A proof of projective determinacy

Derived Models Associated to Mice

“…There are no interesting homogeneously Suslin sets unless there are measurable cardinals. For the most part, we shall be working under the assumption that there are infinitely many Woodin cardinals, in which case one has: Theorem 2.3 Let λ be a limit of Woodin cardinals; then (a) Hom <λ is closed under complements and real quantification ( [10]), (b) every Hom <λ set has a Hom <λ scale ( [25]). …”

“…The results of [9], [30], and [10] show that if λ is a limit of Woodin cardinals, then the Hom <λ sets are precisely the < λ-universally Baire sets. (See [25].)…”

“…The following carries over a basic fact about ENF's to this fine-structural setting. [10]] Let N be a premouse, and N |= ZFC+ " λ is a limit of cardinals which are Woodin via extenders on my sequence". Let g be N -generic over some poset of size < λ; then working inside N [g], the following are equivalent, for any set C of reals:…”

Bounding by Canonical Functions, With Ch

“…Recall that a sequence of measures µ i : i < ω such that each µ i concentrates on κ i is a countably complete tower if for any sequence A i : i < ω such that each A i ∈ µ i there is a sequence z ∈ κ ω such that each z i ∈ A i . See [6,14] for more detail.…”

Large Cardinals, Inner Models, and Determinacy: An Introductory Overview

“…However one important feature is that they do have complements defined as projections of treesT with the latter definable from their weakly homogenously tree T : However on its own this is no help. The breakthrough was: Theorem 3.17 (Martin-Steel [19]) Suppose that λ is a Woodin cardinal and T is a λ + weakly homogeneously Suslin tree. Then for γ < λ, theT above is γ-homogeneously Suslin.…”