Definable set without parameters Can someone help me to solve the following problem:
Suppose that $T$ is a set of sentences and that there is an $\mathcal{N} = (N,J)$ such that $\mathcal{N} \vDash T$ and $N$ is infinite. Show that there is an $\mathcal{M} = (M,I)$ and an element $a \in M$ such that $M \vDash T$ and $a$ is not definable in $\mathcal{M}$ without parameters.
The idea I had was the following: Fix $a \in N$. If $a$ is not definable in $\mathcal{N}$ without parameters, then we're done ($M = N$). If this is not the case, then let $M = N \cup \{z\}$, where $z$ is going to be an element that is going to "mirror" the behavior of $a$; i.e. whichever predicate $a$ satisfies, $z$ is going to satisfies it as well, but I am not sure. Please any hint would be awesome!
 A: Let $\kappa$ be the cardinality of the language used in the theory $T$. Then, the number of elements in models of $T$ is bounded by $\kappa$, as every $\emptyset$-definable set is definable using a formula (without parameters) from $L$.
By upward Löwenheim-Skolem theorem, since $T$ has infinite models, there is a model $(M,I)$ of $T$ of cardinality $|M|= \kappa^+>\kappa$, and by the pigeonhole principle, $M$ should contain elements that cannot be defined using a formula without parameters.
You can also argue by compactness as follows: suppose $\{\phi_i(y):i<\lambda\}$ is an enumeration of all formulas in the language $L$ that are consistent with $T$ and defined elements without parameters. That is, for every $\phi_i(y)$ the set of formulas $T\cup \{\exists y(\phi_i(y)\wedge \forall z (\phi_i(z)\rightarrow z=y))\}$ is consistent.
Let $L'=L\cup \{c\}$ be an extension of $L$ by a new constant symbol, and consider  the $L'$-theory $$\Gamma=T\cup \{\neg \phi_i(c):i<\lambda\}$$
Since $T$ has infinite models, it is easy to show that $\Gamma$ is finitely satisfiable. Hence, by compactness, there is an $L'$-structure $M$ satisfying $\Gamma$. In particular, $M$ is a model of $T$, and by construction the element $a=c^M$ (the interpretation of the constant $c$ in $M$) will be an element that is not definable without parameters in the language $L$.
