Definition:Order Indiscernible

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

Let $\left({I, \le}\right)$ be an infinite ordered set.

Let $X = \left\{ {x_i \in \mathcal M: i \in I}\right\}$ 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 (an) order indiscernible (set) over $A$ in $\mathcal M$ :
 * 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:
 * $\mathcal M \models \phi \left({x_{i_1}, \ldots, x_{i_n} }\right) \iff \phi \left({x_{j_1}, \ldots, x_{j_n} }\right)$
 * for all $\mathcal L$-formulas $\phi$ with $n$ free variables and parameters from $A$.

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

Also known as
Elements of an order indiscernible set are often called order indiscernibles.

Also see

 * Definition:Indiscernible