Mutual Contradiction of Two Self-Consistent Abstractions

Abstract

A vast class of abstractions are proved self-contradictory by Russell-type paradoxes in the sense that the negation of any one of them can be proved tautologically. On the other hand, there are a vast class of abstractions, each being self-consistent. A simple criterion for abstractions to be self-consistent (a sufficient condition) can be given. However, even a fairly restricted class of abstractions, each satisfying the criterion to be self-consistent, may contradict to each other.