Heinz Bachmann's function was the first true ordinal collapsing function, described by Rathjen as a "novel idea" for 1950.[1] An English translation of Bachmann's original paper was submitted to ArXiv in 2019 by Martin Dowd.[2]
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[3] (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[4] 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[5], the proof-theoretic ordinal of Kripke-Platek set theory with axiom of infinity.
Sources[]
- ↑ M. Rathjen, "A history of ordinal representations" (notes) (p.9) Archived 2007-06-12.
- ↑ M. Dowd, A Translation of "Die Normalfunktionen und das Problem der ausgezeichneten Folgen von Ordnungszahlen" by Heinz Bachmann
- ↑ M. Rathjen, Proof theory: From arithmetic to set theory (p.13). Accessed 2021-06-19.
- ↑ Rathjen, Michael. "The Art of Ordinal Analysis".
- ↑ M. Rathjen, Relativized ordinal analysis: The case of Power Kripke-Platek set theory (p.8). Accessed 2021-07-17.
See also[]
Basics: cardinal numbers · ordinal numbers · limit ordinals · fundamental sequence · normal form · transfinite induction · ordinal notation · Absolute infinity
Theories: Robinson arithmetic · Presburger arithmetic · Peano arithmetic · KP · second-order arithmetic · ZFC
Model Theoretic Concepts: structure · elementary embedding
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\eta_0\) · \(\Gamma_0\) (Feferman–Schütte ordinal) · \(\varphi(1,0,0,0)\) (Ackermann ordinal) · \(\psi_0(\Omega^{\Omega^\omega})\) (small Veblen ordinal) · \(\psi_0(\Omega^{\Omega^\Omega})\) (large Veblen ordinal) · \(\psi_0(\varepsilon_{\Omega + 1}) = \psi_0(\Omega_2)\) (Bachmann-Howard ordinal) · \(\psi_0(\Omega_\omega)\) with respect to Buchholz's ψ · \(\psi_0(\varepsilon_{\Omega_\omega + 1})\) (Takeuti-Feferman-Buchholz ordinal) · \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\) (countable limit of Extended Buchholz's function) · \(\omega_1^\mathfrak{Ch}\) (Omega one of chess) · \(\omega_1^{\text{CK}}\) (Church-Kleene ordinal) · \(\omega_\alpha^\text{CK}\) (admissible ordinal) · recursively inaccessible ordinal · recursively Mahlo ordinal · reflecting ordinal · stable ordinal · \(\lambda,\gamma,\zeta,\Sigma\) (Infinite time Turing machine ordinals) · gap ordinal · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Hardy hierarchy · Slow-growing hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal functions: enumeration · normal function · derivative · Veblen function · ordinal collapsing function · Weak Buchholz's function · Bachmann's function · Madore's function · Feferman's theta function · Buchholz's function · Extended Weak Buchholz's function · Extended Buchholz's function · Jäger-Buchholz function · Jäger's function · Rathjen's psi function · Rathjen's Psi function · Stegert's Psi function · Arai's psi function
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal · Berkeley cardinal · (Club Berkeley cardinal · limit club Berkeley cardinal)
Classes: \(\textrm{Card}\) · \(\textrm{On}\) · \(V\) · \(L\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)