It’s ill-defined in the sense that it doesn’t uniquely define the set. There are at least two different sets that D could be (one containing it and one not containing it), hence the expression doesn’t denote a well-defined set. (*)

The axioms of ZF do not allow to form that expression, so the set doesn’t exist in ZF.

(*) This is from a universist view. In a pluralist view, one wouldn’t say that the fact of the matter of whether D contains itself or not is independent from naive set theory, and that there are set universes where it is the case and others where it isn’t. But I would hold that naive set theory starts from a universist view.

... and, as such, it doesn't contain itself!

> category theory can use set theory, but does not depend on it

But aren't, say, the morphisms between two objects necessarily a set (termed "hom-set")?

> In one system, a set can contain itself

But doesn't this lead to a contradiction (or to making the system of little use)?

> 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. ””

TFA is right. Parent comment is not really rebutting in any meaningful way. Your rebuttal makes less sense.

>> "a term can have only one type... Due to this law, types cannot contain themselves"

> types are not sets and sets are not types therefore it makes no sense to link these two statements/judgements in the way you are linking them

> in the way you are linking them

This is what the text says, not me.

Except such set is empty and thus does not contain itself.