CONSTRUCTIVE VERSIONS OF ORDINAL NUMBER CLASSES

In article 1 we give certain terminology and background related to the classial theory of ordinals. In article 2 we discuss S1, S3 and the general notion of system of notations. In article 3 we discuss certain peculiarities of the Addison-Kleene extension of S3. In article 4 we extend S1. In article 5 we extend S3 in continuation of the Addison-Kleene extension. In article 6 we show that the system of article 4 is in both two-function-quantifier forms. In artile 7 we use this result to show that iteration of the hyperjump over the extension of S1 does not lead beyond sets in both twofunction--quantifier forms. In artcle 8 we obtain the reults of article 6-7 for the extension of S3. In article 9 we raise certain open questions and discuss possible directions for further research. Author