Derived Models Associated to Mice

Steel, John R.

doi:10.1142/9789812794055_0003

Cited by 38 publications

(78 citation statements)

References 19 publications

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Section: Covering With Lower Parts Holds At ηmentioning

confidence: 99%

“…Using 16 Notice that M can be obtained from an iterate of N using the procedure we used in Lemma 4.11. 17 This follows from the fact that D(N ) "Σ is super fullness preserving and correctly guided".…”

Section: It Follows From the Claim That For Allmentioning

confidence: 99%

“…The aforementioned fact is an unpublished result due to Woodin. However, the readers familiar with the methods of descriptive inner model theory can dovetail a proof of it by combining the Derived Model Theorem of [18] with Theorem 10.42 of [16] or Theorem 1.2.9 of [7].As is well-known, the K c,Σ construction may not converge for various fine structural reasons. However, as mentioned above, if all countable submodels of the models appearing in the K c,Σ construction are ω 1 + 1-iterable then the K c,Σ construction converges (see [19]).…”

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confidence: 99%

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Covering with universally Baire operators

Sargsyan

2015

Advances in Mathematics

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We introduce a covering conjecture and show that it holds below AD R + "Θ is regular". We then use it to show that in the presence of mild large cardinal axioms, P F A implies that there is a transitive model containing the reals and ordinals and satisfying AD R + "Θ is regular". The method used to prove the Main Theorem of this paper is the core model induction. The paper contains the first application of the core model induction that goes significantly beyond the region of AD + + θ 0 < Θ.One of the central themes in set theory is to identify canonical inner models which compute successor cardinals correctly. A prototype of such results is Jensen's famous covering theorem which in particular implies that provided 0 # doesn't exist, for every cardinal κ ≥ ω 2 , cf((κ + ) L ) ≥ κ where L is the constructible universe.Clearly "canonical inner model" is open for interpretations. For an inner model theorist, the canonical objects of a set theoretic universe are the sets coded by a mixture of fine extender sequences and the universally Baire sets. Recall that a set of reals is universally Baire if its continuous preimages in all compact Hausdorff spaces have the property of Baire 1 . * 2000 Mathematics Subject Classifications: 03E15, 03E45, 03E60. † Keywords: Mouse, inner model theory, descriptive set theory, hod mouse, core model induction, UBH. ‡ This material is partially based upon work supported by the NSF Grant No DMS-1201348. Part of this paper was written while the author was a Leibniz Fellow at the Mathematisches Forschungsinstitut Oberwolfach.1 A set of reals is said to have the property of Baire if it is different from an open set by a meager set. 1In more set theoretic terms, a set of reals A is κ-universally Baire (or simply κ-uB) if there are trees T and S on κ × ω such that p[T ] = A and for every partial ordering P of size < κ and for every generic g ⊆ P, In most cases, to show that a K c construction converges it is enough to show that the countable submodels of the models produced during the K c construction are countably iterable. This is the content of the iterability conjecture of [19] (see Conjecture 6.5 of [19]). The best partial result is that a K c construction converges provided there is no non-domestic mouse (see [1]). One of the modern new techniques in inner model theory is the core model 2 induction and the work in this paper started by asking how the core model induction can be used to prove instances of the iterability conjecture. We remark that from now on by "K c construction" we mean the K c construction intro-Naturally, an inner model theorist will conjecture that transitive inner models that are closed under all the universally Baire sets of the universe and also are closed under robust embeddings have covering properties, i.e., if M is this hypothetical universe then for many cardinals κ, cf ((κ + ) M ) ≥ κ. Taking this intuition seriously, when we try to prove the iterability conjecture via core model induction, instead of proving instances of iterability conjecture, we ...

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Section: Covering With Lower Parts Holds At ηmentioning

confidence: 99%

Section: It Follows From the Claim That For Allmentioning

confidence: 99%

mentioning

confidence: 99%

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Covering with universally Baire operators

Sargsyan

2015

Advances in Mathematics

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“…11 Remark 6.2. We caution the reader that the notation "Env" and the term "envelope" are taken from Steel [23], where they were defined differently. Jackson [5] proved that in the intended context of AD the pointclass Λ(Δ Γ , κ) is equal to the pointclass that Steel calls Env(Γ) (which is a boldface pointclass defined from a lightface inductive-like pointclass Γ).…”

Section: The Envelope In the General Casementioning

confidence: 99%

The envelope of a pointclass under a local determinacy hypothesis

Wilson

2015

Annals of Pure and Applied Logic

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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.

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“…It is shown in [21][section 4] that under AD, every mouse operator on H(ω 1 ) determines itself on generic extensions in some H(ω 1 ) cone.…”

Section: Proofmentioning

confidence: 99%