi < \alpha}\), and \(\psi_\Omega(\alpha) = \min \{\rho < \Omega : C^\Omega(\alpha, \rho) \cap \Omega = \rho\}\).
\(\psi_\Omega(\varepsilon_{\Omega + 1})\) is the Bachmann-Howard ordinalM. Rathjen, Relativized ordinal analysis: The case of Power Kripke-Platek set theory (p.8). Accessed 2021-07-17., the proof-theoretic ordinal of Kripke-Platek set theory with axiom of infinity.
==Sources==
==See also==
Category:Ordinal collapsing functions
Category:Set theory