I'm wondering if there is a version of Zorn's lemma that applies to collections that are "small" in a sense I'll describe below, and which true independent of the axiom of choice. Specifically, say I have a collection of sets such that each set in it is countable, but the collection as a whole may be uncountable. I order this collection by inclusion, and show that every chain has an upper bound. I can apply Zorn's lemma and get that there's a maximal element. But what I'm wondering is if I can do this without the usual version Zorn's lemma, which requires the axiom of choice. Every chain must be countable, since otherwise its bound would be uncountable. Is there a way to do this?