I was reading this document on axiomatic set theory, and on page 4, it defines $\exists \{x:P\}$ as an alternative notation for $\exists y \forall x P$. Since the first notation looks very much like a predicate, I was wondering if it was possible to treat the existential quantifier as a predicate (at least in set theory?).
For example, suppose $E(s)$ is the predicate "the set $s$ exists". This could be equivalent to $\exists xP(x)$, where $P(x)$ is the property of the set. This in turn could be equivalent to $\exists x \forall y [Q(y) \rightarrow (y \in x)]$, where $Q(y)$ is the property that the elements of the set must have.
To make things concrete:
$E(\text{positive even integers})$ $\iff$ "the set of positive even integers exists"
$\exists xP_{\text{positive even integers}}(x)$ $\iff$ "there is a set $x$ such that the statement '$x$ is the set of positive even integers' is true"
$\exists x \forall y [Q_{\text{positive even integer}}(y) \rightarrow (y \in x)]$ $\iff$ "there is a set $x$ such that for all sets $y$, the statement 'if $y$ is an even positive integer, then $y$ is in $x$' is true"
If the above is possible, then $\exists s E(s)$ would mean "some set exists". Can this be expressed in the other two forms?
I apologize if this is nonsense, but I'm just curious.
I think you are right about your understanding of predicates. I think in the notation P is not a predicate, instead it is a Property. There exists a set y such that for all sets x, there is a property P which holds about x and y.