Finite tuple ($(a_0,\dots ,a_{n-1})$)

A finite tuple is an ordered list of finitely many entries; coordinates can repeat, and their positions matter. A tuple of length $n$ from a set $M$ is an element of the Cartesian product $M^n$. The empty tuple has length zero. Finite tuples support Scott formulas in a back-and-forth method construction, where matching tuples determine a partial isomorphism of structures.

An ordered triple is a finite tuple with three ordered coordinates. Repeated coordinates are allowed; equality requires equality at each of the three positions.

Using distinct fixed tags $t_1,\dots ,t_n$, encode a tuple $(a_1,\dots ,a_n)$ by the set $\{⟨t_i,a_i⟩:1\le i\le n\}$. Injectivity of the ordered-pair construction makes the unique member with first coordinate $t_i$ recover $a_i$. Thus a set can retain the order of a finite tuple when its entries are tagged, even if some coordinates coincide.