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Some Interesting Connections between the Slow Growing Hierarchy and the Ackermann Function

Andreas Weiermann
The Journal of Symbolic Logic
Vol. 66, No. 2 (Jun., 2001), pp. 609-628
DOI: 10.2307/2695032
Stable URL: http://www.jstor.org/stable/2695032
Page Count: 20
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Some Interesting Connections between the Slow Growing Hierarchy and the Ackermann Function
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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 Γ0. Let T be the set-theoretic ordinal notation system for Γ0 and Ttree the tree ordinal representation for Γ. It is shown in this paper that (Gα)α ∈ T matches up with the class of functions which are elementary recursive in the Ackermann function as does (Gα)α ∈ Ttree (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 ε0) and prove that (Gα)α ∈ Ttree* still matches up with (Gα)α ∈ Ttree but (Gα)α ∈ T* now consists of elementary recursive functions only. We discuss the relationship between these results and the Γ0-based termination proof for the standard rewrite system for the Ackermann function.

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