Is a set containing itself already a paradox?

Question

This is inspired by Russel's paradox stating there is not set of all sets. It uses the presupposition that set can contain itself. However, this already seems paradoxical.

Suppose a set A = {}. Then the smallest set containing it will be {{}}. Let's call it B. Since {} != {{}}, we can say that A != B. Namely, we can say that {} does not belong to {}. Therefore there is no set containing itself. According to this Russel's paradox itself is indefinable, since any set will not contain itself.

Of course, maybe in naive set theory it was presupposed that a set belongs to itself, but then we should say {{}} = {} and, for example, {a, {a}} = {a}.

Therefore, I can't understand how is Russel's paradox really a paradox if a set containing itself isn't. Am I missing something? Is this issue recognized and if so how is it dealt with?

Answer 1

Russell's paradox arises within naïve set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself. Hence the paradox.

The "root" of the paradox is the so-called unrestriceted Comprehension Principle of naïve set theory:

for every property φ(x) expressible in the language, there is the set { x: φ(x)} of all and only those objects satisfying that property.

The paradox arises considering as φ(x) the property “~(x ∈ x)”.

Zermelo's solution to the paradox is built on the replacement of Comprehension principle with the Axiom schema of specification:

in order to assert the existence of the set B satisfying the property φ(x), we have to "separate" it from an already existsing set A.

How does Zermelo's theory avoid Russell’s paradox?

Assuming the existence of V – the whole universe of sets – and let φ be x ∉ x, a contradiction again appears to arise. But in this case, all the contradiction shows is that V is not a set.

All the contradiction shows is that “V” is an empty name, i.e., that it has no reference, that it does not exist.

In conclusion, the non-existence of the set of all sets is a feature of Zermelo-Fraenkel Set Theory.

There are Alternative Axiomatic Set Theories, like Quine's New Foundations, where the universal set V exists.

The impossibility of "circular" sets, like e.g. x ∈ x, is due to the Axiom of regularity.

The axiom implies that no set is an element of itself, and that there is no infinite sequence {a(n)} such that a(i+1) is an element of a(i).

Ther are other axiomatic set theories that are non-well-founded, i.e. that allow sets to contain themselves and otherwise violate the rule of well-foundedness.

---

Conclusion : a set containing itself is not per se a paradox. Paradoxes can arise in cooperation with other basic assumptions regarding sets.

The universal set is an example of "set containing itself" and its conception is quite natural.

Answer 2

For any binary relation R (not just "is an element of"), it is trivial to prove, using the ordinary rules of logic, that:

~Ex: Ay: [yRx <=> ~yRy]

(Assume the contrary and obtain a contradiction in 1 or 2 lines.)

So, the root of the problem was not sets being elements of themselves, whatever that may mean. The problem was that early attempts to axiomatize set theory allowed you to prove that, contrary to the above mentioned rules of logic:

Ex: Ay: [y in x <=> ~ y in y]

This "bug" was fixed by various means in subsequent versions of the axioms of set theory (or logic in some cases) that would, in one way or another, disallow this theorem.