The well ordering principle, the axiom of choice, and Zorn's Lemma are all "equivalent", meaning you can pick any one as an axiom and prove the other two.

So some text books may pick one as the axiom and others pick a different axiom.

The crazy thing about the well-ordering principle: It states that a well ordering exists on the reals, which means that you can find an ordering such that any open set has a minimum. Apparently, elsewhere in mathematics, they've proven that even though it exists, you cannot articulate that ordering.

There's a common joke:

"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?"

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kccqzy a day ago | parent [-]

You are talking about the well-ordering theorem, not the similarly named well-ordering principle. That’s exactly my confusion when I first opened this PDF.

	▲	BeetleB a day ago \| parent [-]
		Different folks use different conventions. When I was taught it, they called it the principle, not theorem. You can find similar comments on the Internet (e.g. math subreddit). Here's one that acknowledges it: https://math.stackexchange.com/questions/1837836/well-orderi... > The "well-ordering principle" has (at least) two different meanings. The first meaning is just another name for the well-ordering theorem. The second meaning is the statement that the usual relation < on the set N is a well-ordering. This statement is equivalent to the statement that ordinary induction on the natural numbers works.