Definition:Indiscernible

Definition
Let $\mathcal{M}$ be an $\mathcal{L}$-structure.

Let $I$ be an infinite set.

Let $X = \{x_i \in \mathcal{M} : i \in I\}$ be an infinite subset of the universe of $\mathcal{M}$ indexed by $I$.

$X$ is indiscernible in $\mathcal{M}$ if for every $n\in \mathbb N$ and every pair of subsets $\{i_1, \dots, i_n\}$ and $\{j_1, \dots, j_n\}$ of $I$ each with $n$ distinct elements, we have $\mathcal{M} \models \phi(x_{i_1}, \dots, x_{i_n}) \leftrightarrow \phi(x_{j_1},\dots,x_{j_n})$ for all $\mathcal{L}$-formulas $\phi$ with $n$ free variables.

Informally, $X$ is indiscernible if $\mathcal{M}$ cannot distinguish between same-sized tuples over $X$ using $\mathcal{L}$-formulas.

Elements of an indiscernible set are often called indiscernibles.

Also see
The definition of an order indiscernible set.