Model given for a formula The formula in question
$$ \forall x \exists y [\ \ P(x,y) \rightarrow \exists z \forall u (\lnot P(u,z))\ \ ] $$
according to my text-book, has the following model
$ \text{Domain} = \mathbb{N} $
$ \lvert P\rvert = \{ (n,m): n,m \ \in \mathbb{N},\ \  n =2m \} $
According to its description, I'd assume the set $ \lvert P\rvert $ to have the following elements:
$$  \lvert P\rvert = \{ \\(0, 0) \\ (2, 1) \\ (4,2) \\ (6,3) \\ ... \} $$
So there has to exist a $z$ , which applied to any possible $u$ is not part of set  $\lvert P\rvert$ so that $\exists z \forall u (\lnot P(u,z))$ this condition can hold.
But inside $\lvert P\rvert$ every possible $z \in \mathbb{N}$ is inside the set, meaning there doesn't exist a $z$ in common to every $u$ which could satisfy the model.
Am I missing something? 

Your statement says $\forall x\, \exists y\, \big( P(x,y) \longrightarrow \cdots\cdots\big).$
For every value of $x,$ there is some value of $y$ such that $x$ is not $2$ times $y.$ Thus for that value of $y,$ the relation $P(x,y)$ is false. Since it is false, the statement $P(x,y)\longrightarrow\cdots\cdots$ is true.
