Deriving X ∉ X through an axiomatic ∊ relation
No claim of originality is offered. From a modern vantage point, we need not resort to a theory of types or an axiomatic ban on X ∊ X. We are not in the same position as Russell, Whitehead, Zermelo and other pioneers in wrestling with how to best ground mathematics in some overall system. The element or membership relation ∊ pairs an element with a set. We adopt the convention that the left-hand part of the pair is an element and the right-hand part is a set. So x ∊ X is ∊ . X is defined by the collection of x's with which it is paired, whether intensionally (by rule) or extensionally (by particulars). We must beware regarding the left part of the relation as a set A and the right part as a set B, making the relation a subset of the cross product set A X B. For then we face infinite regress. This I suppose could be handled by an infinite sequence of axioms. But better, I think, is to say that A and B are primitive sets (or collections), which are defined so as to not form a ...