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David Aspero
University of East Anglia
  1.  25
    Large Cardinals and Locally Defined Well-Orders of the Universe.David Asperó & Sy-David Friedman - 2009 - Annals of Pure and Applied Logic 157 (1):1-15.
    By forcing over a model of with a class-sized partial order preserving this theory we produce a model in which there is a locally defined well-order of the universe; that is, one whose restriction to all levels H is a well-order of H definable over the structure H, by a parameter-free formula. Further, this forcing construction preserves all supercompact cardinals as well as all instances of regular local supercompactness. It is also possible to define variants of this construction which, in (...)
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  2. Definable Well-Orders of $H(\Omega _2)$ and $GCH$.David Asperó & Sy-David Friedman - 2012 - Journal of Symbolic Logic 77 (4):1101-1121.
    Assuming ${2^{{N_0}}}$ = N₁ and ${2^{{N_1}}}$ = N₂, we build a partial order that forces the existence of a well-order of H(ω₂) lightface definable over ⟨H(ω₂), Є⟩ and that preserves cardinal exponentiation and cofinalities.
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  3.  11
    Separating Club-Guessing Principles in the Presence of Fat Forcing Axioms.David Asperó & Miguel Angel Mota - 2016 - Annals of Pure and Applied Logic 167 (3):284-308.
  4.  8
    Adding Many Baumgartner Clubs.David Asperó - 2017 - Archive for Mathematical Logic 56 (7-8):797-810.
    I define a homogeneous \–c.c. proper product forcing for adding many clubs of \ with finite conditions. I use this forcing to build models of \=\aleph _2\), together with \\) and \ large and with very strong failures of club guessing at \.
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  5.  25
    Forcing Lightface Definable Well-Orders Without the GCH.David Asperó, Peter Holy & Philipp Lücke - 2015 - Annals of Pure and Applied Logic 166 (5):553-582.
  6.  9
    Guessing and Non-Guessing of Canonical Functions.David Asperó - 2007 - Annals of Pure and Applied Logic 146 (2):150-179.
    It is possible to control to a large extent, via semiproper forcing, the parameters measuring the guessing density of the members of any given antichain of stationary subsets of ω1 . Here, given a pair of ordinals, we will say that a stationary set Sω1 has guessing density if β0=γ and , where γ is, for every stationary S*ω1, the infimum of the set of ordinals τ≤ω1+1 for which there is a function with ot)<τ for all νS* and with {νS*:gF} (...)
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  7.  24
    A Maximal Bounded Forcing Axiom.David Asperó - 2002 - Journal of Symbolic Logic 67 (1):130-142.
    After presenting a general setting in which to look at forcing axioms, we give a hierarchy of generalized bounded forcing axioms that correspond level by level, in consistency strength, with the members of a natural hierarchy of large cardinals below a Mahlo. We give a general construction of models of generalized bounded forcing axioms. Then we consider the bounded forcing axiom for a class of partially ordered sets Γ 1 such that, letting Γ 0 be the class of all stationary-set-preserving (...)
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  8.  6
    Coding by Club-Sequences.David Asperó - 2006 - Annals of Pure and Applied Logic 142 (1):98-114.
    Given any subset A of ω1 there is a proper partial order which forces that the predicate xA and the predicate xω1A can be expressed by -provably incompatible Σ3 formulas over the structure Hω2,,NSω1. Also, if there is an inaccessible cardinal, then there is a proper partial order which forces the existence of a well-order of Hω2 definable over Hω2,,NSω1 by a provably antisymmetric Σ3 formula with two free variables. The proofs of these results involve a technique for manipulating the (...)
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  9.  27
    Dense Non-Reflection for Stationary Collections of Countable Sets.David Asperó, John Krueger & Yasuo Yoshinobu - 2010 - Annals of Pure and Applied Logic 161 (1):94-108.
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  10.  9
    Bounded Forcing Axioms and the Continuum.David Asperó & Joan Bagaria - 2001 - Annals of Pure and Applied Logic 109 (3):179-203.
    We show that bounded forcing axioms are consistent with the existence of -gaps and thus do not imply the Open Coloring Axiom. They are also consistent with Jensen's combinatorial principles for L at the level ω2, and therefore with the existence of an ω2-Suslin tree. We also show that the axiom we call BMM3 implies 21=2, as well as a stationary reflection principle which has many of the consequences of Martin's Maximum for objects of size 2. Finally, we give an (...)
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  11.  4
    Bounded Martin's Maximum, Weak Erdӧs Cardinals, and Ψ Ac.David Asperó & Philip D. Welch - 2002 - Journal of Symbolic Logic 67 (3):1141-1152.
  12.  19
    A Forcing Notion Collapsing $\Aleph _3 $ and Preserving All Other Cardinals.David Asperó - 2018 - Journal of Symbolic Logic 83 (4):1579-1594.
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  13.  9
    Baumgartnerʼs Conjecture and Bounded Forcing Axioms.David Asperó, Sy-David Friedman, Miguel Angel Mota & Marcin Sabok - 2013 - Annals of Pure and Applied Logic 164 (12):1178-1186.
