Arch. math. Logik 18 (1976), 47--53 BUILT-UP SYSTEMS OF FUNDAMENTAL SEQUENCES AND HIERARCHIES OF NUMBER-THEORETIC FUNCTIONS* By Diana Schmidt In [2], L6b and Wainer introduced a general procedure for generating hierarchies which can be used for classifying a wide variety of classes of number-theoretic functions. Similar hierarchies were also studied by Robbin [3], Rose [4] and Schwichtenberg [5]. The basic ingredient in all these cases was a transfinite sequence (F~)~ A of number-theoretic functions, indexed by an initial segment A of the second number class and defined inductively as follows: F0 = some strictly monotonic function; F~÷I is defined from F~ so that F~÷I is strictly monotonic if F, is and grows faster than F~; Fa=Fzx (x) if 2 is a limit ordinal, where (2x)~o, is a fundamental sequence for 2. Clearly, this sort of definition depends essentially on the fundamental sequences chosen. In [2] it was shown that, for the ordinals below eo, fundamental sequences can be chosen so that all the F~'s are strictly monotonic, and the following questions were raised (question B on p. 113): 1. Is there an upper bound on the ordinals fl for which there are fundamental sequences for all limit ordinals < fl such that F~ is strictly monotonic for each ~</~7 2. What conditions must the fundamental sequences to all limit ordinals < fl satisfy in order to ensure that F~ is strictly monotonic for each c~ < fl? In § 1 we introduce the concept of built-up systems of fundamental sequences. We answer question 2 by showing that, for a wide variety of sequences (F~), each F~ is strictly monotonic (and, in fact, the F~'s have an additional "mono- tonicity" property) if the system of fundamental sequences used is built-up. In § 2 we give a negative answer to question 1 by showing that there is a built-up system of fundamental sequences for every initial segment of the second number class. In § 3, as an illustration, we give a system of fundamental sequences for all ordinals below Feferman's F o and sketch a proof that the system is built-up. * Eingegangen am26.2.1975.