Set Theory Question -- Which one is correct?

[Heisenberg7]

I can't decide which one is better to use. I know for a fact that the second one is correct, but I would like to know if I can use the first one too. Which one would you use?
$$\forall x \in \mathbb{Z} (\exists y \in \mathbb{Z} : x > y)$$
Or
$$\forall x \in \mathbb{Z} (\exists y \in \mathbb{Z} , x > y)$$
Is there a different way to group these expressions?

 

[FactChecker]

I prefer the first version. The ':' means only 'such that' to me and is less ambiguous than ','. But I think this is just a matter of taste.

 

[fresh_42]

I sometimes use
$$
\substack{\forall\\x\in \mathbb{Z}}\quad\substack{\exists\\y\in \mathbb{Z}}\quad x>y
$$
to avoid exactly this question. The comma notation is quite unusual. Another parenthesis would be better
$$
\left(\forall\;x\in \mathbb{Z}\right)\;\left(\exists\;y\in \mathbb{Z}\right)\;x>y
$$
A textbook on logic normally doesn't use any of them. Logic has its own notations like $\dashv.$ I once saw how Russell dealt with set theory. It was barely readable.

 

[SSequence]

I would write:
$\forall x \in \mathbb{Z} \, \exists y \in \mathbb{Z} \,\, [ x>y ]$
$\forall x \in \mathbb{Z} \, \exists y \in \mathbb{Z} \,\, ( x>y )$

It is also OK to write something like:
$\forall x \in \mathbb{Z} \, [\, \exists y \in \mathbb{Z} \,\, ( x>y ) ]$
$\forall x \in \mathbb{Z} \, (\, \exists y \in \mathbb{Z} \,\, ( x>y ) \,)$

Since the expression $x>y$ is really short, it seems to me that it should also be fine to write:
$\forall x \in \mathbb{Z} \, \exists y \in \mathbb{Z} \,\,\,\,\, x>y$


I haven't seen a symbol like $:$ used in logical statements (but maybe it is used commonly and I don't know it). Normally I think the symbol $:$ is widely used [in place of $|$ ] in defining specific sets. For example, the set of even integers $E \subseteq \mathbb{Z}$:
$E=\{x \in \mathbb{Z}: \exists k \in \mathbb{Z} (x=2k) \}$

 

[SSequence]

Not sure I should bump the thread for a small point. But I think I kind of get how the symbol $:$ seems pretty reasonable for use in logical statements (or representing predicates etc.). Though for longer expressions, personally I think it might be easier to use brackets (at least for me).

Regarding the quesion in OP, I think the original expression (the first one) as written is fine [though this is also mentioned in the very first reply]. I would say it would also be OK to write:
$\forall x \in \mathbb{Z} \, \exists y \in \mathbb{Z} \,\,:\,\, x>y$