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  1. Some applications of coarse inner model theory.Greg Hjorth - 1997 - Journal of Symbolic Logic 62 (2):337-365.
    The Martin-Steel coarse inner model theory is employed in obtaining new results in descriptive set theory. $\underset{\sim}{\Pi}$ determinacy implies that for every thin Σ 1 2 equivalence relation there is a Δ 1 3 real, N, over which every equivalence class is generic--and hence there is a good Δ 1 2 (N ♯ ) wellordering of the equivalence classes. Analogous results are obtained for Π 1 2 and Δ 1 2 quasilinear orderings and $\underset{\sim}{\Pi}^1_2$ determinacy is shown to imply that (...)
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  • A boundedness lemma for iterations.Greg Hjorth - 2001 - Journal of Symbolic Logic 66 (3):1058-1072.
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  • The comparison lemma.John R. Steel - forthcoming - Annals of Pure and Applied Logic.
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  • A Long Pseudo-Comparison of Premice in L[x].Farmer Schlutzenberg - 2018 - Notre Dame Journal of Formal Logic 59 (4):599-604.
    A significant open problem in inner model theory is the analysis of HODL[x] as a strategy premouse, for a Turing cone of reals x. We describe here an obstacle to such an analysis. Assuming sufficient large cardinals, for a Turing cone of reals x there are proper class 1-small premice M,N, with Woodin cardinals δ,ε, respectively, such that M|δ and N|ε are in L[x], M and N are countable in L[x], and the pseudo-comparison of M with N succeeds, is in (...)
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  • On the prewellorderings associated with the directed systems of mice.Grigor Sargsyan - 2013 - Journal of Symbolic Logic 78 (3):735-763.
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  • Inner model operators in L.Mitch Rudominer - 2000 - Annals of Pure and Applied Logic 101 (2-3):147-184.
    An inner model operator is a function M such that given a Turing degree d, M is a countable set of reals, d M, and M has certain closure properties. The notion was introduced by Steel. In the context of AD, we study inner model operators M such that for a.e. d, there is a wellorder of M in L). This is related to the study of mice which are below the minimal inner model with ω Woodin cardinals. As a (...)
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  • In inner models with Woodin cardinals.Sandra Müller & Grigor Sargsyan - 2021 - Journal of Symbolic Logic 86 (3):871-896.
    We analyze the hereditarily ordinal definable sets $\operatorname {HOD} $ in $M_n[g]$ for a Turing cone of reals x, where $M_n$ is the canonical inner model with n Woodin cardinals build over x and g is generic over $M_n$ for the Lévy collapse up to its bottom inaccessible cardinal. We prove that assuming $\boldsymbol \Pi ^1_{n+2}$ -determinacy, for a Turing cone of reals x, $\operatorname {HOD} ^{M_n[g]} = M_n,$ where $\mathcal {M}_{\infty }$ is a direct limit of iterates of $M_{n+1}$, (...)
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  • Scales in K(R) at the End of a Weak Gap.J. R. Steel - 2008 - Journal of Symbolic Logic 73 (2):369 - 390.
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  • Determinacy and Jónsson cardinals in L.S. Jackson, R. Ketchersid, F. Schlutzenberg & W. H. Woodin - 2014 - Journal of Symbolic Logic 79 (4):1184-1198.
    Assume ZF + AD +V=L and letκ< Θ be an uncountable cardinal. We show thatκis Jónsson, and that if cof = ω thenκis Rowbottom. We also establish some other partition properties.
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  • Maximal almost disjoint families, determinacy, and forcing.Karen Bakke Haga, David Schrittesser & Asger Törnquist - 2022 - Journal of Mathematical Logic 22 (1):2150026.
    We study the notion of [Formula: see text]-MAD families where [Formula: see text] is a Borel ideal on [Formula: see text]. We show that if [Formula: see text] is any finite or countably iterated Fubini product of the ideal of finite sets [Formula: see text], then there are no analytic infinite [Formula: see text]-MAD families, and assuming Projective Determinacy and Dependent Choice there are no infinite projective [Formula: see text]-MAD families; and under the full Axiom of Determinacy [Formula: see text][Formula: (...)
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  • Maximal almost disjoint families, determinacy, and forcing.Karen Bakke Haga, David Schrittesser & Asger Törnquist - 2021 - Journal of Mathematical Logic 22 (1).
    We study the notion of ????-MAD families where ???? is a Borel ideal on ω. We show that if ???? is any finite or countably iterated Fubini product of the ideal of finite sets Fin, then there are no analytic...
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  • Is there a set of reals not in K(R)?Daniel W. Cunningham - 1998 - Annals of Pure and Applied Logic 92 (2):161-210.
    We show, using the fine structure of K, that the theory ZF + AD + X R[X K] implies the existence of an inner model of ZF + AD + DC containing a measurable cardinal above its Θ, the supremum of the ordinals which are the surjective image of R. As a corollary, we show that HODK = K for some P K where K is the Dodd-Jensen Core Model relative to P. In conclusion, we show that the theory ZF (...)
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  • Hod up to A D R + Θ is measurable.Rachid Atmai & Grigor Sargsyan - 2019 - Annals of Pure and Applied Logic 170 (1):95-108.
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  • An analysis of the models.Rachid Atmai - 2019 - Journal of Symbolic Logic 84 (1):1-26.
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