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    Friedman, Sy-David Rathjen, Michael and Weiermann, Andreas 2013. Slow consistency. Annals of Pure and Applied Logic, Vol. 164, Issue. 3, p. 382.

    De Smet, Michiel and Weiermann, Andreas 2012. Goodstein sequences for prominent ordinals up to the Bachmann–Howard ordinal. Annals of Pure and Applied Logic, Vol. 163, Issue. 6, p. 669.

    Fairtlough, Matt and Wainer, Stanley S. 1998. Handbook of Proof Theory.

    Cichon, E.A. and Weiermann, A. 1997. Term rewriting theory for the primitive recursive functions. Annals of Pure and Applied Logic, Vol. 83, Issue. 3, p. 199.

    Weiermann, Andreas 1995. Investigations on slow versus fast growing: How to majorize slow growing functions nontrivially by fast growing ones. Archive for Mathematical Logic, Vol. 34, Issue. 5, p. 313.

    Handley, W.G. and Wainer, S.S. 1994. Equational derivation vs. computation. Annals of Pure and Applied Logic, Vol. 70, Issue. 1, p. 17.

    H. Gallier, Jean 1991. What's so special about Kruskal's theorem and the ordinal Γo? A survey of some results in proof theory. Annals of Pure and Applied Logic, Vol. 53, Issue. 3, p. 199.

    Pohlers, W. 1991. Proof theory and ordinal analysis. Archive for Mathematical Logic, Vol. 30, Issue. 5-6, p. 311.

    1988. Theories of Computational Complexity.

    Crossley, John N. and Kister, Jane Bridge 1987. Natural well-orderings. Archiv für Mathematische Logik und Grundlagenforschung, Vol. 26, Issue. 1, p. 57.


The slow-growing and the Graegorczyk hierarchies

  • E.A. Cichon (a1) and S.S. Wainer (a2)
  • DOI:
  • Published online: 01 March 2014

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.

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[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.

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