Expressing the nth element of a set

Question

Let's say I have a set $S$ of infinite length. How can I express a function that returns the nth element in the set?

Answer 1

A "set" doesn't come with any notion of order among its elements, unless you explicitly equip it with one.

If you want to express a collection of elements arranged in a particular order such that "the $n$th element" makes sense, then what you want is not a set, but a sequence. Sequences are usually identified with functions $\mathbb N\to X$ where $X$ is the set of elements in the sequence. The notation for a particular element in the sequence can either be the usual $a_n$, or if you want to be more formal but less readable, $f(n)$ where $f$ is the function that represents the sequence.