\begin{definition}[Definition:Order Indiscernible]
Let $\MM$ be an $\LL$-structure.
Let $\struct {I, \le}$ be an infinite ordered set.
Let $X = \set {x_i \in \MM: i \in I}$ be an infinite subset of the universe of $\MM$ indexed by $I$.
Let $A$ be a subset of the universe of $\MM$.
$X$ is '''(an) order indiscernible (set) over $A$ in $\MM$''' {{iff}}:
:For every $n \in \N$ 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:
::$\MM \models \map \phi {x_{i_1}, \ldots, x_{i_n} } \iff \map \phi {x_{j_1}, \ldots, x_{j_n} }$
:for all $\LL$-formulas $\phi$ with $n$ free variables and parameters from $A$.
{{Disambiguate|Definition:Logical Formula}}
Informally, $X$ is '''order indiscernible''' {{iff}} $\MM$ cannot distinguish between same-sized increasing ordered tuples over $X$ using $\LL$-formulas.
\end{definition}