Coding over Core Models

Early in their careers, both Peter Koepke and Philip Welch made major contributions to two important areas of set theory, core model theory and coding, respectively. In this article we aim to survey some of the work that has been done which combines these two themes, extending Jensen's original Coding Theorem from $L$ to core models witnessing large cardinal properties

Easton supported Jensen coding and projective measure without projective Baire

We prove that it is consistent relative to a Mahlo cardinal that all sets of reals definable from countable sequences of ordinals are Lebesgue measurable, but at the same time, there is a $\Delta^1_3$ set without the Baire property. To this end, we introduce a notion of stratified forcing and stratified extension and prove an iteration theorem for these classes of forcings. Moreover we introduce a variant of Shelah's amalgamation technique that preserves stratification. The complexity of the set which provides a counterexample to the Baire property is optimal.Comment: 142 page

Strong coding

AbstractWe present here a refinement of the method of Jensen coding [7] and apply it to the study of admissible ordinals. An ordinal α is recursively inaccessible if it is both admissible and the limit of admissible ordinals. Solovay asked if it is consistent to have a real R such that the R-admissible ordinals equal the recursively inaccessible ordinals. This is a problem in class forcing as any real in a set generic extension of L must preserve the admissibility of a final segment of the admissible ordinals.Our main theorem provides an affirmative solution to Solovay's problem