Any ideas on how to do this one?

Let A be a set. Show that a complement of A does not exist. So I need to show that there isn't a set of all x not in A.

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- Sep 10th 2009, 08:48 AMspaceship42Axioms
Any ideas on how to do this one?

Let A be a set. Show that a complement of A does not exist. So I need to show that there isn't a set of all x not in A. - Sep 10th 2009, 08:55 AMPlato
- Sep 10th 2009, 08:57 AMspaceship42
- Sep 10th 2009, 09:13 AMPlato
- Sep 10th 2009, 11:15 AMspaceship42
Oh, I'm sorry. I forgot that I'd called this thread "Axioms". This problem is about sets, not axioms. I mislabeled it accidentally.

- Sep 10th 2009, 11:28 AMTaluivren
Hi,

$\displaystyle A$ is a set and $\displaystyle B=\{x:\, x \not \in A\}$ ?

Assuming axioms of ZF set theory, $\displaystyle B$ can't be a set: If $\displaystyle B$ were a set, then since $\displaystyle A$ is a set, axiom of pairing gives that $\displaystyle \{A,B\}$ is a set. Axiom of union then tells us that that there exist a set $\displaystyle C$ whose elements are elements of $\displaystyle A$ and elements of $\displaystyle B$. But from the definition of $\displaystyle B$ we get that $\displaystyle C$ is the universal class of all sets, which is a proper class, i.e. is not a set. This is a contradiction, thus $\displaystyle B$ can't be a set.

Can you see why the class of all sets is not a set?