Definition:Order Indiscernible

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

Let $(I, \leq)$ be an infinite ordered set.

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

Let $A$ be a subset of the universe of $\mathcal{M}$.

$X$ is order indiscernible over $A$ in $\mathcal{M}$ if for every $n\in \mathbb N$ and every pair of chains $i_1 < \cdots < i_n$ and $j_1 < \cdots < j_n$ in $I$ 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 and parameters from $A$.

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

Elements of an order indiscernible set are often called order indiscernibles.

Also see

 * The definition of an indiscernible set.