It might be worth saying explicitly what these three (equivalent) axioms say.
The best way to explain what Zorn's lemma is saying is to give an example, so let me show that Zorn's lemma implies the ("obviously false") well-ordering principle. Let A be any set. We'll try to find a well-ordering of it. Let O be the set of well-orderings of subsets of A. Given two of these -- say, o1 and o2 -- say that o1 <= o2 if o2 is an "extension" of o1 -- that is, o2 is a well-ordering of a superset of whatever o1 is a well-ordering of, and o1 and o2 agree where both are defined. Now, this satisfies the condition in Zorn's lemma: if you have a subset of O on which <= is a total order, this means that for any two things in the subset one is an extension of the other, and then the union of all of them is an upper bound. So if Zorn's lemma is true then O has a maximal element, i.e. a well-ordering of some subset of A that extends every possible well-ordering of any subset of A. Call this W. Now W must actually be defined on the whole of A, because for every element a of A there's a "trivial" well-ordering of {a}, and W must extend this, which requires a to be in W's domain.
(A few bits of terminology that I didn't digress to define above. A total ordering on a set is a relation < for which if a<b and b<c then a<c, and for which exactly one of a<b, b<a, a=b holds for any a,b. OR a relation <= for which if a<=b and b<=c then a<=c, and for which for any a,b either a<=b or b<=a, and for which a<=b and b<=a imply a=b. A partial ordering is similar except that you're allowed to have pairs for which a<b and b<a (OR: a<=b and b<=a) both fail. We can translate between the "<" versions and the "<=" versions: "<" means "<= but not =", or "<=" means "< or =". Given a partial ordering, an upper bound for a set A is an element b for which a<=b for every a in A. A maximal element in a partially ordered set is an element of the set that's an upper bound for the whole set.)
Imagine someone finding out that "Physics professors fail on basic physics problems". This, of course, would never happen. To become a physicist in academia, one has to (among million other things) demonstrate proficiency on far harder problems than that.
Philosophy professors, however, are a different story. Cosmologist Sean Carroll tweeted a link to a paper from the Harvard Moral Psychology Research Lab, which found that professional moral philosophers are no less subject to the effects of framing and order of presentation on the Trolley Problem than non-philosophers. This seems as basic an error as, say, confusing energy with momentum, or mixing up units on a physics test.
Abstract:
We examined the effects of framing and order of presentation on professional philosophers’ judgments about a moral puzzle case (the “trolley problem”) and a version of the Tversky & Kahneman “Asian disease” scenario. Professional philosophers exhibited substantial framing effects and order effects, and were no less subject to such effects than was a comparison group of non-philosopher academic participants. Framing and order effects were not reduced by a forced delay during which participants were encouraged to consider “different variants of the scenario or different ways of describing the case”. Nor were framing and order effects lower among participants reporting familiarity with the trolley problem or with loss-aversion framing effects, nor among those reporting having had a stable opinion on the issues before participating the experiment, nor among those reporting expertise on the very issues in question. Thus, for these scenario types, neither framing effects nor order effects appear to be reduced even by high levels of academic expertise.
Some quotes (emphasis mine):
When scenario pairs were presented in order AB, participants responded differently than when the same scenario pairs were presented in order BA, and the philosophers showed no less of a shift than did the comparison groups, across several types of scenario.
[...] we could find no level of philosophical expertise that reduced the size of the order effects or the framing effects on judgments of specific cases. Across the board, professional philosophers (94% with PhD’s) showed about the same size order and framing effects as similarly educated non-philosophers. Nor were order effects and framing effects reduced by assignment to a condition enforcing a delay before responding and encouraging participants to reflect on “different variants of the scenario or different ways of describing the case”. Nor were order effects any smaller for the majority of philosopher participants reporting antecedent familiarity with the issues. Nor were order effects any smaller for the minority of philosopher participants reporting expertise on the very issues under investigation. Nor were order effects any smaller for the minority of philosopher participants reporting that before participating in our experiment they had stable views about the issues under investigation.
I am confused... I assumed that an expert in moral philosophy would not fall prey to the relevant biases so easily... What is going on?