Sometimes slow growing is fast growing

Abstract

The slow growing hierarchy is commonly defined as follows: G 0( x) = 0, G x−1 ( x) := G x ( x) + 1 and G λ ( x) := G λ[ x] ( x) where λ< ε 0 is a limit and ·[·]: ε 0∩ Lim × ω → ε 0 is a given assignment of fundamental sequences for the limits below ε 0. The first obvious question which is encountered when one looks at this definition is: How does this hierarchy depend on the choice of the underlying system of fundamental sequences? Of course, it is well known and easy to prove that for the standard assignment of fundamental sequence the hierarchy ( G x ) x< ε 0 is slow growing, i.e. each G x is majorized by a Kalmar elementary recursive function. It is shown in this paper that the slow growing hierarchy ( G x ) x< ε 0 — when it is defined with respect to the norm-based assignment of fundamental sequences which is defined in the article by Cichon (1992, pp. 173–193) — is actually fast growing, i.e. each PA-provably recursive function is eventually dominated by G x for some α< ε 0. The exact classification of this hierarchy, i.e. the problem whether it is slow or fast growing, has been unsolved since 1992. The somewhat unexpected result of this paper shows that the slow growing hierarchy is extremely sensitive with respect to the choice of the underlying system of fundamental sequences. The paper is essentially self-contained. Only little knowledge about ordinals less than ε 0 — like the existence of Cantor normal forms, etc. and the beginnings of subrecursive hierarchy theory as presented, for example, in the 1984 textbook of Rose — is assumed.