$\exists y, x\forall z: y - x > z$
How come this mean: the difference between two number can be arbitrarily large?
Can we replace the > sign by =? and why?
$\exists y, x\forall z: y - x > z$
How come this mean: the difference between two number can be arbitrarily large?
Can we replace the > sign by =? and why?
The statement does not say that the difference between two numbers can be arbitrarily large: it says that there are two numbers, $x$ and $y$, whose difference is bigger than any number. This is of course false, since whatever $y-x$ is, it’s a number, and it’s certainly not bigger than itself.
To say that there are numbers with arbitrarily large differences, you must reverse the quantifiers:
$\forall z~\exists x,y~(y-x>z)\;.$
In words: for each $z$ there are numbers $x$ and $y$ whose difference is greater than $z$.