Some interesting connections between the slow growing hierarchy and the Ackermann function

Journal of Symbolic Logic 66 (2):609-628 (2001)
Abstract
It is shown that the so called slow growing hierarchy depends non trivially on the choice of its underlying structure of ordinals. To this end we investigate the growth rate behaviour of the slow growing hierarchy along natural subsets of notations for Γ 0 . Let T be the set-theoretic ordinal notation system for Γ 0 and T tree the tree ordinal representation for Γ. It is shown in this paper that (G α ) α ∈ T matches up with the class of functions which are elementary recursive in the Ackermann function as does (G α ) α ∈ T tree (by folklore). By thinning out terms in which the addition function symbol occurs we single out subsystems $T* \subseteq T$ and $T^{tree*} \subseteq T^{tree}$ (both of order type not exceeding ε 0 ) and prove that (G α ) α ∈ T tree* still matches up with (G α ) α ∈ T tree but (G α ) α ∈ T* now consists of elementary recursive functions only. We discuss the relationship between these results and the Γ 0 -based termination proof for the standard rewrite system for the Ackermann function
Keywords No keywords specified (fix it)
Categories (categorize this paper)
DOI 10.2307/2695032
Options
 Save to my reading list
Follow the author(s)
My bibliography
Export citation
Find it on Scholar
Edit this record
Mark as duplicate
Revision history Request removal from index
 
Download options
PhilPapers Archive


Upload a copy of this paper     Check publisher's policy on self-archival     Papers currently archived: 22,184
External links
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
Through your library
References found in this work BETA
Jean-Yves Girard (1981). Π12-Logic, Part 1: Dilators. Annals of Mathematical Logic 21 (2-3):75-219.
Wilfried Buchholz (1995). Proof-Theoretic Analysis of Termination Proofs. Annals of Pure and Applied Logic 75 (1-2):57-65.
Toshiyasu Arai (1991). A Slow Growing Analogue to Buchholz' Proof. Annals of Pure and Applied Logic 54 (2):101-120.

View all 8 references / Add more references

Citations of this work BETA

Add more citations

Similar books and articles

Monthly downloads

Added to index

2009-01-28

Total downloads

16 ( #239,842 of 1,934,791 )

Recent downloads (6 months)

1 ( #434,672 of 1,934,791 )

How can I increase my downloads?

My notes
Sign in to use this feature


Discussion
Start a new thread
Order:
There  are no threads in this forum
Nothing in this forum yet.