Eric, it's a bit misleading to say that we copy the argument from Zorn's lemma. For most people who are untrained in the Way of the Choice, Zorn's lemma is just this magical box where you have a partial order and you conclude there is a maximal under some condition. What we do here is either appeal to something like Koenig's lemma, or essentially reprove Zorn's lemma for countable partial orders. But a lot of people are not really aware of that because they don't have a trained mind for these sort of arguments. So "copying the argument from Zorn's lemma" can be quite confusing. — Asaf Karagila 22 hours ago