Measurable cardinals and the reals of L

• At stage 0, M0 = M , U0 = U and κ0 = κ. • At stage α+1, if it is well-founded, we build Ult(Mα, Uα) and let Mα+1 be its transitive collapse, where the ultrapower is computed from the point of view of Mα. Let jα,α+1 : Mα → Mα+1 be the ultrapower embedding. Then set κα+1 = j(κα) and Uα+1 = j(Uα). • At limit stages λ, if it is well-founded, we define Mλ as the direct limit of the directed system of embeddings generated by (Mα, jα,α+1 :α < λ), set κλ to be the image of the κα under this limit and Uλ to be the limit of the Uα.