What's the difference between the two statements
$$ \forall x \exists y:P(x, y) $$
and
$$ \forall x: P(x, y) $$
My understanding the first means $ y $ does not have to be the same for every $ x $, while the second means one $ y $ satisfies $ P $ for every $ x $.
For example, can we write $ \forall n \in Z \exists x \in R : n = x $ as follows
$$ \forall n \in Z (x \in R \, and \, n = x) $$
Thanks
On
Consider $A:=~\forall x\exists y~P(x,y)$ and $B:=~\forall x~P(x,y)$.
The entity $y$ is bound within $A$, where as it occurs free within $B$.
$A$ is claiming that each $x$ has its own $y$ which makes the predicate hold. That may or not be the same value for each $x$, but there is some such value for each $x$.
$B$ is claiming that each $x$ makes the predictate hold for some constant $y$.
For example, can we write $ \forall n \in Z~\exists x \in R : n = x $ as follows
$$\forall n \in Z (x \in R \, and \, n = x)$$
No. They are not equivalent statements.
The first says that every integer has a real number of equal value.
The second statement says that $x$ is a real number and is simultaneously equal to every integer.
The first says: "For whatever x you give me, I can always find a y such that the statement is true."
The second says: "For whatever x you give me, the statement is true." In this case, the predicate doesn't depend on y. So you don't need P(x,y); just P(x) is ok.