A Survey on Ordinal Notations Around the Bachmann-Howard Ordinal

Part of the Progress in Computer Science and Applied Logic book series (PCS, volume 28)


Various ordinal functions which in the past have been used to describe ordinals not much larger than the Bachmann-Howard ordinal are set into relation.


Ordinal Notations Proof-theoretic Applications Veblen Hierarchy Additive Principal Numbers Feferman 
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  1. 1.
    P. Aczel, An new approach to the Bachmann method for describing countable ordinals (Preliminary Summary). Unpublished NotesGoogle Scholar
  2. 2.
    P. Aczel, Describing ordinals using functionals of transfinite type. J. Symbolic Logic 37(1), 35–47 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    H. Bachmann, Die Normalfunktionen und das Problem der ausgezeichneten Folgen von Ordnungszahlen. Vierteljschr. Naturforsch. Ges. Zürich 95, 115–147 (1950)MathSciNetzbMATHGoogle Scholar
  4. 4.
    J. Bridge, Some problems in mathematical logic. Systems of ordinal functions and ordinal notations. Ph.D. thesis. Oxford University, 1972Google Scholar
  5. 5.
    J. Bridge, A simplification of the Bachmann method for generating large countable ordinals. J. Symbolic Logic 40, 171–185 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    U.T. Buchholtz, Unfolding of systems of inductive definitions. Ph.D thesis. Stanford University, 2013Google Scholar
  7. 7.
    W. Buchholz, Normalfunktionen und konstruktive Systeme von Ordinalzahlen. Proof theory symposion Kiel 1974. Springer Lecture Notesin Mathmetical, vol. 500. pp. 4–25 (1975)Google Scholar
  8. 8.
    W. Buchholz, Collapsingfunktionen. Unpublished Notes (1981).
  9. 9.
    W. Buchholz, A new system of proof-theoretic ordinal functions. Ann. Pure Appl. Logic 32(3), 195–207 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    W. Buchholz, K. Schütte, Die Beziehungen zwischen den Ordinalzahlsystemen \(\Sigma \) und \(\overline{\theta }(\omega )\). Arch. Math. Logik und Grundl. 17, 179–189 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    W. Buchholz, K. Schütte, Ein Ordinalzahlensystem für die beweistheoretische Abgrenzung der \(\Pi ^1_1\)-Separation und Bar-Induktion. Bayr. Akad. Wiss. Math.-Naturw. Kl. 99–132 (1983)Google Scholar
  12. 12.
    W. Buchholz, K. Schütte, Proof Theory of Impredicative Subsystems of Analysis. No. 2 in Studies in Proof Theory, Monographs. Bibliopolis (1988)Google Scholar
  13. 13.
    J.N. Crossley, J. Bridge-Kister, Natural well-orderings. Arch. Math. Logik und Grundl. 26, 57–76 (1986/87)Google Scholar
  14. 14.
    S. Feferman, Proof theory: a personal report, in Proof theory, 2nd edn., ed. by G. Takeuti, (North-Holland 1987) pp. 447–485Google Scholar
  15. 15.
    H. Gerber, An extension of Schütte’s Klammersymbols. Math. Ann. 174, 203–216 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    D. Isles, Regular ordinals and normal forms, in Intuitionism and Proof Theory (Proc. Conf., Buffalo N.Y., 1968), pp. 339–362, North-Holland (1970)Google Scholar
  17. 17.
    H. Pfeiffer. Ausgezeichnete Folgen für gewisse Abschnitte der zweiten und weiterer Zahlenklassen, Dissertation, Hannover, 1964Google Scholar
  18. 18.
    M. Rathjen, A. Weiermann, Proof-theoretic investigations on Kruskal’s theorem. Ann. Pure Appl. Logic 60(1), 49–88 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    K. Schütte, Kennzeichnung von Ordnungszahlen durch rekursiv erklärte Funktionen. Math. Ann. 127, 15–32 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    K. Schütte, Proof Theory. No. 225 in Grundlehren der Mathematischen Wissenschaften (Springer, 1977)Google Scholar
  21. 21.
    K. Schütte, Beziehungen des Ordinalzahlensystems \({\rm {OT}}(\vartheta )\) zur Veblen-Hierarchie. Unpublished notes (1992)Google Scholar
  22. 22.
    O. Veblen, Continous increasing functions of finite and transfinite ordinals. Trans. Amer. Math. Soc. 9, 280–292 (1908)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    R. Weyhrauch, Relations between some hierarchies of ordinal functions and functionals. Completed and circulated in 1972. Ph.D. thesis, Stanford University, 1976Google Scholar

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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Mathematisches Institut, Ludwig-Maximilians-Universität MünchenMunichGermany

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