$$\mathcal{S} = \{1, 2, ..., k\};\, k \in \mathbb{N}$$,
Let's say $k = 1$, would the set $\mathcal{S}$ be $\{1, 2, 1\} = \{1, 2\}$ or would it be $\{1,1\} = \{1\}$.
In other words, I know that the 1, 2 is there in the definition to establish that you start at 1 and go up to $k$ by increments of 1. Is the 2 there only to establish this fact, or does its presence mean it must be in any set $\mathcal{S}$?
On
When $k=1$ the notation is meant to tell you that $S=\{1\}$. The dots tell you it's informal, and may be a little confusing in edge cases.
A formal way to write that definition would be $$ S = \{ n \in \mathbb{N} \ | \ 1 \le n \le k \}. $$
That would even give the right answer when $k=0$: the empty set.
We write the informal definition because it's easier to read.
In this context, the $2$ is only suggestive of the pattern. It's not in the set if $k < 2$.
A more precise version of $\{1,2,...,k\}$ would be $\{x \in \mathbb{Z}\mid 1 \le x \le k\}$.