Well ordering principle in Folland chapter 0

Question

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.

Answer 1

The statements “Every nonempty set can be well ordered.” and “Every non-empty set of positive integers contains a least element.” are distinct ones. The second one is what is commonly called well ordering principle. It states the that $\Bbb Z_+$, under its usual order relation, is well-ordered. It has little to do with the fact that, on any set, you can define an order relation with respect to which that set is well ordered.

Answer 2

$\Bbb R$ is not well ordered under its usual ordering. The well-ordering principle (in its first form) states only that it's possible to impose some ordering on $\Bbb R$ that makes it well ordered. This requires some form of the Axiom of Choice, and in fact the well-ordering principle as stated above is equivalent to the Axiom of Choice.

If this is confusing, consider instead $\Bbb Z$, the set of all integers. It's easy to see that $\Bbb Z$ under its usual ordering is not well ordered because, for example, the set of negative integers has no least element. But it's easy to construct (even without the Axiom of Choice) an ordering on $\Bbb Z$ that is well ordered. $x R y \iff (\vert x \vert \lt \vert y \vert \lor (x \lt y \land x = -y))$. This essentially folds the integers in half at the origin with each negative integer slightly below its positive counterpart.

Answer 3

The set $\mathbb{R}$ with the standard order $a<b$ iff $b-a$ is a positive number is not well ordered as you pointed out. However, the axiom of choice and its equivalent manifestations (the well order principle) states that there is a total order relation$\prec$ on $\mathbb{R}$ such that under this order, $\mathbb{R}$ is well ordered, that is, for any nonempty set $A\subset \mathbb{R}$ there is $a\in A$ such that for any $b\in A$, $b=a$ or $a\prec b$.

No-one has found an explicit well order relation on $\mathbb{R}$. It is a theoretical consequence of the Axiom of Choice (or rather an equivalence to the Axiom of Choice)