P.R.-regulated systems of notation and the subrecursive hierarchy equivalence property

Abstract:

We can attempt to extend the Grzegorczyk Hierarchy transfinitely by defining a sequence of functions indexed by the elements of a system of notation $\mathcal {S}$, using either iteration (majorization) or enumeration techniques to define the functions. (The hierarchy is then the sequence of classes of functions elementary in the functions of the sequence of functions.) In this paper we consider two sequences ${\{ {F_s}\} _{s \in \mathcal {S}}}$ and ${\{ {G_s}\} _{s \in \mathcal {S}}}$ defined by iteration and a sequence ${\{ {E_s}\} _{s \in \mathcal {S}}}$ defined by enumeration; the corresponding hierarchies are $\{ {\mathcal {F}_s}\} ,\{ {\mathcal {G}_s}\} ,\{ \mathcal {E}{_s}\}$. We say that $\mathcal {S}$ has the subrecursive hierarchy equivalence property if these two conditions hold: (I) ${\mathcal {E}_s} = {\mathcal {F}_s} = {\mathcal {G}_s}$ for all $s \in \mathcal {S}$; (II) ${\mathcal {E}_s} = {\mathcal {E}_t}$ for all $s,t \in \mathcal {S}$ such that $|s| = |t|(|s|$ is the ordinal denoted by s). We show that a certain type of system of notation, called p.r.-regulated, has the subrecursive hierarchy equivalence property. We present a nontrivial example of such a system of notation, based on Schütte’s Klammersymbols. The resulting hierarchy extends those previously in print, which used the so-called standard fundamental sequences for limits $< {\varepsilon _0}$.