Takeuti–Feferman–Buchholz ordinal explained

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, after Gaisi Takeuti, Solomon Feferman and Wilfried Buchholz. It is written as

\psi _{0(\varepsilon} ^{\Omega _{\omega} +1} ) using Buchholz's psi function,^[3] an ordinal collapsing function invented by Wilfried Buchholz,^{[4] [5] [6]} and \theta _{\varepsilon _{\Omega _{\omega} +1}} ( 0 ) 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:\Pi ^{1} _{1} - CA + BI ,^[9] a subsystem of second-order arithmetic\Pi ^{1} _{1} -comprehension + transfinite induction

• ID_ω, 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

\Omega _{\alpha} represent the smallest uncountable ordinal with cardinality \aleph _{\alpha} .

• Let

\varepsilon _{\beta} represent the \beta th epsilon number, equal to the 1+ \beta th fixed point of \alpha \mapsto \omega ^{\alpha}

• Let

\psi represent Buchholz's psi function

References

1. Web site: Buchholz's ψ functions. 2021-08-10. cantors-attic. en-US.
2. Web site: Buchholz's ψ functions. 2021-08-17. cantors-attic. en-US.
3. Web site: 2017-07-29. A Zoo of Ordinals. 2021-08-10. Madore.
4. Web site: 1981. Collapsingfunktionen. 2021-08-10. University of Munich.
5. 1986-01-01. A new system of proof-theoretic ordinal functions. Annals of Pure and Applied Logic. en. 32. 195–207. 10.1016/0168-0072(86)90052-7. 0168-0072. Buchholz. W.. free.
6. Book: Buchholz . Wilfried . Schütte . Kurt . 88-7088-166-0 . Naples, Italy . Bibliopolis . Studies in Proof Theory, Monographs . Proof Theory of Impredicative Subsystems of Analysis . 2 . 1988.
7. Book: Takeuti, Gaisi . Proof Theory . 2nd. Dover Publications . 978-0-486-32067-0 . 2013.
8. Book: Buchholz . W. . ⊨ISILC Proof Theory Symposion . Normalfunktionen und Konstruktive Systeme von Ordinalzahlen . Lecture Notes in Mathematics . 1975 . 500 . 4–25 . 10.1007/BFb0079544 . Springer . 978-3-540-07533-2 . de.
9. Book: Buchholz . Wilfried . Feferman . Solomon . Solomon Feferman . Pohlers . Wolfram . Sieg . Wilfried . 10.1007/bfb0091894 . 3-540-11170-0 . 655036 . Springer-Verlag, Berlin-New York . Lecture Notes in Mathematics . Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies . 897 . 1981.
10. Web site: ordinal analysis in nLab. 2021-08-28. ncatlab.org.
11. Web site: number theory - Can PA prove very fast growing functions to be total?. 2021-08-17. Mathematics Stack Exchange.