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Write a such SO ( second order) sentence that:

$$\text{ directed } G = (V,E ) \models \phi \iff \text{ G is directed infinite tree where every vertex has finite degree} $$

My solution:

$$\phi = \phi_c \wedge \phi_r \wedge \phi_b \wedge \neg \phi_s$$

$r(p,q) \Leftarrow \exists R R(p,q) \wedge \ R\text{ is transitive } \wedge \forall x \forall y E(x,y) \iff R(x,y)$

says: there exists path between $p,q$

$\phi_c : \forall x \forall y r(x,y) \implies \neg r(y,x)$

There is no cycle.

$neq(p,q) \iff \text{ neq is antysymmetric and transitive } \wedge p \neq q$ $\phi_s: \exists x \exists C \forall c_1 \exists c_2 c_1 \in C \implies e(x,c) \wedge c_2 \in C \wedge neq (c_1, c_2)$

There exists a vertex with infinite degree.

$\phi_r : \exists r \forall x x \neq y \implies R(r,x)$

There exists a root.

$\phi_b: \exists B \forall x x \in B \implies \exists y \in B \wedge (x,y) \wedge y \neq x$

There exists an infinite branch.

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    For the "there is a path" relation, you've written an equivalence where you mean an implication, but even that won't work. You want $R$ to be the least relation which is transitive and which includes the edge relation. (Alternatively, you could define "there is a path from $p$ to $q$" as "there are no $A,B$ disjoint with $p \in A, q\in B$, such that there is no arrow from $A$ to $B$.) – 2017-02-12
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    "but even that won't work.", why? – 2017-02-12
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    Because then $R$ being the total relation satisfies the requirements, so that (with your definition) there is a path from any node to any other. – 2017-02-12
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    I don't see that. Please convince me. – 2017-02-12
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    If $R$ is the total relation, then $R(p,q)$ (trivially), $R$ is transitive (trivially), and $E(x,y) \to R(x,y)$ (trivially). Hence, $\exists R(\text{conditions})$ is always true. What is there to understand? – 2017-02-12
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    Yes, it is trivial is $R(p,q)$ is total relation. But why it is? – 2017-02-12
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    Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/53501/discussion-between-mees-de-vries-and-logic). – 2017-02-12

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