  14.  6
    Bounded Martin’s Maximum with an Asterisk.David Asperó & Ralf Schindler - 2014 - Notre Dame Journal of Formal Logic 55 (3):333-348.
    We isolate natural strengthenings of Bounded Martin’s Maximum which we call ${\mathsf{BMM}}^{*}$ and $A-{\mathsf{BMM}}^{*,++}$, and we investigate their consequences. We also show that if $A-{\mathsf{BMM}}^{*,++}$ holds true for every set of reals $A$ in $L$, then Woodin’s axiom $$ holds true. We conjecture that ${\mathsf{MM}}^{++}$ implies $A-{\mathsf{BMM}}^{*,++}$ for every $A$ which is universally Baire.
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  15.  30
    Bounded Martin's Maximum, Weak Erdős Cardinals, and $\Psi\Sb {AC}$.David Asperó & Philip Welch - 2002 - Journal of Symbolic Logic 67 (3):1141-1152.
  16.  9
    Dependent Choice, Properness, and Generic Absoluteness.David Asperó & Asaf Karagila - forthcoming - Review of Symbolic Logic:1-25.
    We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to $\mathsf {DC}$ -preserving symmetric submodels of forcing extensions. Hence, $\mathsf {ZF}+\mathsf {DC}$ not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of (...)
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  17.  18
    Forcing Notions in Inner Models.David Asperó - 2009 - Archive for Mathematical Logic 48 (7):643-651.
    There is a partial order ${\mathbb{P}}$ preserving stationary subsets of ω 1 and forcing that every partial order in the ground model V that collapses a sufficiently large ordinal to ω 1 over V also collapses ω 1 over ${V^{\mathbb{P}}}$ . The proof of this uses a coding of reals into ordinals by proper forcing discovered by Justin Moore and a symmetric extension of the universe in which the Axiom of Choice fails. Also, using one feature of the proof of (...)
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  18.  9
    On a Convenient Property About $${[\Gamma]^{\Aleph_0}}$$.David Asperó - 2009 - Archive for Mathematical Logic 48 (7):653-677.
    Several situations are presented in which there is an ordinal γ such that ${\{ X \in [\gamma]^{\aleph_0} : X \cap \omega_1 \in S\,{\rm and}\, ot(X) \in T \}}$ is a stationary subset of ${[\gamma]^{\aleph_0}}$ for all stationary ${S, T\subseteq \omega_1}$ . A natural strengthening of the existence of an ordinal γ for which the above conclusion holds lies, in terms of consistency strength, between the existence of the sharp of ${H_{\omega_2}}$ and the existence of sharps for all reals. Also, an (...)
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  19.  1
    On a Convenient Property About [FORMULA].David Asperó - 2009 - Archive for Mathematical Logic 48 (7):653-677.
    Several situations are presented in which there is an ordinal γ such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\{ X \in [\gamma]^{\aleph_0} : X \cap \omega_1 \in S\,{\rm and}\, ot \in T \}}$$\end{document} is a stationary subset of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${[\gamma]^{\aleph_0}}$$\end{document} for all stationary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${S, T\subseteq \omega_1}$$\end{document}. A natural strengthening of the existence of an ordinal γ for which the above (...)
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  20.  7
    Reducibility of Equivalence Relations Arising From Nonstationary Ideals Under Large Cardinal Assumptions.David Asperó, Tapani Hyttinen, Vadim Kulikov & Miguel Moreno - 2019 - Notre Dame Journal of Formal Logic 60 (4):665-682.
    Working under large cardinal assumptions such as supercompactness, we study the Borel reducibility between equivalence relations modulo restrictions of the nonstationary ideal on some fixed cardinal κ. We show the consistency of Eλ-clubλ++,λ++, the relation of equivalence modulo the nonstationary ideal restricted to Sλλ++ in the space λ++, being continuously reducible to Eλ+-club2,λ++, the relation of equivalence modulo the nonstationary ideal restricted to Sλ+λ++ in the space 2λ++. Then we show that for κ ineffable Ereg2,κ, the relation of equivalence modulo (...)
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  21.  17
    Bounded Martin's Maximum, Weak [Image] Cardinals, and [Image].David Asperó & Philip D. Welch - 2002 - Journal of Symbolic Logic 67 (3):1141 - 1152.
    We prove that a form of the $Erd\H{o}s$ property (consistent with $V = L\lbrack H_{\omega_2}\rbrack$ and strictly weaker than the Weak Chang's Conjecture at ω1), together with Bounded Martin's Maximum implies that Woodin's principle $\psi_{AC}$ holds, and therefore 2ℵ0 = ℵ2. We also prove that $\psi_{AC}$ implies that every function $f: \omega_1 \rightarrow \omega_1$ is bounded by some canonical function on a club and use this to produce a model of the Bounded Semiproper Forcing Axiom in which Bounded Martin's Maximum (...)
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