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 using Buchholz's psi function,[3] an ordinal collapsing function invented by Wilfried Buchholz,[4][5][6] and 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:
- ,[9] a subsystem of second-order arithmetic
- -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
This article is missing information about the definition of the Takeuti-Feferman-Buchholz ordinal.(April 2024) |
- Let represent the smallest uncountable ordinal with cardinality .
- Let represent the th epsilon number, equal to the th fixed point of
- Let represent Buchholz's psi function
References
- ^ "Buchholz's ψ functions". cantors-attic. Retrieved 2021-08-10.
- ^ a b "Buchholz's ψ functions". cantors-attic. Retrieved 2021-08-17.
- ^ a b "A Zoo of Ordinals" (PDF). Madore. 2017-07-29. Retrieved 2021-08-10.
- ^ "Collapsingfunktionen" (PDF). University of Munich. 1981. Retrieved 2021-08-10.
- ^ 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.
- ^ 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.
- ^ Takeuti, Gaisi (2013). Proof Theory (2nd ed.). Dover Publications. ISBN 978-0-486-32067-0.
- ^ 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.
- ^ 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.
- ^ "ordinal analysis in nLab". ncatlab.org. Retrieved 2021-08-28.
- ^ "number theory - Can PA prove very fast growing functions to be total?". Mathematics Stack Exchange. Retrieved 2021-08-17.