would a set of all countable sets have any paradoxical properties?

Question

I recently talked with a friend about set theory and he mentioned "set of all countable sets". I think that such set does not exist (just like "set of all sets" does not exist) and I would like to explain to him why I think so. I could just ask him to prove, using axioms of set theory, that it exists - and reject the existence of this set until its existence is proven. However, I would prefer to show that assumption of its existence would lead to some paradoxes. So, would a set of all countable sets have any paradoxical properties?

Answer 1

As @noah-schweber mentioned in comments, supposing X were the set of all countable sets, let's consider ⋃X (a union of all elements of X). Let's notice that each set is a member of a countable set (for instance, a member of one-element set for which it is the only element). So ⋃X would contain - as its subset - a set of all sets, but we know that the set of all sets does not exist, so the set of all countable sets also does not exist.