Abstract

For α < ε0, Nα denotes the number of occurrences of ω in the Cantor normal form of α with the base ω. For a binary number-theoretic function f let B denote the length n of the longest descending chain of ordinals <ε0 such that for all i < n, Nαi ≤ f . Simpson [2] called ε0 as slowly well ordered when B is totally defined for f = K · . Let |n| denote the binary length of the natural number n, and |n|k the k-times iterate of the logarithmic function |n|. For a unary function h let L denote the function B ) with h0 = K + |i| · |i|h. In this note we show, inspired from Weiermann [4], that, under a reasonable condition on h, the functionL is primitive recursive in the inverse h–1 and vice versa