#what i'm trying to justify is my intuition that zorn's lemma is constructively stronger than the well-ordering principle

[bubbloquacious]

Hmmmmm, so using Zorn's lemma you can prove the well-ordering principle quite easily. It's a straightforward application; you consider the poset of all partial well-orders of the set, ordered by inclusion. But, do you need the law of excluded middle (or more specifically the principle of double negation elimination, so that proofs by contradiction are valid) to show that a maximal element in this poset is in fact a well-order? Normally you go well assume there's two incomparable elements, then we can expand the well-order, so our original well-order was not maximal, which contradicts the assumption. However, it feels, vibically, like this appeal to LEM is not strictly necessary... Data-wise it seems like you don't need to create anything more out of thin air than the given maximal partial well-order.

Follower challenge: prove the well-ordering principle without the law of excluded middle using only a single appeal to Zorn's lemma.