Contradiction about the set of all sets

Question

How is it paradoxal that a set of all sets exists in set theory?

Russel's paradox is about the set of all sets that do not contain themselves cannot exist, that I understand.

But what about the set of all sets, in a unrestricted manner? Is it the fact that it contains itself (which may be what allows to define the paradoxal set of Russel's paradox) that is problematic?

It is "problematic" in connection with other intuitive assumptions (axioms) about sets. There are some set theories where the universal set is not problematic. — Mauro ALLEGRANZA, Oct 23 '23 at 14:08
In ZF, the Universe V being a set is easily shown to be contradictory with the Foundation Axiom. As the Foundation Axiom prevents any sets from containing themselves. — Michael Carey, Oct 24 '23 at 01:22

score 3 · Answer 1 · answered Oct 23 '23 at 20:10

3

The simplest answer is because in the standard axioms for set theory, one has the axiom of restricted compression, which says that for any set $S$ and predicate $\alpha$ the collection

$$ S' = \{x\in S|\alpha(x)\} $$

Forms a set.

If we assume the set of all sets exists, call it $\mathcal{S}$ then we can create the collection

$$ S = \{x\in \mathcal{S}|x\notin x\}$$

And this must be a set, but of course you know that this leads to a contradiction, so our original assumption that $\mathcal{S}$ exists is incorrect.

answered Oct 23 '23 at 20:10

Carlyle

3,865

much clearer, thanks a lot!! – Brahim Khalil Abid Oct 24 '23 at 05:26
I am failing to understand this a bit. We could also define S(the one inside the set definition) as any other set instead of the set of all sets and the result would be the same, no? – pdaranda661 Mar 14 '24 at 00:26

score 0 · Answer 2 · answered Oct 23 '23 at 16:40

Cantor Theorm is used to prove the inexistence of a set of all sets:

Another difficulty with the idea of a universal set concerns the power set of the set of all sets. Because this power set is a set of sets, it would necessarily be a subset of the set of all sets, provided that both exist. However, this conflicts with Cantor's theorem that the power set of any set (whether infinite or not) always has strictly higher cardinality than the set itself.

Contradiction about the set of all sets

2 Answers2