Definition:Indiscernible

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Definition

Let $\MM$ be an $\LL$-structure.

Let $I$ be an infinite set.

Let $X = \set {x_i \in \MM: i \in I}$ be an infinite subset of the universe of $\MM$ indexed by $I$.

For:

every $n \in \N$

and:

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

Let:

$\MM \models \map \phi {x_{i_1}, \ldots, x_{i_n} } \leftrightarrow \map \phi {x_{j_1}, \ldots, x_{j_n} }$ for all $\LL$-formulas $\phi$ with $n$ free variables.

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

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

Also known as

Elements of an indiscernible set are often called indiscernibles.

Also see

• Definition:Order Indiscernible

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