Abstract
From a set theoretical system with element relation ∋ and rank relation ⊏ a convenient axiom system of set theory is developed for mathematicians (specially for categorists) by postulating a new axiom of universe. There exists the class V of all usual sets (and urelements) and above V there are hyper classes as auxiliary totalities, arranged in\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega } ^2\) stages. With classes and hyper classes one can work nearly as with sets. But classes and hyper classes differ from sets because there is no scheme of replacement for them. From the basic concepts and the axioms of the system of axioms one gets the present mathematical concepts and results by logical composition or logical deduction respectively.
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