Nested Quantifiers and Equivalences I'm a little confused with nested quantifiers when it comes to comparing two statements. I have this for example:
Assume: $$ P(x,y,z) : x + y = z $$
$$\forall x \forall y \exists z : P(x,y,z)$$
The domain is real numbers for all variables. Now I know the above statement evaluates to true since adding any two real numbers, you'll always have a value $z$  for which it is true.
Now the confusion arises with the following:
$$\exists z \forall x \forall y : P(x,y,z)$$
The way I read this: "There is a real number $z$ for every real number $x$ and for every real  number $y$". This statement is False but don't see why... wouldn't there be a value for $z$ for which this is true? I'm stuck on how to interpret these. They sound so similar yet I cannot see the difference.

The second statement $$\exists z \forall x \forall y \;P(x, y, z)$$ is false because there is no ONE $z \in \mathbb R$ for which $P(x, y, z)$ is true for every possible pair of reals $(x, y)$. 
I.e., there is no ONE $z \in \mathbb R$ such that $P(x, y, z)$ is true no matter what the values of $x$ and $y$.
Is there any $z$ for which $1 + 2 = z$ and $2 + 3 = z$? If there were such a $z$, we'd have $1+2 = 2 + 3$, which is absurd.
