Definition:Order Indiscernible

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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 $\mathcal M$ if and only if:

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 if and only if $\mathcal M$ 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