Some Restricted Lindenbaum Theorems Equivalent to the Axiom of Choice

David Miller

⟨S, Cn⟩ is a deductive calculus if Cn is a function from ℘(S) to ℘(S) for which (a) if X ⊆ S then X ⊆ Cn(X) ⊆ S; (b) if X ⊆ S then Cn(Cn(X)) = Cn(X); and (c) if X ⊆ S then Cn(X) = ⋃{Cn(Z) | Z ⊆ X and |Z| < ℵ_{0}}. A set Y ⊆ S is a theory in ⟨S, Cn⟩ if Y = Cn(Y); a theory Y is y-saturated in ⟨S, Cn⟩ if (a) y ∉ Y and (b) if x ∉ Y then y ∈ Cn(Y ∪ {x}). The generalized Lindenbaum extension theorem LET (for whose proof the axiom of choice AC is needed) says that if y ∉ Y then Y has a y-saturated extension.

Dzik [1981] gives a direct proof of AC from a premise let that attributes only a weak version of the Lindenbaum property, namely the existence of one saturated theory, to a restricted set of deductive calculi (in particular Cn(Y) takes only the values Y and S). Within ZF, plainly, let and LET are equivalent to AC. The question arises whether the conditional let ⇒ LET can be proved in ZF without in effect first proving AC from let, and then proving LET from AC. The paper provides a partial answer to this question.

Dzik, W. [1981]. ‘The Existence of Lindenbaum’s Extensions is Equivalent to the Axiom of Choice’. Reports on Mathematical Logic 12, pp. 29-31.

Address: Department of Philosophy, University of Warwick, Coventry CV4 7AL, UK

E-mail: d.w.miller@warwick.ac.uk