Classes

Abstract

We pointed out in the introduction that one objective of axiomatic set theory is to avoid the classical paradoxes. One such paradox, the Russell paradox, arose from the naive acceptance of the idea that given any property there exists a set whose elements are those objects having the given property, i.e., given a wff φ (x) containing no free variables other than x, there exists a set that contains all objects for which φ (x) holds and contains no object for which φ(x) does not hold. More formally $(\exists a)(\forall x)[x \in a \leftrightarrow \varphi (x)]$.