A Simpler proof of Jensen's Coding Theorem

[82] provides a proof of Jensen's remarkable Coding Theorem , which demonstrates that the universe can be included in L[R] for some real R, via class forcing. The purpose of this article is to present a simpler proof of Jensen's theorem, obtained by implementing some changes first developed for the theory of strong coding (Friedman [87]). The basic idea is to first choose A ⊆ ORD so that V = L[A] and then generically add sets G α ⊆ α + , α O or an infinite cardinal (O + denotes ω) so that G α codes both G α + and A ∩ α +. Also for limit cardinals α, G α is coded by G ¯ α | ¯ α < α. Thus there are two " building blocks " for the forcing, the successor coding and the limit coding. We modify the successor coding so as to eliminate Jensen's use of " generic codes " (this improves an earlier modification of this type, due to Welch and Donder). And we thin out the limit coding so as to eliminate the technical problems causing Jensen's split into cases according to whether or not O # exists.