Source
Turner, “Ultrafilters as Propositional Theories”, LIT-145, read in full in NOTE-177 from the author’s preprint of 20 November 2021, every page and every proof. The published version (Philosophy Compass, 2025) was seen only through its abstract, which omits compactness.
What was actually shown
The dictionary (§1.2, pp. 3–4). Take a non-empty set W of indices (“worlds”); a W-proposition is a subset of W, with ∩ as conjunction, complement as negation and ⊆ as implication. A filter over W, a family closed under intersection and superset and not containing ∅, is then a propositional theory closed under pairwise conjunction and implication that does not contain the impossible proposition. An ultrafilter also contains p or W − p for every p.
Theorem 2 (p. 5). For a filter, maximal, prime and ultra are equivalent.
Theorem 3 (pp. 6–8). Every filter extends to an ultrafilter, by transfinite recursion over a well-ordering of 𝒫(W), which needs choice.
Theorem 9, Łoś (pp. 18–20). Over an “intensional space” (each world assigned a model of a fixed first-order language), the ultraproduct M_T, whose domain is equivalence classes of world-to-object functions, satisfies M_T, [A] ⊨ φ iff ⟦φ⟧_A ∈ T, for an ultrafilter T. Only the negation step uses ultra-ness; for a mere filter the construction is a reduced product.
Theorem 12, compactness (pp. 20–21). Let W be the finite subsets of Γ, each with a model. The sets containing some ⟦⋀Δ⟧ form a filter; extend it to an ultrafilter and apply Łoś. So a finitely satisfiable Γ has a model.
What could have come out otherwise. These are standard theorems and could not. What NOTE-177 checked is that each proof goes through. They do, with slips it lists: a gappy maximal-to-prime step in Theorem 2, a wrong condition cited in Theorem 9, and Δ ∩ Σ for Δ ∪ Σ in Theorem 12.
What this does not say
- That a filter is a consistent theory. It is only finitely consistent. It can contain an “essentially infinite contradiction” (p. 3), and that is why compactness needs the extension to an ultrafilter.
- That the extension theorem is Lindenbaum’s lemma, or Łoś’s theorem Henkin’s proof. Those are informal analogies (C4, C6), not formal reductions.
- That the “propositions” are propositions. They are sets of indices; the author calls the reading a “pretense” that helps the model theory (p. 15) and disclaims modal semantics (p. 11).
- That philosophers have a use for it. The paper says they can put ultrafilters “to good use” and names no application (C9).
- That the published version says the same. Only the preprint was read.
Connections
- The same ultrafilter machinery builds Wenmackers’s numerosity probabilities for a fair lottery on ℕ (THEORY-401), where the choice of free ultrafilter is what the solution is not unique up to.
- Stone’s representation of a Boolean algebra by its prime ideals (THEORY-020) is the algebraic face of the same equivalence of prime and ultra filters.