In this book set theory INC# based on intuitionistic logic with restricted modus ponens rule is proposed. It proved that intuitionistic logic with restricted modus ponens rule can to safe Cantor naive set theory from a triviality Considering only pure sets, the naive set comprehension principle says, for anycondition, that there is a set containing all and only the sets satisfying this condition. Infirst-order logic, this can be formulated as the following schematic principle, where may be any formula in whichy does not occur freely: yxx y . 1.1 Russell's paradox shows that the instance...