Morse-Kelley Class Comprehension axiom and Russell's paradox

[nomadreid]

As I understand the ZFC solution to Russell's paradox, since {x|x$\notin$x} must be {x|x$\notin$x}$\cap$S for some set S, the paradox goes away, but in Morse-Kelley, if I understand Class Comprehension correctly, although again there must be some M such that {x|x$\notin$x}$\cap$M, this M may be a proper class, which no longer is as limiting as the ZFC version, and hence no longer gives the same solution. So either I am going wrong somewhere, or MK solves Russell's Paradox in a different way. I would be grateful for enlightenment. Thanks.

 

[lugita15]

The Class comprehension Axiom schema states that for any formula phi (with one free variable), there exists a class of all sets satisfying phi. It does not say that there exists a class of all classes satisfying phi. So there's a (proper) class of all sets which do not contain themselves, but there's no class of all classes which do not contain themselves. So Russell's paradox doesn't occur.