Talk:Category of small categories

Question on Russell's paradox
Russell's paradox (a collection of a certain kind, of all such collections that do not contain themselves as members) seems to require a "NOT" or "exclusion" or "subtraction" operation, as in complex propositions of the form "(1st simpler proposition) but NOT (2nd simpler proposition)", or set-subtraction, conventionally written A \ B.

This is why there cannot be a set of all sets.

But categories do not, AFAIK, require that such an exclusion constructor exist. As such, categories should therefore be immune to Russell's paradox, shouldn't they? That is, the category of ALL categories (not just small ones) and functors between them would be a member of itself, without engendering any contradictions.

(Sociological disproof: If they were immune, then people would not have invented 2-Cats. So they must not be immune! ;-)

Can anyone clarify? Jmacwiki (talk) 22:31, 26 September 2012 (UTC)