Rudimentary recursion, provident sets and forcing

and call a set A provident if it is transitive and closed under all p-rud-rec functions with p a member of A. * If ζ is the least ordinal not in a provident set A, then ζ is indecomposable, that is, that the sum of two ordinals less than ζ is less than ζ. Conversely, if ζ is indecomposable, η > ζ and p ∈ Jζ , then the Jensen set Jη is is closed under all p-rud rec functions; in particular Jζ is provident. Let c be a transitive set. A modification of the usual hierarchy defining the constructible closure L(c) of c proves desirable: define, by a simultaneous rudimentary recursion on ordinals, sets cν , P c ν thus: c0 = ∅ cν+1 = c ∩ {x | x ⊆ cν} cλ = ⋃ ν<λcν P c 0 = ∅ P c ν+1 = {cν} ∪ cν+1 ∪ T(P c ν ) P c λ = ⋃ ν<λP c ν