Thread: a problem about set theory

1. a problem about set theory

Let A be a set,show that a "complement" of A does not exists, i.e. show that the set of all x does not belong to A does not exist.

how to prove it? give me some hints

2. Are you doing Zermelo-Fraenkel theory?

3. yes,this is a question in <<introduction to set theory>>, do you have any idea about it?

4. That doesn't really answer my question. You have naive set theory, Zermelo-Fraenkel set theory, von Neumann set theory, etc. In which kind of set theory are you working?

5. In Zermelo-Fraenkel theory, you would need to prove

$(\exists !B)(\forall x)(x\in B\iff x\not\in A),$

after which you could define complementation as follows:

$A^{c}=y\iff (\forall x)(x\in y\iff x\not\in A)\land y\;\text{is a set}.$

So you're thinking of $B$ as the universal set. Any ideas here?

6. In Z set theory (or its extensions) we will have previously proven:

~ExAy y in x

Also, we we will have previously proven that for any two sets S and T there exists the union of them (SuT) such that Ay(y in SuT <-> (y in S or y in T)).

Then, toward a contradiction, suppose Az(z in C <-> z not in A).

Then Ay y in AuC.