In Folland chapter 0.2 Orderings, the well-ordering principle is stated as:
Every nonempty set $X$ can be well ordered.
I read over the proof, which seems fine to me. However, I also understand that, for example, $R$ is not well ordered, because we cannot find a minimal in the subset, for example, $(0,1)$. And this sees implies that not every set is well ordered.
I then look up online and found a different version of well-ordering principle:
In mathematics, the well-ordering principle states that every non-empty set of positive integers contains a least element.
I'm confused. So well-ordering only apply to positive integers? not a general set $X$? Why Follad stated the principle without clarifying that $X$ has to be a set of positive integer? Moreover, I also notice that there is no where using the assumption that $X$ is a set of positive integers in the proof given by Folland. Thank you.