Coding over a measurable cardinal

Abstract

The purpose of this paper is to extend the coding method (see Beller, Jensen and Welch [82]) into the context of large cardinals.Theorem. Suppose μ is a normal measure on κ in V and 〈 V, A〉 ⊨ ZFC. Then there is a 〈V, A〉-definable forcing for producing a real R such that:(a) V[R] ⊨ ZFC and A is V[R]-definable with parameter R.(b) V[R] = L[μ*, R], where μ* is a normal measure on κ in V[R] extending μ.(c) V ⊨ GCH → is cardinal and cofinality preserving.Corollary. It is consistent that μ is a normal measure, R ⊆ ω is not set-generic over L[μ] and 0+ ∉ L[μ, R].Some other corollaries will be discussed in §4 of the paper.The main difficulty in L[μ]-coding lies in the problem of “stationary restraint”.As in all coding constructions, conditions will be of the form belonging to an initial segment of the cardinals, where p(γ) is a condition for almost disjoint coding into a subset of γ+. In addition for limit cardinals γ in Domain(p), 〈pγ′∣γ′ < γ〉 serves to code pγ.An important restriction in coding arguments is that for inaccessible for only a nonstationary set of γ′ < γ. The reason is that otherwise there are conflicts between the restraint imposed by the different and the need to code extensions of pγ below γ.