In "Mathematical Logic" second edition written by H-D Ebbinghaus, J.Flum and W.Thomas, in chapter 9 "Extensions of First-Order Logic", page 140, in the prooof of theorem 1.5 (The Lowenheim-Skolem theorem does not hold for $\mathfrak{L}_{II}$, there's a sentence saying "To define $\varphi_{unc}$ we use an $L_{II}^{\emptyset}$-formula $\psi_{fin}(X)$, similar to $\varphi_{fin}$, with just one free unary relation variable $X$, for which $(\mathfrak{A},\gamma)\models\psi_{fin}(X)$ iff $\gamma(X)$ is finite."
Right after that sentence there's a note saying "We leave it to the reader to write down such a formula.".
Does anyone know how to write down such a formula? I have no clue.
Let $S$ be a symbol set, that is, a set of relation symbols, function symbols and constants. The alphabet of $L_{II}^S$ contains, in addition to the symbols of $L^S$, for each $n\geq1$ countably many $n$-ary relation variables $V_0^n,V_1^n,V_2^n,...$.
To denote relation variables we use letters $X,Y,...$, where we indicate the arity by superscripts, if necessary. We define the set $L_{II}^S$ of second-order $S$-formulas to be the set generated by the rules of the calculus for first-order furnulas extended by the following two rules:
1. if $X$ is an $n$-ary relation variable and $t_1,\ldots,t_n$ are $S$-terms, then $Xt_1 \ldots t_n$ is an $S$-formula.
2. If $\varphi$ is an $S$-formula and $X$ is a relation variable, then $\exists X\varphi$ is an $S$-formula.
In order to be clearer, my question is part of the following proof of the follwoing theorem:
"Theorem: The Lowenheim-Skolem theorem does not hold for $\mathfrak{L}_{II}$.
proof: We give a setence $\varphi \in L^\emptyset_{II}$ such that for all structures $\mathfrak{A}$, $\mathfrak{A}\models \varphi_{unc}$ iff $A$ is uncountable.
Then $\varphi_{unc}$ is satisfiable but it has no model that is at most countable.
To define $\varphi_{unc}$ we use an $L^\emptyset_{II}$-formula $\psi_{fin}(X)$, similar to $\varphi_{fin}$, with just one free unary relation variable $X$, for which
$(\mathfrak{A},\gamma)\models \psi_{fin}(X)$ iff $\gamma (X)$ is finite.
(We leave it to the reader to write down such a formula.)"