Boolean Extensions which Efface the Mahlo Property

The results that follow are intended to be understood as informal counterparts to formal theorems of Zermelo-Fraenkel set theory with choice. Basic notation not explained here can usually be found in [5]. It will also be necessary to assume a knowledge of the fundamentals of boolean and generic extensions, in the style of Jech's monograph [3]. Consistency results will be stated as assertions about the existence of certain complete boolean algebras, B, C , etc., either outright or in the sense of a countable standard transitive model M of ZFC augmented by hypotheses about the existence of various large cardinals. Proofs will usually be phrased in terms of the forcing relation ⊩ over such an M , especially when they make heavy use of genericity. They are then assertions about Shoenfield-style P -generic extensions M ( G ), in which the ‘names’ are required without loss of generality to be elements of M B = ( V B ) M , B is the boolean completion of P in M (cf. [3, p. 50]: the notation there is RO ( P )), the generic G is named by Ĝ ∈ M B such that (⟦ p ∈ Ĝ ⟧ B = p and (cf. [11, p. 361] and [3, pp. 58–59]), and for p ∈ P and c 1 , …, c n ∈ M B , p ⊩ φ( c 1 , …, c n ) iff ⟦φ( c 1 , …, c n )⟧ B ≥ p (cf. [3, pp. 61–62]). Some prior acquaintance with large cardinal theory is also needed. At this writing no comprehensive introductory survey is yet in print, though [1], [10], [12]and [13] provide partial coverage. The scheme of definitions which follows is intended to fix notation and serve as a glossary for reference, and it is followed in turn by a description of the results of the paper. We adopt the convention that κ, λ, μ, ν, ρ and σ vary over infinite cardinals, and all other lower case Greek letters (except χ, φ, ψ, ϵ ) over arbitrary ordinals.