For each of the following formulae ϕ(x), show why the set {x : ϕ(x)} does or does not exist:
(i) ∀y (x ∈ y).
My thinking here is that the empty set is a subset of all sets
(ii) ∃y (x ∈ y).
My thinking here is to use axiom of pairing to get y= {x,x}
(iii) ∀y (y ∈ x).
My thinking here is that there is no set of all sets
(iv) ∃y (y ∈ x).
My thinking here is that if x is the empty set it can have no members
Any help as to whether I'm on the right lines here would be greatly appreciated!!
I was wondering if using the subset axiom would be a better way of dealing with these?
Your ideas seem not aiming the original intention of the question.
Hint: Each of the given formulas is either valid for all $x$ or is false for all $x$.
We get $\{x:\Phi(x)\}$ as a set only in the latter case.