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I have the following formula: $\exists y \forall x ((x \geq y) \wedge \neg(y \geq x))$

This essentially boils down to: $\exists y \forall x (x > y)$

I have to check whether this applies to certain universes. But my result depends on whether I can assume that x can also equal y, because in that case the condition will never be true (how could a number be smaller than and at the same time equal to another number?).

Can you give me a hint?

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@Franz: "For every $x$" means for every $x$. Period. But also keep in mind that when checking if it applies to "certain universes", the meaning of the variables and the symbols will depend on the universe. Not every object in the world is a number, and not every interpretation of $\geq$ is the same. – Arturo Magidin Nov 2 '10 at 2:12
Which means my assumption that x can also be y is valid, right? Just makin' sure... ;) – Franz Nov 2 '10 at 2:15
@Franz: Yes; there are no restrictions on the quantification. But the meaning of the symbols $\geq$ and $\gt$ (and even whether you can "boil down" the condition to the conjunction to $x\gt y$) will depend on the universe in which you are checking. Don't be misled into thinking that the relation that $\geq$ represents in every universe necessarily has the same properties as the relation that it represents in the "usual" one. – Arturo Magidin Nov 2 '10 at 2:18
Yup, makes sense. – Franz Nov 2 '10 at 2:19
I've validated my transformations for my universes. Thanks for the answer. I'd vote you up, if you'd make it an answer ;) – Franz Nov 2 '10 at 2:20
up vote 3 down vote accepted

I'll put the answer here so the question can be marked as answered.

"For every $x$" means for every $x$, period. So, yes, there are no restrictions on $x$ not being equal to $y$; any such restrictions would have to be given as predicates (a clause $x\neq y$).

I'll note (as I did in the comments) that whether the transformation from $(x\geq y) \wedge \neg(y\geq x)$ is equivalent to $x\gt y$ (and whether $\gt$ is areflexive) depends on the interpretation of $\geq$ and $\gt$ on the universe in question. The specification of the universe should not be only the set, but also the meaning of the relational and functional symbols like $\geq$. Only if there is some working convention can you simply assume that the meaning will be "the usual one".

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