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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: Note that to interpret the transformations, you must specify more than just the set, you must also specify the meaning of the symbol $\geq$. I've repeated my comment as an answer so the question can be marked as answered.2010-11-02

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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".