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Heinz Bachmann's function was the first true ordinal collapsing function, described by Rathjen as a "novel idea" for 1950.[1]

## 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 ordinals[2] (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[3] 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 ordinal[4], the proof-theoretic ordinal of Kripke-Platek set theory with axiom of infinity.

## Sources

1. M. Rathjen, "A history of ordinal representations" (notes) (p.9) Archived 2007-06-12.
2. M. Rathjen, Proof theory: From arithmetic to set theory (p.13). Accessed 2021-06-19.
3. Rathjen, Michael. "The Art of Ordinal Analysis"
4. M. Rathjen, Relativized ordinal analysis: The case of Power Kripke-Platek set theory (p.8). Accessed 2021-07-17.