The envelope of a pointclass under a local determinacy hypothesis

Abstract

Given an inductive-like pointclass Γ and assuming the Axiom of Determinacy, Martin identified and analyzed a pointclass that contains the prewellorderings of the next scale beyond Γ if such a scale exists. We show that much of Martin's analysis can be carried out assuming only ZF + DC R and Δ Γ determinacy by adapting arguments of Kechris and Woodin [10] and Martin [13]. This generalization can be used to show that every set of reals is Suslin in the intersection of two divergent models of AD + , giving a new proof of a theorem of Woodin, as well as to show that every set of reals is Suslin in the derived model at an indestructibly weakly compact limit of Woodin cardinals.

Determinacy from strong compactness of $ω_1$

“…] is a strong gap, by the Kechris-Woodin theorem, AD + holds in sLp G Ω (R, Code(Ω))|α, and again by results of [16], [10], and [22], we also get a self-justifying-system A Wadge-cofinal in sLp G Ω (R, Code(Ω))|α ∩ ℘(R).…”

“…21 The initial segment may be strict. 22 Ordinal definability from Σ in the definition of MC(Σ) is in the language of set theory, not in the language of sLp G Ω (R, Code(Ω)), but by the paragraph above 4.2, this will not make a difference.…”

“…This notion was used by Martin to identify the next scaled pointclass after an inductive-like scaled pointclass in the AD context; see Jackson [1]. We will need its adaptation to the partial determinacy context as defined in the second author's thesis [21] (see also the subsequent paper [22]. )…”

“…It is essentially proved in the thesis [21] (which deals with generic large cardinal properties of ω 1 in ZFC rather than with large cardinal properties of ω 1 , but the argument carries over to the present context.) An easier version with "scale" replaced by "semiscale" is proved in the paper [22], and a special case of the scale construction appears in another paper [23].…”

Universally Baire Sets and Generic Absoluteness

“…Two related theorems are that if κ is supercompact then every tree becomes weakly homogeneous in some small forcing extension ( Another difference from the results mentioned above is that because our large cardinal hypothesis of measurability is so weak, we will need to augment it with a smallness assumption about the tree, namely that not too many sets are constructible from it; this is made precise in the statement of the lemma. A related argument appears in Wilson [18,Lemma 9.4], where the large cardinal hypothesis is weak compactness and the smallness assumption is about a pointclass called the envelope.…”