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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. It was named by David Madore, after Gaisi Takeuti, Solomon Feferman and Wilfried Buchholz. It is written as ${\displaystyle \psi _{0}(\varepsilon _{\Omega _{\omega }+1})}$ using Buchholz's psi function, an ordinal collapsing function invented by Wilfried Buchholz, and ${\displaystyle \theta _{\varepsilon _{\Omega _{\omega }+1}}(0)}$ in Feferman's theta function, an ordinal collapsing function invented by Solomon Feferman. It is the proof-theoretic ordinal of several formal theories:

• ${\displaystyle \Pi _{1}^{1}-CA+BI}$, a subsystem of second-order arithmetic
• ${\displaystyle \Pi _{1}^{1}}$-comprehension + transfinite induction
• ID_ω, the system of ω-times iterated inductive definitions

Definition

This article is missing information about the definition of the Takeuti-Feferman-Buchholz ordinal. Please expand the article to include this information. Further details may exist on the talk page. (April 2024)

• Let ${\displaystyle \Omega _{\alpha }}$ represent the smallest uncountable ordinal with cardinality ${\displaystyle \aleph _{\alpha }}$.
• Let ${\displaystyle \varepsilon _{\beta }}$ represent the ${\displaystyle \beta }$th epsilon number, equal to the ${\displaystyle 1+\beta }$th fixed point of ${\displaystyle \alpha \mapsto \omega ^{\alpha }}$
• Let ${\displaystyle \psi }$ represent Buchholz's psi function

References

1. "Buchholz's ψ functions". cantors-attic. Retrieved 2021-08-10.
2. ^ "Buchholz's ψ functions". cantors-attic. Retrieved 2021-08-17.
3. ^ "A Zoo of Ordinals" (PDF). Madore. 2017-07-29. Retrieved 2021-08-10.
4. "Collapsingfunktionen" (PDF). University of Munich. 1981. Retrieved 2021-08-10.
5. Buchholz, W. (1986-01-01). "A new system of proof-theoretic ordinal functions". Annals of Pure and Applied Logic. 32: 195–207. doi:10.1016/0168-0072(86)90052-7. ISSN 0168-0072.
6. Buchholz, Wilfried; Schütte, Kurt (1988). Proof Theory of Impredicative Subsystems of Analysis. Studies in Proof Theory, Monographs. Vol. 2. Naples, Italy: Bibliopolis. ISBN 88-7088-166-0.
7. Takeuti, Gaisi (2013). Proof Theory (2nd ed.). Dover Publications. ISBN 978-0-486-32067-0.
8. Buchholz, W. (1975). "Normalfunktionen und Konstruktive Systeme von Ordinalzahlen". ⊨ISILC Proof Theory Symposion. Lecture Notes in Mathematics (in German). Vol. 500. Springer. pp. 4–25. doi:10.1007/BFb0079544. ISBN 978-3-540-07533-2.
9. Buchholz, Wilfried; Feferman, Solomon; Pohlers, Wolfram; Sieg, Wilfried (1981). Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies. Lecture Notes in Mathematics. Vol. 897. Springer-Verlag, Berlin-New York. doi:10.1007/bfb0091894. ISBN 3-540-11170-0. MR 0655036.
10. "ordinal analysis in nLab". ncatlab.org. Retrieved 2021-08-28.

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