Postscript to “built-up systems of fundamental sequences and hierarchies of number-theoretic functions”

We refer the reader to [2] for all details and definitions. In [2] we had answered a question of L f b and Wainer by showing that: a) if an assignment of fundamental sequences to an initial segment A of the second number class is built-up, then F~ is strictly monotonic for each e~ A ; b) there is a built-up assignment of fundamental sequences for every proper initial segment of the second number class. However, we had left unanswered the question whether there is a built-up assignment of fundamental sequences for the whole of the second number class. We have since noticed that the notion "built-up" coincides with a notion studied in Bachmann [1], and thus a theorem of Bachmann [ I ] answers the above question negatively. The purpose of this note is to prove the equivalence of Bachmann's notion with ours. We also give Bachmann's proof of the negative answer and note a further property of built-up assignments of fundamental sequences (see Remark).