Use cardinal notation to denote the last element in a finite set

Question

In set theory, is it correct/okay/reasonable to denote the last element in a finite set using cardinal notation?

Example:

$$ X=\{x_1,x_2,...,x_{|X|}\} $$

Answer 1

There's no last element of a set, because a set is unordered. If you want to talk about the "last element", there are two ways to do it:

• define the set as $\{x_1,x_2,\ldots,x_n\}$, and then refer to $x_n$. This requires you to have already defined an order at the start, and it doesn't work well with other constructions of sets.
• use a tuple instead. A element of the set $X^n$ is an ordered collection of $n$ elements of $X$: we call this an "$n$-tuple". For instance, an element of $\mathbb{R}^2$ is an ordered pair of the form $(x,y)$, where both $x$ and $y$ are real numbers. In this case, you could make a note saying something like "We use subscripts to pick out elements of a tuple: for instance, $T_2$ is the second element of the tuple $T$. We also use $|T|$ to denote the length of $T$." Then you can go ahead and use $T_{|T|}$ without any problems.