In this paper we continue the investigations begun in [2] of equivalence classes of (denumerable) well-orderings under one-one partial recursive, order-preserving maps (recursive isotonisms). The numbering and notation are continued from [2] except that we now use lower case Greek letters for classical ordinals and upper case Roman letters for sets and we slightly modify the definition of (see §IX.3).

In section IX, after proving some lemmata which we use repeatedly in this paper, we establish the existence of Cantor Normal Forms for all co-ordinals less than a co-ordinal of the form WA (a fortiori for all co-ordinals less than a principal number for exponentiation).