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$\exists y, x\forall z: y - x > z$

How come this mean: the difference between two number can be arbitrarily large?

Can we replace the > sign by =? and why?

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    That statement is false. You need to change the order of the quantifiers.2012-11-03

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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 $z$.

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    @Gladstone: But *the difference* doesn’t appear in either quantifier. It’s *for any specified amount of separation there are two numbers further apart than that*. I really think that any attempt to associate quantifiers to parts of speech will hurt more often than it helps.2012-11-04