Takeuti–Feferman–Buchholz ordinal
In the mathematical fields of set theory and proof theory, the Takeuti–Feferman–Buchholz ordinal (TFBO) is a large countable ordinal, which acts as the limit of the range of Buchholz's psi function and Feferman's theta function.[1][2] It was named by David Madore,[2] after Gaisi Takeuti, Solomon Feferman and Wilfried Buchholz. It is written as [math]\displaystyle{ \psi_0(\varepsilon_{\Omega_\omega + 1}) }[/math] using Buchholz's psi function,[3] an ordinal collapsing function invented by Wilfried Buchholz,[4][5][6] and [math]\displaystyle{ \theta_{\varepsilon_{\Omega_\omega + 1}}(0) }[/math] in Feferman's theta function, an ordinal collapsing function invented by Solomon Feferman.[7][8] It is the proof-theoretic ordinal of several formal theories:
- [math]\displaystyle{ \Pi_1^1 -CA + BI }[/math],[9] a subsystem of second-order arithmetic
- [math]\displaystyle{ \Pi_1^1 }[/math]-comprehension + transfinite induction[3]
- IDω, the system of ω-times iterated inductive definitions[10]
Despite being one of the largest large countable ordinals and recursive ordinals, it is still vastly smaller than the proof-theoretic ordinal of ZFC.[11]
Definition
- Let [math]\displaystyle{ \Omega_\alpha }[/math] represent the smallest uncountable ordinal with cardinality [math]\displaystyle{ \aleph_\alpha }[/math].
- Let [math]\displaystyle{ \varepsilon_\beta }[/math] represent the [math]\displaystyle{ \beta }[/math]th epsilon number, equal to the [math]\displaystyle{ 1+\beta }[/math]th fixed point of [math]\displaystyle{ \alpha \mapsto \omega^\alpha }[/math]
- Let [math]\displaystyle{ \psi }[/math] represent Buchholz's psi function
References
- ↑ "Buchholz's ψ functions" (in en-US). https://neugierde.github.io/cantors-attic/Buchholz%27s_%CF%88_functions#takeuti-feferman-buchholz-ordinal.
- ↑ 2.0 2.1 "Buchholz's ψ functions" (in en-US). https://neugierde.github.io/cantors-attic/Buchholz%27s_%CF%88_functions#takeuti-feferman-buchholz-ordinal.
- ↑ 3.0 3.1 "A Zoo of Ordinals". 2017-07-29. http://www.madore.org/~david/math/ordinal-zoo.pdf.
- ↑ "Collapsingfunktionen". 1981. https://www.mathematik.uni-muenchen.de/~buchholz/articles/Collapsing.pdf.
- ↑ Buchholz, W. (1986-01-01). "A new system of proof-theoretic ordinal functions" (in en). Annals of Pure and Applied Logic 32: 195–207. doi:10.1016/0168-0072(86)90052-7. ISSN 0168-0072.
- ↑ Buchholz, W.; Schütte, K. (1988). "Proof Theory of Impredicative Subsystems of Analysis" (in en).
- ↑ "[PDF Proof Theory Second Edition by Gaisi Takeuti | Perlego"]. https://www.perlego.com/book/112275/proof-theory-second-edition-pdf.
- ↑ Buchholz, W. (1975). "Normalfunktionen und Konstruktive Systeme von Ordinalzahlen" (in de). ⊨ISILC Proof Theory Symposion. Lecture Notes in Mathematics. 500. Springer. pp. 4–25. doi:10.1007/BFb0079544. ISBN 978-3-540-07533-2.
- ↑ Buchholz, Wilfried (1981). Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies. Lecture Notes in Mathematics. 897. Springer-Verlag, Berlin-New York. doi:10.1007/bfb0091894. ISBN 3-540-11170-0.
- ↑ "ordinal analysis in nLab". https://ncatlab.org/nlab/show/ordinal+analysis#table_of_ordinal_analyses.
- ↑ "number theory - Can PA prove very fast growing functions to be total?". https://math.stackexchange.com/q/1080924.
 |  |
