Mathematics of Totalities: an alternative to mathematics of sets

Abstract

I dare say, a set is contranatural if some pair of its elements has a nonempty intersection. So, we consider only collections of disjoint nonempty elements and call them totalities. We propose the propositional logicTT, where a proposition letters some totality. The proposition is true if it letters the greatest totality. There are five connectives inTT: ∧, ∨, ∩, ⌉, # and the last is called plexus. The truth of σ # π means that any element of the totality σ has a nonempty intersection with any element of the totality π. An imbeddingG of the classical predicate logicCPL inTT is defined. A formulaf ofCPL is a classical tautology if and only ifG(f) is always true inTT. So, mathematics may be expounded inTT, without quantifiers.