Here’s one version of the axiom of choice for pluralities in terms of superplurals:
- AC(plural,superplurals): For a superplurality xxx of disjoint pluralities, there is a plurality zz that contains exactly one item from each plurality yy in xxx.
And here’s Zorn’s Lemma for pluralities and and partial order formulas:
- Zorn(plural,formulas): Given a partial order formula ϕ(x,y) on the items of a plurality xx, if every subplurality that is a ϕ-chain has a ϕ-upper bound in xx, then xx has a ϕ-maximal element.
(A partial order formula ϕ(x,y) is a formula such that the expression ϕ(x,y) satisfies the axioms of x ≤ y for a partial order ≤. A ϕ-chain and ϕ-upper bound are defined in the natural way.)
Interestingly, AC(plural,superplurals) does not entail Zorn(plural,formulas). For consider a model of plurals and superplurals where the domain is the natural numbers, the plurals are interpreted as non-empty sets of naturals, the superplurals as non-empty sets of non-empty sets of naturals, and we have all the arithmetical predicates, all within a model of ZF where the Axiom of Dependent Choice fails for some relation and there is a predicate expressing that relation. Then AC(plural,superplurals) holds in this model, because the naturals are well-ordered. However, Zorn(plural,formulas) with the naturals being the domain yields Dependent Choice for that relation (since we can encode finite sequences of naturals as naturals).
