In this context I believe that the author is thinking of AC as Zorn's Lemma. I will try to make a draft of a possible expansion of the reasoning. Take the set

In this context I believe that the author is thinking of AC as Zorn's Lemma. I will try to make a draft of a possible expansion of the reasoning. Take the set V of functions P(N) -> {Ma,Mi,Un} that for each subset of the naturals classifies it as a majority, a minority, or undecided respecting the conditions in the article. P(A) = Ma ⇔ P(A') = Mi P(A) = Ma ⇔ P(A ∪ B) = Ma P(A) = Ma ⇒ P(A \ {n}) = Ma P(A) = Ma ∧ P(B) = Ma => P(A ∩ B) = Ma It is easy to prove that such a function exists f_0 (A) = Mi if |A| ∈ N, f_0 (A) = Ma if |A| = ∞, and f_0 (A) = Un otherwise. It is harder to prove that there is a such a function that also respects P(A) = Ma ∨ P(A) = Mi for subsets of the naturals. I believe that to prove this you pretty much have to use Zorn's Lemma with the order f ≤ g ⇔ (f(A) ≤ g(A)) for all A where Un ≤ Ma, Un ≤ Mi, but Mi ≰ Ma and Ma ≰ Mi. I feel like you can prove that the hypothesis of Zorn's Lemma are satisfied in this case. But is is easy to see how without ZL or something similar you would have to decide one-by-one how whether each non-finite non-cofinite set maps to Mi or Ma, but there are uncountably many of them to do by simple induction.