Appendix B - Proofs of Theorems 2 and 3 from Chapter 6

Summary

Theorem 2.Every fundamentally complete situation s actual in a semantically well-founded model A and such that P(s) contains no A-inexpressible propositions can be extended to a Burgean situation s* actual in A.

Proof. Let s₀ be the maximal subsituation of s containing no semantic SOAS, and let P be the smallest set of propositions p such that 〈Tr, p; i〉 ∈ s or 〈Tr, p : i〉 ∈ Par(q) for some q ∈ P. Consider the unique Burgean series S based on s₀ and P, S = {s₀, s₁, …, s_a, …}. Claim: s ⊆ ∪S.

Since s is semantically well founded, there is a foundation series S′ = {s₀, s_1′, S_2′, …} such that s ⊆ ∪ S'. It is sufficient to show that, for all a, there is a γ such that s′_a ⊆ s_γ. Assume for a reductio that a is the least ordinal such that, for all λ, ∼ s′_a ⊆ s_γ. Clearly, a > 0. By inductive hypothesis, for all δ < a, there exists an ∈ such that s′_δ ⊆ s_∈. Let ζ be the supremum of all such ∈, that is, for all δ < a, s′_δ ⊆ s_ζ.

a is either a limit ordinal or a successor. If a is a limit ordinal, then it immediately follows that s′_a ⊆ s_ζ, contradicting the hypothesis. So a is a successor, and s′_a-1 ⊆ s_ζ. Every nonsemantic SOA in s′_a–s′_a-1 belongs to s₀. Suppose that 〈Tr, p; i〉 ∈ s′_a. We can show that 〈Tr, p; i〉 ∈ s_ζ+1 by induction on the complexity of p.