Alternatives and Successors of Hypercomplete Sets

Abstract

As it was foreshadowed in 11.0, we shall use μ-hypercomplete sets (instead of μ-complete ones) in proving that any, μ-consistent set is embeddable into a structure 〈W, R, Φ〉 satisfying the conditions outlined in 10.0 already. In this §, we shall prove (in Theorem 12.2) that if α is a μ-hypercomplete set, and Mg₀∈Co(α), then there exists a μ-hypercomplete set β (as we shall call it, the Mg₀-successor of α) such that g _o ∈Co(β), and Co(β) is a μ-alternative to Co(α); moreover, if μ=5, then Co(α) is a μ-alternative to Co(β) as well. Furthermore, Theorem 12.4 gives sufficient conditions for granting the transitivity of alternativeness for μ=4, 5.