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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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    the first quantifier(s) = subject of the verb and the last quantifier(s) = object?2012-11-03
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    @Gladstone: No; what would the verb be? $\exists x$ contains the copular verb, and $\forall x$ is a prepositional phrase.2012-11-03
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    @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.2012-11-03
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    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 $z$.2012-11-03
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    the difference (subject) of two numbers (second quantifiers) can (verb) be arbitrarily large.2012-11-04
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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