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Well, i try to found an example of sentence $\Psi$ which satisfiable by any finite models, but there exist some infinite model that doesn't satisfies it.
We can choose any language we want, no restrictions.

So far i only think about a sentence that say that in finite model there must be a maximum. But, how to write it in a way that in every model (that can give any ןnterpretation it want to the language) it will say this? tnx!

user2637293
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Let $\psi$ be in the language $L=\{ f\}$ where $f$ is a unary function symbol.

Then write "$f$ is an injection $\to$ $f$ is a surjection".

Show that this will hold on any finite structure, but you can clearly find an infinite counter example

Kyle Gannon
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  • first tnx. i wrote it like that : $$\left(\forall :x\forall :y\left(f\left(x\right)=f\left(y\right)\rightarrow :x=y\right)\right)\rightarrow \left(\forall y\exists x\left(f\left(x\right)=y\right)\right)$$ But why it holds for finite models? – user2637293 Feb 03 '15 at 23:46
  • Proof by Induction! Every injection from $M \to M$ where $|M|=n<\omega$ is a surjection. – Kyle Gannon Feb 03 '15 at 23:58
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    @user2637293 http://math.stackexchange.com/questions/63072/surjectivity-implies-injectivity – the gods from engineering Feb 03 '15 at 23:58