I am reading chapter 2 of Elements of set theory by Herbert Enderton and I have a confusion.
Can we contruct a set from subset axiom of ZF set theory, such that the set of all sets which does not belongs to itself. I think its true ( $\{ x \mid x \text { does not belongs to itself} \}$ ) and if it can be constructed than such a set should exist as we are constructing the set from subset axiom.
But I have also heard that ZF set theory avoids Russell's paradox through its axioms.
Please explain, where am I going wrong. Thanks. Forgive my latex as I am typing this via phone.
Where you are going wrong is that the subset axiom requires that you start with a set. If there were such a thing as the set of all sets, you could take the subset that are not members of themselves, and get the problem of Russell's paradox. But first you would have to show that there is a set of all sets, which you can't do in ZF. The subset axiom doesn't allow you to construct the set of all objects that have property P. It allows you to construct the set of all elements in set X that have property P.