In naieve set theory, you quickly run into existence trouble if you try to do meta-things things like take "the set of all sets". For example, does the set of all sets that don't contain themselves contain itself? (see, Russell)
On the other hand, people have no trouble casually talking about the "category of all categories", etc. How do we know there isn't a contradiction lurking somewhere here - especially since many books define categories in terms of sets?
