i'm confused about the definition of indiscernibles in the book Marker: Model Theory: An Introduction.
Let (I,<) be an ordered set, and let (xi : i ∈ I) be a sequence of distinct elements of M. We say that (xi : i ∈ I) is a sequence of order indiscernibles if whenever i1
And, if claim from the definition holds for a fixed m and some set I, does it hold for every i
The question also goes for diagonal indiscernibles.
We say that I ⊆ M is a sequence of diagonal indiscernibles for Γ if whenever φ(u1,...,um,v1,...,vn) ∈ Γ x0,...,xn,y1,...,yn ∈ I with x0
If m and n from the definition are fixed, does the claim from the definition holds for all i
Yes, indiscernibility is for all formulas in all arities (although one can speak of restricted forms of indiscernibility).
For example, in the structure $(\mathbb{Q}, <)$, any (monotonic) sequence is a sequence of indiscernibles, because all the structure can "say" is whether one element is above another.
By contrast, in $(\mathbb{Z}, <)$ there are no indiscernible sequences. This is because we have formulas expressing "There are exactly $n$ elements between $x$ and $y$" for each $n\in\mathbb{N}$, and for any $a, b\in\mathbb{Z}$, exactly one of those formulas holds. So, for example, why can't $(1, 2, 3)$ be the beginnining of an indiscernible sequence? Well, let $\varphi(x, y)=\neg\exists z(x<z<y)$; then $\varphi(1, 2)$ holds but $\varphi(1, 3)$ doesn't, even though $(1, 2)$ and $(1, 3)$ are tuples "of the same type" from the point of view of the sequence $(1, 2, 3)$.