Definition:Order Indiscernible

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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 $\MM$ 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:
$\MM \models \map \phi {x_{i_1}, \ldots, x_{i_n} } \iff \map \phi {x_{j_1}, \ldots, x_{j_n} }$
for all $\LL$-formulas $\phi$ with $n$ free variables and parameters from $A$.

Informally, $X$ is order indiscernible if and only if $\MM$ cannot distinguish between same-sized increasing ordered tuples over $X$ using $\LL$-formulas.

Also known as

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

Also see