A Uniform Approach to Fundamental Sequences and Hierarchies

Abstract

Abstract. In this article we give a unifying approach to the theory of fundamental sequences and their related Hardy hierarchies of number-theoretic functions and we show the equivalence of the new approach with the classical one.Mathematics Subject Classification: 03D20, 03F15, 03E10.

Multiply-Recursive Upper Bounds with Higman’s Lemma

“…This defines an ordinal-indexed family of functions (M α ) α∈CNF(ω ω ω ) similar to some classical subrecursive hierarchies, with the added twist of the max operation-see (Buchholz et al, 1994;Moser and Weiermann, 2003) for somewhat similar hierarchies. This is a real issue and one cannot replace a "max α∈... {M α (x)}" with "M sup{α∈...} (x)" since M α is not always bounded by…”

“…We discuss in this subsection an alternative proof of Proposition 5.2 (with an additional hygienic condition on g), which relies on the work of Buchholz et al (1994) on alternative definitions of hierarchies, and in particular on their Theorem 4. The proof reuses some results given in Appendix B.4 (namely (52) and Proposition B.2), and the interplay between leanness and the predecessor function (Lemma B.1).…”

“…This makes the translation of bounds in terms ofH α into more "standard" hierarchies significantly harder (in fact Weiermann does not provide any)-our particular variations of the ideas present in the work of Buchholz et al (1994) might actually be of independent interest.…”

A classification of rapidly growing Ramsey functions

“…H 0 (n) := n, H α+1 (n) = H α (n + 1) and H α (n) := H α[n] (n + 1) if α is a limit. The theory of these functions is developed, for example, in [4], [8], [16]. …”

“…For further information on Ramsey theory the reader is referred e.g. to [9] and for further information on subrecursive hierarchies the reader may consult, for example, [4], [8], [15]. This paper is based in part on a modification of Buchholz' technical report [3].…”