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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?

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  • $\begingroup$ Who cares what you think? Take your thinking and use it to create a proof or a counter example. $\endgroup$ – William Elliot Apr 14 '18 at 22:13
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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.

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