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]

Definition

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  • 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

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  1. ^ "Buchholz's ψ functions". cantors-attic. Retrieved 2021-08-10.
  2. ^ a b "Buchholz's ψ functions". cantors-attic. Retrieved 2021-08-17.
  3. ^ a b "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.