I'm working through David Marker's Model Theory, and I'm stuck on Exercise 1.4.2 b. It states
Let $\mathcal L$ be any finite language and let $\mathcal M$ be a finite $\mathcal L$-structure. Show that there is an $\mathcal L$-sentence $\phi$ such that $\mathcal N \vDash \phi$ if and only if $\mathcal N$ is isomorphic to $\mathcal M$.
I think I understand the idea here, we want to basically reproduce the interpretation of each $n$-ary function $f \in \mathcal F(\mathcal L)$ and each $n$-ary relation $R \in \mathcal R(\mathcal L)$, and the cardinality of $\mathcal M$ into an $\mathcal L$-sentence.
My main issue is that I don't know how to refer to arbitrary elements in $\mathcal M$ that aren't symbols in $\mathcal L$. For example, in the language of fields, in the structure $(\mathbb R, +, -, \cdot, \div, 0, 1)$, I don't know how to refer to the number $\pi$, for example. Even if the structure is finite, I don't see how to refer to arbitrary elements in the underlying set.
Any advice would be appreciated.
The idea is to use quantification to require the cardinality of the satisfying model to be $|\mathcal M|$, and to require that the functions, relations, and constants behave the same as in $\mathcal M$. So something like:
$$\phi = \exists m_1, \dots, m_k \forall m_0 \left(\text{$m_0$ must be one of the $m_i$'s}\right)\land \\\left( \bigwedge_{f \in \mathcal F} \text{describe the function $f$} \right) \land \\\left( \bigwedge_{r \in \mathcal R} \text{describe the relation $r$} \right) \land \\\left( \bigwedge_{c \in \mathcal C} \text{describe the constant $c$} \right) \quad$$
Where $k = |\mathcal M|$. These four "components" of the formula describe
Any structure that satisfies these four conditions is isomorphic to $\mathcal M$.