Hypercomplete and Compossible Sets

Abstract

In Section 10.0, some problems of the completeness proof of the Q calculi were outlined. It was mentioned there that the difficulties are especially high ones in the cases µ=4, and µ=5. The main problem is as follows. Let α be a µ-consistent set of sentences (where µ=4, 5). One has to prove that α is embeddable into a structure 〈W, R, Φ〉 satisfying certain conditions outlined in 10.0. Here R must be reflexive and transitive, moreover, if µ=5, then R must be, in addition, symmetric. For a w ₀∈ W, Φ(w ₀) must be a µ-complete superset of α. The obvious strategy for defining 〈W, R, Φ〉 inductively is as follows. One starts by defining Φ(w ₀) as a µ-complete superset of α. Then one proceeds by showing that if Φ(w) is defined, and Mg∈ Φ(w), then there exists a µ-complete set β such that g∈ ß,and β is an “alternative” to Φ(w) (in the sense that Lf∈Φ(w) implies f∈β, and Nf∈Φ(w) implies ~f∉β).By introducing a new index w ₁ and postulating 〈w, w ₁〉∈ R and Φ(w ₁)=ß, the possibility of the induction step is proved, and the definition is (seemingly) completed. The crux of the matter is that—as it was illustrated in 10.0—the transitivity of R (and, if µ=5, the symmetry of R)will not hold automatically.