In this paper we shall answer some questions in the set theory of L, the universe of all sets constructible from the reals. In order to do so, we shall assume ADL, the hypothesis that all 2-person games of perfect information on ω whose payoff set is in L are determined. This is by now standard practice. ZFC itself decides few questions in the set theory of L, and for reasons we cannot discuss here, ZFC + ADL yields the most (...) interesting “completion” of the ZFC-theory of L.ADL implies that L satisfies “every wellordered set of reals is countable”, so that the axiom of choice fails in L. Nevertheless, there is a natural inner model of L, namely HODL, which satisfies ZFC.. The superscript “L” indicates, here and below, that the notion in question is to be interpreted in L.) HODL is reasonably close to the full L, in ways we shall make precise in § 1. The most important of the questions we shall answer concern HODL: what is its first order theory, and in particular, does it satisfy GCH?These questions first drew attention in the 70's and early 80's. (shrink)

In this paper we shall answer some questions in the set theory of L, the universe of all sets constructible from the reals. In order to do so, we shall assume ADL, the hypothesis that all 2-person games of perfect information on ω whose payoff set is in L are determined. This is by now standard practice. ZFC itself decides few questions in the set theory of L, and for reasons we cannot discuss here, ZFC + ADL yields the most (...) interesting “completion” of the ZFC-theory of L.ADL implies that L satisfies “every wellordered set of reals is countable”, so that the axiom of choice fails in L. Nevertheless, there is a natural inner model of L, namely HODL, which satisfies ZFC.. The superscript “L” indicates, here and below, that the notion in question is to be interpreted in L.) HODL is reasonably close to the full L, in ways we shall make precise in § 1. The most important of the questions we shall answer concern HODL: what is its first order theory, and in particular, does it satisfy GCH?These questions first drew attention in the 70's and early 80's. (shrink)

The purpose of this paper is to outline some recent progress in descriptive inner model theory, a branch of set theory which studies descriptive set theoretic and inner model theoretic objects using tools from both areas. There are several interlaced problems that lie on the border of these two areas of set theory, but one that has been rather central for almost two decades is the conjecture known as the Mouse Set Conjecture. One particular motivation for resolving MSC is that (...) it provides grounds for solving the inner model problem which dates back to 1960s. There have been some new partial results on MSC and the methods used to prove the new instances suggest a general program for solving the full conjecture. It is then our goal to communicate the ideas of this program to the community at large. (shrink)