# Definition:Indiscernible

## Definition

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

Let $I$ be an infinite 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 $I$.

For:

every $n \in \N$

and:

every pair of subsets $\left\{ {i_1, \ldots, i_n}\right\}$ and $\left\{ {j_1, \ldots, j_n}\right\}$ of $I$ each with $n$ distinct elements,

let:

$\mathcal M \models \phi \left({x_{i_1}, \ldots, x_{i_n} }\right) \leftrightarrow \phi \left({x_{j_1}, \ldots, x_{j_n} }\right)$ for all $\mathcal L$-formulas $\phi$ with $n$ free variables.

Then $X$ is (an) indiscernible (set) in $\mathcal M$.

Informally, $X$ is indiscernible (set) if $\mathcal M$ cannot distinguish between same-sized ordered tuples over $X$ using $\mathcal L$-formulas.

## Also known as

Elements of an indiscernible set are often called indiscernibles.