Skip to main content
×
Home
    • Aa
    • Aa

The slow-growing and the Graegorczyk hierarchies

  • E.A. Cichon (a1) and S.S. Wainer (a2)
Abstract

We give here an elementary proof of a recent result of Girard [4] comparing the rates of growth of the two principal (and extreme) examples of a spectrum of “majorization hierarchies”—i.e. hierarchies of increasing number-theoretic functions, indexed by (systems of notations for) initial segments I of the countable ordinals so that if α < βI then the βth function dominates the αth one at all but finitely-many positive integers x.

Hardy [5] was perhaps the first to make use of a majorization hierarchy—the Hα's below—in “exhibiting” a set of reals with cardinality ℵ1. More recently such hierarchies have played important roles in mathematical logic because they provide natural classifications of recursive functions according to their computational complexity. (All the functions considered here are “honest” in the sense that the size of their values gives a measure of the number of steps needed to compute them.)

The hierarchies we are concerned with fall into three main classes depending on their mode of generation at successor stages, the other crucial parameter being the initial choice of a particular (standard) fundamental sequence λ0 < λ1 < λ2 < … to each limit ordinal λ under consideration which, by a suitable diagonalization, will then determine the generation at stage λ.

Our later comparisons will require the use of a “large” initial segment I of proof-theoretic ordinals, extending as far as the “Howard ordinal”. However we will postpone a precise description of these ordinals and their associated fundamental sequences until later.

Copyright
References
Hide All
[7] M.H. Löb and S.S. Wainer , Hierarchies of number-theoretic functions. I, II, Archiv für Mathematische Logik und Grundlagenforschung, vol. 13 (1970), pp. 39–51 and pp. 97113.

[8] J. Paris and L. Harrington , A mathematical incompleteness in Peano arithmetic, Handbook of Mathematical Logic ( J. Barwise , Editor), North-Holland, Amsterdam, 1977, pp. 11331142.

[10] J. Ketonen and R.M. Solovay , Rapidly growing Ramsey functions, Annals of Mathematics, vol. 113 (1981), pp. 267314.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 4 *
Loading metrics...

Abstract views

Total abstract views: 118 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 17th October 2017. This data will be updated every 24 hours.