=’ not equal to, ϵ’ not a “member” of

1) x=y: undefined except x=x (axiom), x=’x false (logic).

2) xϵy: Principal Primitive Undefined concept*

3) x and y undefined.

Theorem: xϵx iff x=x.

Proof: x and x are identical so there can be no other relation between them.

Theorem: xϵ’x iff x=’x.

Proof:

If xϵ’x and x=x then xϵx. Therefore xϵ’x → x=’x

If x=’x and xϵx then x=x. Therefore x=’x → xϵ’x

Conclusion: xϵx is x=x. xϵ’x is x=’x which is a false statement.

Ex: Suppes obtains Russel’s Paradox by using the function φ(x) is (xϵ’x) which is (x=’x) which is false and inadmissible in the”axiom of abstraction,” Suppes pg 6.

Any derivation which leads to Russel’s Paradox in the form “(xϵx) iff (xϵ’x)” should be “(x=x) iff (x=’x)” where (x=’x) is false and inadmissible so there is no paradox.

Patrick SuppesAxiomatic Set Theory, pg 5. Dover, 1972

*Paul HalmosNaive Set Theory, pg 2, Van Nostrand, 1960

Willard Quine Set Theory and its Logic, Harvard, 1969

Personally, I can bareley scratch out a few nuggets from the first few pages of these references.