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A Survey on Ordinal Notations Around the Bachmann–Howard Ordinal

  • Wilfried Buchholz
Chapter
Part of the Outstanding Contributions to Logic book series (OCTR, volume 13)

Abstract

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. Special efforts are made to reveal the intrinsic connections between Feferman’s \(\theta \)-functions and the Bachmann hierarchy.

Keywords

Bachmann–Howard ordinal Bachmann hierarchy Fundamental sequence Klammersymbol Normal function 

2010 Mathematics Subject Classification

03F15 

References

  1. 1.
    Aczel, P.: A New Approach to the Bachmann Method for Describing Large Countable Ordinals (Preliminary Summary). unpublishedGoogle Scholar
  2. 2.
    Aczel, P.: Describing ordinals using functionals of transfinite type. J. Symbol. Logic 37(1), 35–47 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bachmann, H.: Die Normalfunktionen und das Problem der ausgezeichneten Folgen von Ordnungszahlen. Vierteljschr. Naturforsch.Ges. Zürich, pp. 115–147 (1950)Google Scholar
  4. 4.
    Bridge, J.: A simplification of the Bachmann method for generating large countable ordinals. J. Symbol. Logic 40, 171–185 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Buchholtz, U.T.: Unfolding of systems of inductive definitions. PhD thesis, Stanford University (2013)Google Scholar
  6. 6.
    Buchholz, W.: Normalfunktionen und konstruktive Systeme von Ordinalzahlen. In Diller, J., Müller, G.H. (eds.), Proof Theory Symposium, Kiel 1974. Lecture Notes in Mathematics, vol. 500, pp. 4–25. Springer, Berlin (1975)Google Scholar
  7. 7.
    Buchholz, W.: Collapsingfunktionen (1981). http://www.mathematik.uni-muenchen.de/~buchholz/Collapsing.pdf
  8. 8.
    Buchholz, W.: A survey on ordinal notations around the Bachmann-Howard ordinal. In: Kahle, R., Strahm, T., Studer, T. (eds.) Advances in Proof Theory. Progress in Computer Science and Applied Logic, vol. 28, pp. 1–29. Birkhäuser, Basel (2016)Google Scholar
  9. 9.
    Buchholz, W., Schütte, K.: Die Beziehungen zwischen den Ordinalzahlsystemen \(\Sigma \) und \(\overline{\theta }(\omega )\). Arch. Math. Logik und Grundl. 17, 179–189 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Buchholz, W., Schütte, K.: Ein Ordinalzahlensystem für die beweistheoretische Abgrenzung der \(\Pi ^1_1\)-Separation und Bar-Induktion. Bayr. Akad. Wiss. Math.-Naturw. Kl., pp. 99–13 (1983)Google Scholar
  11. 11.
    Buchholz, W., Schütte, K.: Proof Theory of Impredicative Subsystems of Analysis. Studies in Proof Theory Monographs, vol. 2. Bibliopolis, Napoli (1988)Google Scholar
  12. 12.
    Crossley, J.N., Bridge-Kister, J.: Natural well-orderings. Arch. Math. Logik und Grundl. 26, 57–76 (1986/87)Google Scholar
  13. 13.
    Feferman, S.: Proof theory: a personal report. In: Takeuti, G. (ed.) Proof Theory. Studies in Logic and the Foundations of Mathematics, 2nd edn. pp. 447–485. North-Holland (1987)Google Scholar
  14. 14.
    Gerber, H.: An extension of Schütte’s Klammersymbols. Math. Ann. 174, 203–216 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Howard, W.: A system of abstract constructive ordinals. J. Symbol. Logic 37, 355–374 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Isles, D.: Regular ordinals and normal forms. In: Kino, A., Myhill, J., Vesley, R.E. (eds.), Intuitionism and Proof Theory (Proceedings Conference, Buffalo N.Y., 1968), pp. 339–362. North-Holland, 1970Google Scholar
  17. 17.
    Jäger, G.: \(\rho \)-inaccessible ordinals, collapsing functions and a recursive notation system. Archiv für mathematische Logik 24, 49–62 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Pfeiffer, H.: Ausgezeichnete Folgen für gewisse Abschnitte der zweiten und weiterer Zahlenklassen. PhD thesis, Hannover (1964)Google Scholar
  19. 19.
    Pohlers, W.: Ordinal notations based on a hierarchy of inaccessible cardinals. Ann. Pure Appl. Logic 33, 157–179 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Rathjen, M.: Ordinal notations based on a weakly Mahlo cardinal. Arch. Math. Logic 29, 249–263 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rathjen, M.: Proof theory of reflection. Ann. Pure Appl. Logic 68, 181–224 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Rathjen, M.: Recent advances in ordinal analysis: \({\Pi }^1_2\)-CA and related systems. Bull. Symbol. Logic 1(4), 468–485 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rathjen, M., Weiermann, A.: Proof-theoretic investigations on Kruskal’s theorem. Ann. Pure Appl. Logic 60(1), 49–88 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Schütte, K.: Kennzeichnung von Ordnungszahlen durch rekursiv erklärte Funktionen. Math. Ann. 127, 15–32 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Schütte, K.: Proof Theory, Grundlehren der Mathematischen Wissenschaften, vol. 225. Springer, Berlin (1977)Google Scholar
  26. 26.
    Schütte, K.: Beziehungen des Ordinalzahlensystems \({{\rm OT}}(\vartheta )\) zur Veblen-Hierarchie. unpublished (1992)Google Scholar
  27. 27.
    Veblen, O.: Continous increasing functions of finite and transfinite ordinals. Trans. Am. Math. Soc. 9, 280–292 (1908)CrossRefzbMATHGoogle Scholar
  28. 28.
    Weyhrauch, R.: Relations between some hierarchies of ordinal functions and functionals. PhD thesis, Stanford University (1976). (Completed and circulated in 1972)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Mathematisches Institut der Universität MünchenMünchenGermany

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