Set theory INC^# based on intuitionistic logic with restricted modus ponens rule

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 ϕmay be any formula in whichy does not occur freely: yxx y ϕ. 1.1 Russell’s paradox shows that the instance obtained by letting ϕ be x x isinconsistent in classical logic. One response to the paradox is to restrict naive setcomprehension by ruling out this and other problematic instances: only for each of somespecial conditions is it claimed there is a set containing all and only the sets satisfyingthe condition.