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  1.  28
    Determinacy in L.Nam Trang - 2014 - Journal of Mathematical Logic 14 (1):1450006.
    Assume V = L ⊨ ZF + DC + Θ > ω2 + μ is a normal fine measure on.
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  2.  28
    $$I_0$$ I 0 and combinatorics at $$\lambda ^+$$ λ +.Nam Trang & Xianghui Shi - 2017 - Archive for Mathematical Logic 56 (1-2):131-154.
    We investigate the compatibility of I0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_0$$\end{document} with various combinatorial principles at λ+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^+$$\end{document}, which include the existence of λ+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^+$$\end{document}-Aronszajn trees, square principles at λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}, the existence of good scales at λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}, stationary reflections (...)
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  3.  21
    On a class of maximality principles.Daisuke Ikegami & Nam Trang - 2018 - Archive for Mathematical Logic 57 (5-6):713-725.
    We study various classes of maximality principles, \\), introduced by Hamkins :527–550, 2003), where \ defines a class of forcing posets and \ is an infinite cardinal. We explore the consistency strength and the relationship of \\) with various forcing axioms when \. In particular, we give a characterization of bounded forcing axioms for a class of forcings \ in terms of maximality principles MP\\) for \ formulas. A significant part of the paper is devoted to studying the principle MP\\) (...)
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  4.  9
    Divergent Models with the Failure of the Continuum Hypothesis.Nam Trang - forthcoming - Journal of Symbolic Logic:1-11.
    We construct divergent models of $\mathsf {AD}^+$ along with the failure of the Continuum Hypothesis ( $\mathsf {CH}$ ) under various assumptions. Divergent models of $\mathsf {AD}^+$ play an important role in descriptive inner model theory; all known analyses of HOD in $\mathsf {AD}^+$ models (without extra iterability assumptions) are carried out in the region below the existence of divergent models of $\mathsf {AD}^+$. Our results are the first step toward resolving various open questions concerning the length of definable prewellorderings (...)
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  5.  13
    Derived models and supercompact measures on.Nam Trang - 2015 - Mathematical Logic Quarterly 61 (1-2):56-65.
    The main result of this paper is Theorem, which shows that it is possible for derived models to satisfy “ω1 is ‐supercompact”. Other constructions of models of this theory are also discussed; in particular, Theorem constructs a normal fine measure on and hence a model of “Θ is regular”+“ω1 is ‐supercompact” from a model of “Θ is measurable”.
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  6.  23
    Structure theory of L and its applications.Nam Trang - 2015 - Journal of Symbolic Logic 80 (1):29-55.
    In this paper, we explore the structure theory ofL under the hypothesisL ⊧ “AD +μis a normal fine measure on” and give some applications. First we show that “ ZFC + there existω2Woodin cardinals”1has the same consistency strength as “ AD +ω1is ℝ-supercompact”. During this process we show that ifL ⊧ AD then in factL ⊧ AD+. Next we prove important properties ofL including Σ1-reflection and the uniqueness ofμinL. Then we give the computation of full HOD inL. Finally, we use (...)
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  7.  10
    Countable Length Everywhere Club Uniformization.William Chan, Stephen Jackson & Nam Trang - 2023 - Journal of Symbolic Logic 88 (4):1556-1572.
    Assume $\mathsf {ZF} + \mathsf {AD}$ and all sets of reals are Suslin. Let $\Gamma $ be a pointclass closed under $\wedge $, $\vee $, $\forall ^{\mathbb {R}}$, continuous substitution, and has the scale property. Let $\kappa = \delta (\Gamma )$ be the supremum of the length of prewellorderings on $\mathbb {R}$ which belong to $\Delta = \Gamma \cap \check \Gamma $. Let $\mathsf {club}$ denote the collection of club subsets of $\kappa $. Then the countable length everywhere club uniformization (...)
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  8.  8
    More definable combinatorics around the first and second uncountable cardinals.William Chan, Stephen Jackson & Nam Trang - 2023 - Journal of Mathematical Logic 23 (3).
    Assume [Formula: see text]. If [Formula: see text] is an ordinal and X is a set of ordinals, then [Formula: see text] is the collection of order-preserving functions [Formula: see text] which have uniform cofinality [Formula: see text] and discontinuous everywhere. The weak partition properties on [Formula: see text] and [Formula: see text] yield partition measures on [Formula: see text] when [Formula: see text] and [Formula: see text] when [Formula: see text]. The following almost everywhere continuity properties for functions on (...)
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  9.  8
    Sealing of the universally baire sets.Grigor Sargsyan & Nam Trang - 2021 - Bulletin of Symbolic Logic 27 (3):254-266.
    A set of reals is universally Baire if all of its continuous preimages in topological spaces have the Baire property. ${\sf Sealing}$ is a type of generic absoluteness condition introduced by Woodin that asserts in strong terms that the theory of the universally Baire sets cannot be changed by set forcings. The ${\sf Largest\ Suslin\ Axiom}$ is a determinacy axiom isolated by Woodin. It asserts that the largest Suslin cardinal is inaccessible for ordinal definable surjections. Let ${\sf LSA}$ - ${\sf (...)
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  10.  9
    Tame failures of the unique branch hypothesis and models of ADℝ + Θ is regular.Grigor Sargsyan & Nam Trang - 2016 - Journal of Mathematical Logic 16 (2):1650007.
    In this paper, we show that the failure of the unique branch hypothesis for tame iteration trees implies that in some homogenous generic extension of [Formula: see text] there is a transitive model [Formula: see text] containing [Formula: see text] such that [Formula: see text] is regular. The results of this paper significantly extend earlier works from [Non-tame mice from tame failures of the unique branch bypothesis, Canadian J. Math. 66 903–923; Core models with more Woodin cardinals, J. Symbolic Logic (...)
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  11.  15
    Determinacy from strong compactness of ω1.Nam Trang & Trevor M. Wilson - 2021 - Annals of Pure and Applied Logic 172 (6):102944.
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  12.  9
    Supercompactness Can Be Equiconsistent with Measurability.Nam Trang - 2021 - Notre Dame Journal of Formal Logic 62 (4):593-618.
    The main result of this paper, built on previous work by the author and T. Wilson, is the proof that the theory “ADR+DC + there is an R-complete measure on Θ” is equiconsistent with “ZF+DC+ ADR + there is a supercompact measure on ℘ω1(℘(R))+Θ is regular.” The result and techniques presented here contribute to the general program of descriptive inner model theory and in particular, to the general study of compactness phenomena in the context of ZF+DC.
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