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Let $T$ be a complete, infinite binary tree with any one-place relationship $U$. Let $L(x,y)$ be a such relationship that $$L(x,y) \text{ iif x is a father of y}$$ $R(x,y)$ is defined respectively.

So, our $T_u = (T, L, P, U)$

Construct a such sentence MSO ( Monadic Second Order Logic) $\phi$ that:
$$T_u \models \phi \iff \text{ there exists a such path in T that it contains infinite elements from U}$$

My solution:

$$\exists U \forall Ux \forall Uy \ S(x,y)$$ $$ S(x,y) = \exists X \forall Xa \exists b ((L(a,b) \vee P(a,b)) \vee a=y ) \wedge \exists Xc L(x,c) \vee P(x,c) $$

And my friend told me that it is incorrect but he didn't say me why. Please help.

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    A few of preliminary questions: Isn't $U$ a relation symbol instead of a variable? Is $P$ just another name for $R$? By $\exists U \forall Ux \forall Uy S(x,y)$ are you abbreviating $\exists U \forall x \forall y ((U(x) \wedge U(y)) \rightarrow S(x,y))$? – 2017-01-18
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    As a minor note, you mean "infinitely many" elements rather than "infinite" elements. The elements themselves are not likely to be infinite. – 2017-01-18

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The existence of a path that contains infinitely many elements from $U$ can be stated by saying that there is a set $\Pi$ of nodes such that for every node $x$ in $\Pi$, there is another node $y$ in $\Pi$ that is in $U$ and is a (proper) descendant of $x$. If you denote "$y$ is a proper descendant of $x$" by $x < y$, you can write the S2S formula

$$\exists \Pi \,. \forall x \,. \Pi(x) \rightarrow \exists y \,. \Pi(y) \wedge U(y) \wedge x < y \enspace.$$

Defining $x < y$ in S2S requires some thought, but here's a standard solution, in which $L(z,c)$ is true if and only if $c$ is the left child of $z$, and $R(z,c)$ is true if and only if $c$ is the right child of $z$:

$$ x < y := \forall A \,. (A(y) \wedge (\forall z \,. \forall c \,. (A(c) \wedge (L(z,c) \vee R(z,c))) \rightarrow A(z)) \rightarrow A(x) \enspace.$$

It says that $x < y$ if and only if every set $A$ that contains $y$ and is closed under immediate predecessors also contains $x$.

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    what does it mean $\Pi .$? ( pi and dots)? – 2017-01-18
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    $\Pi$ is just the name of a set-valued variable. The dot is used to separate the quantifier to the formula to which it applies. For example, $\exists y \,. \Pi(y) \wedge U(y) \wedge x < y$ is also written $\exists y (\Pi(y) \wedge U(y) \wedge x < y)$ (and in other ways). – 2017-01-18