Set Theories: Their Philosophic Issues and Foundations

This paper traces the main philososophic issues and the foundations of Cantorian set theory and the contemporary cumulative hieararchy theory (c. h. t.). From a platonistic view point, we formulate two theories of c.h. t, that is, stage theory and level theory. The latter is characteristically stronger than the former. Axioms of Zermelo Fraenkel set theory, ZF, that can be deduced within two theories, are explained and left as logical exercises. We believe that, for a deep understanding of model-theoretic results in axiomatic set theory, we must capture fully the philosophy (and the picture) of the cumulative hieararchy