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.

Let T 0?T 1 denote that each computable function, which is provable total in the first order theory T 0, is also provable total in the first order theory T 1. Te relation ? induces a degree structure on the sound finite Π2 extensions of EA (Elementary Arithmetic). This paper is devoted to the study of this structure. However we do not study the structure directly. Rather we define an isomorphic subrecursive degree structure <≤,?>, and then we study <≤,?> by (...) ubrecursive and computability-theoretic means. Furthermore, we introduce and investigate some operators on the degrees of <≤,?>. These operators corresponds to inferencerules in formal arithmetic. One operator corresponds to the Σ1 collection rule. Another operator corresponds to the Σ1 induction rule. (shrink)

It is well known that the structure of honest elementary degrees is a lattice with rather strong density properties. Let $\mbox{\bf a} \cup \mbox{\bf b}$ and $\mbox{\bf a} \cap \mbox{\bf b}$ denote respectively the join and the meet of the degrees $\mbox{\bf a}$ and $\mbox{\bf b}$ . This paper introduces a jump operator ( $\cdot'$ ) on the honest elementary degrees and defines canonical degrees $\mbox{\bf 0},\mbox{\bf 0}', \mbox{\bf 0}^{\prime \prime },\ldots$ and low and high degrees analogous to the corresponding (...) concepts for the Turing degrees. Among others, the following results about the structure of the honest elementary degrees are shown: There exist low degrees, and there exist degrees which are neither low nor high. Every degree above $\mbox{\bf 0}'$ is the jump of some degree, moreover, for every degree $\mbox{\bf c}$ above $\mbox{\bf 0}'$ there exist degrees $\mbox{\bf a},\mbox{\bf b}$ such that $\mbox{\bf c}=\mbox{\bf a} \cup \mbox{\bf b} = \mbox{{\bf a}}'=\mbox{\bf b}'$ . We have $\mbox{\bf a}'\cup \mbox{\bf b}' \leq (\mbox{\bf a}\cup\mbox{\bf b})'$ and $\mbox{\bf a}'\cap \mbox{\bf b}' \geq (\mbox{\bf a}\cap \mbox{\bf b})'$ . The jump operator is of course monotonic, i.e. $\mbox{\bf a}\leq\mbox{{\bf b}}\Rightarrow \mbox{\bf a}'\leq \mbox{\bf b}'$ . We prove that every situation compatible with $\mbox{\bf a}\leq\mbox{\bf b}\Rightarrow \mbox{\bf a}'\leq \mbox{\bf b}'$ is realized in the structure, e.g. we have incomparable degrees $\mbox{\bf a},\mbox{\bf b}$ such that $\mbox{\bf a}'<\mbox{\bf b}'$ and incomparable degrees $\mbox{\bf a},\mbox{\bf b}$ such that $\mbox{\bf a}' = \mbox{\bf b}'$ etcetera. We are able to prove all these results without the traditional recursion theoretic constructions. Our proof method relies on the fact that the growth of the functions in a degree is bounded. This technique also yields a very simple proof of an old result, namely that the structure is a lattice. (shrink)