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Lets consider undirected graphs. Prove that there exists formula in MSO such that it is true iff only and only from each node in graph each starting in this node path is finite. Keep in mind that quantification over sets can happend only at front of formula.

It seems to be undoable. After all, we can't express finitness of structure in MSO... Can someone help ?

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    Are you requiring your paths to be Hamiltonian? If not, then any graph with at least one edge has an infinite path. Why do you say that quantification over sets is restricted to the front of a formula? – 2017-02-16
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    Path can contains only different nodes. This requirement about quantification at front is idea of my lecturer. – 2017-02-16
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    OK so you are only looking at Hamiltonian paths. If there were such a formula, you could specialise it to the complete graph on a given structure (the graph in which the adjacency relation is always true) to get a formula that would express finiteness of the structure. – 2017-02-16
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    I have no idea how you think here hammiltonian cycle. For example, connected graph, $n+1$ nodes, exactly one node has degree equals to $n$ rest of them has degree $1$. In this graph each node has only finite path, but there is no Hammiltonian Path – 2017-02-16
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    I simply dont understand your idea. Moreover, it seems to be impossible to create such formula. However, lecturer give us this task. – 2017-02-16
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    Sorry, I've confused things by using the term "Hamiltonian". If your paths are not allowed to visit the same vertex twice, then a solution to the problem you have been set would lead to a sentence that would express finiteness of any structure if you specialised it to the complete graph on the universe of the structure. Hence your intuition that the problem has no solution is correct. – 2017-02-16
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    For me, simply for graph with zero nodes we would be able to check if there is infinitely many nodes (assuming that this procedure exists). What about case when quantification over sets is **not** restricted to the front of a formula? – 2017-02-16
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    I have no idea what the first part of your last comment means. The restriction to quantification at the front of the formula is not relevant to my argument that no such formula exists. – 2017-02-16
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    I mean with zero edges (so we consider number of nodes). – 2017-02-16

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