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Prove that there no exists a such set of sentences FO $\Delta$ that for every directed graph $G$ $$G \models \Delta \iff \text{ G is not a tree *xor* G don't has an infinite path}$$

Alternatively, to make it simpler we can say:

$$(G \models \Delta) \mathrel{\mbox{iff}} (\mbox{$G$ is a tree} \leftrightarrow \mbox{every path in $G$ is finite}).$$

I don't know how to start. Please give me an advice.

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    This seems to be interestingly difficult. Do you perhaps have any model-theoretic tools at your disposal? (i.e. more heavy duty than the usual compactness and Löwenheim-Skolem theorems). – 2017-02-13
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    @HenningMakholm, I suppose that it should be doable with Compactness/Lowenheim-Skolem/EF Games because It is problem from previous exam. – 2017-02-13
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    Hmm, if only "for every directed graph" had been "for every connected graph" -- then I could imagine a solution based on compactness. – 2017-02-13
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    It is written: "for every directed graph". So it must be directed. Perhaps we can assume connectivity, but I think that we cannot resign from directed graph. – 2017-02-13
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    It might help you (and us!) to get rid of all the negatives and the exclusive or from the statement of the question. I think the condition you are trying to state in the displayed bi-implication is $$(G \models \Delta) \mathrel{\mbox{iff}} (\mbox{$G$ is a tree} \leftrightarrow \mbox{every path in $G$ is finite}).$$ Is that right? – 2017-02-13
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    Note that $G \models \Delta$ in two cases: 1) $G$ is not a tree **and** has an infinite path 2) $G$ is a tree **and** $G$ don't has an infinite path So, yes. Your conclusion is OK. – 2017-02-13
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    @Logic: well then I suggest you edit your question and the title to state it that way. Did the exam question really use "xor"? – 2017-02-13
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    @RobArthan I edited. – 2017-02-14
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    Thanks for the clarification. I agree that it looks interestingly difficult. The close votes are probably because you haven't said what you've tried. Maybe you should try a solution based on compactness and tell us what went wrong. – 2017-02-15

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