Definition:Order Indiscernible

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Definition

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 $A$.

The term Definition:Logical Formula as used here has been identified as being ambiguous. If you are familiar with this area of mathematics, you may be able to help improve $\mathsf{Pr} \infty \mathsf{fWiki}$ by determining the precise term which is to be used. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Disambiguate}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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

• Definition:Indiscernible

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