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Slow growing versus fast growing

Published online by Cambridge University Press:  12 March 2014

S. S. Wainer
S. S. Wainer*
Affiliation:
School of Mathematics, University of Leeds, Leeds LS2 9JT, England
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Extract

I falsely claimed, as an aside remark in [8] and also implicitly in the abstract [9], that the slow-growing hierarchy

“catches up” with the fast-growing hierarchy

at level Γ0, i.e. that, for all x > 0,

where x′ is some simple (even linear) function of x.

Girard [4] gave the first correct analysis of the deep relationship which exists between G and F, based on his extensive category-theoretic framework for -logic. This analysis indicates that the first point at which G catches up with F is the ordinal of the theory ID<ω(0 of arbitrary finite iterations of an inductive definition. This is very far beyond Γ0! In particular, in order to capture F at level ∣IDn∣ the slow-growing hierarchy must be generated up to ∣IDn+1∣, i.e. one extra iteration of an inductive definition is needed in order to generate sufficient new ordinals.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 2 , June 1989 , pp. 608 - 614
Copyright
Copyright © Association for Symbolic Logic 1989

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References

REFERENCES

[1]Aczel, P., Another elementary treatment of Girard's result connecting the slow and fast growing hierarchies of number-theoretic functions, manuscript, 1980.Google Scholar
[2]Buchholz, W., Three contributions to the conference on recent advances in proof theory, manuscript, 1980.Google Scholar
[3]Cichon, E. A. and Wainer, S. S., The slow-growing and the Grzegorczyk hierarchies, this Journal, vol. 48 (1983), pp. 399–408.Google Scholar
[4]Girard, J.-Y., logic, Annals of Mathematical Logic, vol. 21 (1981), pp. 75–219.CrossRefGoogle Scholar
[5]Jervell, H. R., Homogeneous trees, Lecture notes, University of Munich, Munich, 1979.Google Scholar
[6]Schmerl, U. R., Über die schwach und die stark wachsende Hierarchie zahlentheoretischer Funktionen, Bayerische Akademie der Wissenschaften, Mathematisch-Naturwissenschaftliche Klasse. Sitzungsberichte, 1981, pp. 1–18.Google Scholar
[7]Schwichtenberg, H., Homogeneous trees and subrecursive hierarchies, Lecture, 1980.Google Scholar
[8]Wainer, S. S., Ordinal recursion and a refinement of the extended Grzegorczyk hierarchy, this Journal, vol. 37 (1972), pp. 287–292.Google Scholar
[9]Wainer, S. S., A subrecursive hierarchy over the predicative ordinals, Conference in mathematical logic—London 70 (Hodges, W., editor), Lecture Notes in Mathematics, vol. 255, Springer-Verlag, Berlin, 1972, pp. 350–351 (abstract #14).Google Scholar
[10]Wainer, S. S., The “slow-growing” approach to hierarchies, Recursion theory (Nerode, A. and Shore, R., editors), Proceedings of Symposia in Pure Mathematics, vol. 42, American Mathematical Society, Providence, Rhode Island, 1985, pp. 487–502.CrossRefGoogle Scholar