I think you're taking this point a little too forcefully; this is meant to informally motivate Russell's paradox, in my reading - which is exactly the title of the section you're referencing.
The point here is a little more subtle; category theory doesn't necessarily rely on sets; the definitions of categories that you often see (involving sets of objects and sets of morphisms) is more axiomatically forceful than the more general definition, which uses the notion of classes; category theory can use set theory, but does not depend on it.
The point here is that type theory offers just such another way to design in an avoidance of Russell's paradox.
You might also want to read about e.g. Grothendieck universes - they're quite relevant here.
> Yes -- in set theory sets can contain themselves
Which set theory? ZFC doesn't permit this.
Non-well-founded set theories are so non-standard that I think it's wrong, or at least misleading, to claim that unqualified "set theory" permits this.
> Yes -- in set theory sets can contain themselves
Hrbacek and Jech would like a word. It is very much not the case that in standard axiomatic set theory sets can contain themselves, precisely because this leads to things like Russell’s paradox. Sets containing themselves is generally prevented by the axiom of regularity. (Every non-empty set S contains an element wihch is disjoint from S) https://en.wikipedia.org/wiki/Axiom_of_regularity
> types are not sets and sets are not types
This is also not true. All types can be expressed as sets but not all sets are types in the standard definitions.
to justify my claim with an excerpt from the article:
““
What is type theory
“Every propositional function φ(x)—so it is contended—has, in addition to its range of truth, a range of significance, i.e. a range within which x must lie if φ(x) is to be a proposition at all, whether true or false. This is the first point in the theory of types; the second point is that ranges of significance form types, i.e. if x belongs to the range of significance of φ(x), then there is a class of objects, the type of x, all of which must also belong to the range of significance of φ(x)” — Bertrand Russell - Principles of Mathematics
In the last section, we almost fell in the trap of explaining types as something that are “like sets, but… “ (e.g. they are like sets, but a term can only be a member of one type). However, while it may be technically true, any such explanation would not be at all appropriate, as, while types started as alternative to sets, they actually ended up being quite different. So, thinking in terms of sets won’t get you far. Indeed, if we take the proverbial set theorist from the previous section, and ask them about types, their truthful response would have to be:
“Have you seen a set? Well, it has nothing to do with it.” [<=== important bit]
So let’s see how we define a type theory in its own right.
””
The modern formulation of functions as sets doesn’t require type theory but is entirely congruent with Russell’s definition, just much less cumbersome. In this view, φ is a relation on the set (D X C) where D and C are the domain and codomain of the function (which he calls the “range of significance of x” and the “range of significance of φ(x)” respectively). So since he’s talking about propositional functions, here C is the set {true, false} and D is all the things that are like whatever x is ie the set {x’: x’ is of the same type as x}.
Now a relation is just a particular type of predicate (ie it too is a set) so here we have x ~ y if φ(x) = y for all (x,y) in (D X C).
Notice here both the propositional function and the type are sets.
The point here is a little more subtle; category theory doesn't necessarily rely on sets; the definitions of categories that you often see (involving sets of objects and sets of morphisms) is more axiomatically forceful than the more general definition, which uses the notion of classes; category theory can use set theory, but does not depend on it.
The point here is that type theory offers just such another way to design in an avoidance of Russell's paradox.
You might also want to read about e.g. Grothendieck universes - they're quite relevant here.