Natural numbers in discrete maths problem

Question

Is true to say that this $$ \forall x \forall y P(x,y) $$ where $$ P(x,y)= x \le y $$ is not correct for the natural numbers. Because there could be an x that is bigger than an y . Or does it mean that for every natural number x , there is a natural number y that will always be bigger? I know that the second one could be written like $$ \forall x \exists y P(x,y) $$ , but for some reason I can't understand what the first one means

Answer 1

$$\forall x\forall y P(x,y)$$

where

$$P(x,y) = x \le y$$

means that "every number is smaller or equal to every number" ... which indeed is false.

Answer 2

Correct.

We have that:

$\mathbb N \nvDash \forall x \forall y \ (x \le y)$,

i.e. the sentence $\forall x \forall y \ (x \le y)$ is not satisfied in the strucute of natural numbers, because it is not true e.g. that $(4 \le 2)$.

Thus, "fixing" e.g. $4$ as value for $x$, it is not true that $\forall y \ (4 \le y)$.