The derived model theorem

Abstract

We shall exposit here one of the basic theorems leading from large cardinals to determinacy, a result of Woodin known as the derived model theorem. The theorem dates from the mid80’s and has been exposited in several sets of informally circulated lecture notes (e.g., [12], [14]), but we know of no exposition in print. We shall also include a number of subsidiary and related results due to various people. We shall use very heavily the technique of stationary tower forcing. The reader should see Woodin’s paper [15] or Larson’s [3] for the basic facts about stationary tower forcing. The second main technical tool needed for a full proof of the derived model theorem is the theory of iteration trees. This is one of the main ingredients in the proof of Theorem 2.1 below, but since we shall simply take that theorem as a “black box” here, it is possible to read this paper without knowing what an iteration tree is. The paper is organized as follows. In §1, we introduce homogeneity, weak homogeneity, and universal Baireness. The main result here is the Martin-Solovay theorem, according to which all weakly homogeneous sets are universally Baire. We give a reasonably complete proof of this theorem. In §2 and §3, we show that in the presence of Woodin cardinals, homogeneity, weak homogeneity, and universal Baireness are equivalent. We also give, in §3, an argument of Woodin’s which shows that strong cardinals yield universally Baire representations after a collapse. In §4 we prove the tree production lemma, according to which sets admitting definitions with certain absoluteness properties are universally Baire. §5 contains a generic absoluteness theorem for (Σ1) Hom∞ statements. In §6 we state and prove the derived model theorem. In §7 we prove that in derived models, the pointclass Σ1 has the Scale Property, and in §8, we use this to produce derived models which satisfy ADR. This paper was written in Fall 2002, and circulated informally. In 2004, Larson’s monograph [3] on stationary tower forcing appeared. Much of the material in §2, §3, and §4 can be found in section 3.3 of [3]. Sections 3.2 and 3.4 of [3] use this machinery to prove important results of Woodin concerning generic absoluteness. In another direction, Neeman’s recent