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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. –  wj32 Nov 3 '12 at 22:48

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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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the first quantifier(s) = subject of the verb and the last quantifier(s) = object? –  Gladstone Asder Nov 3 '12 at 22:55
    
@Gladstone: No; what would the verb be? $\exists x$ contains the copular verb, and $\forall x$ is a prepositional phrase. –  Brian M. Scott Nov 3 '12 at 23:00
    
@Brian I think the OP may be referring to the quantified variables, having learned that the first quantified object(s) refers to the subject of the statement's verb/action, with the second quantified object(s) referring to the object of the statement's verb/action. Just speculating; please clarify Gladstone. –  amWhy Nov 3 '12 at 23:26
    
Gladstone See my comment above. Given @Brian's translation - yes, one could replace the ">" symbol with the "=" symbol. Then it would state, "for each $z$ there are numbers $x$ and $y$ whose difference equals $z$. –  amWhy Nov 3 '12 at 23:30
    
the difference (subject) of two numbers (second quantifiers) can (verb) be arbitrarily large. –  Gladstone Asder Nov 4 '12 at 0:45

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