13 found
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  1.  29
    The exact strength of the class forcing theorem.Victoria Gitman, Joel David Hamkins, Peter Holy, Philipp Schlicht & Kameryn J. Williams - 2020 - Journal of Symbolic Logic 85 (3):869-905.
    The class forcing theorem, which asserts that every class forcing notion ${\mathbb {P}}$ admits a forcing relation $\Vdash _{\mathbb {P}}$, that is, a relation satisfying the forcing relation recursion—it follows that statements true in the corresponding forcing extensions are forced and forced statements are true—is equivalent over Gödel–Bernays set theory $\text {GBC}$ to the principle of elementary transfinite recursion $\text {ETR}_{\text {Ord}}$ for class recursions of length $\text {Ord}$. It is also equivalent to the existence of truth predicates for the (...)
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  2.  16
    Characterizations of pretameness and the Ord-cc.Peter Holy, Regula Krapf & Philipp Schlicht - 2018 - Annals of Pure and Applied Logic 169 (8):775-802.
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  3.  13
    Small embedding characterizations for large cardinals.Peter Holy, Philipp Lücke & Ana Njegomir - 2019 - Annals of Pure and Applied Logic 170 (2):251-271.
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  4.  9
    Small models, large cardinals, and induced ideals.Peter Holy & Philipp Lücke - 2021 - Annals of Pure and Applied Logic 172 (2):102889.
    We show that many large cardinal notions up to measurability can be characterized through the existence of certain filters for small models of set theory. This correspondence will allow us to obtain a canonical way in which to assign ideals to many large cardinal notions. This assignment coincides with classical large cardinal ideals whenever such ideals had been defined before. Moreover, in many important cases, relations between these ideals reflect the ordering of the corresponding large cardinal properties both under direct (...)
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  5.  2
    Ideal Operators and Higher Indescribability.Brent Cody & Peter Holy - forthcoming - Journal of Symbolic Logic:1-39.
    We investigate properties of the ineffability and the Ramsey operator, and a common generalization of those that was introduced by the second author, with respect to higher indescribability, as introduced by the first author. This extends earlier investigations on the ineffability operator by James Baumgartner, and on the Ramsey operator by Qi Feng, by Philip Welch et al., and by the first author.
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  6.  28
    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.
  7.  14
    An axiomatic approach to forcing in a general setting.Rodrigo A. Freire & Peter Holy - 2022 - Bulletin of Symbolic Logic 28 (3):427-450.
    The technique of forcing is almost ubiquitous in set theory, and it seems to be based on technicalities like the concepts of genericity, forcing names and their evaluations, and on the recursively defined forcing predicates, the definition of which is particularly intricate for the basic case of atomic first order formulas. In his [3], the first author has provided an axiomatic framework for set forcing over models of $\mathrm {ZFC}$ that is a collection of guiding principles for extensions over which (...)
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  8.  6
    An ordinal-connection axiom as a weak form of global choice under the GCH.Rodrigo A. Freire & Peter Holy - 2022 - Archive for Mathematical Logic 62 (3):321-332.
    The minimal ordinal-connection axiom $$MOC$$ was introduced by the first author in R. Freire. (South Am. J. Log. 2:347–359, 2016). We observe that $$MOC$$ is equivalent to a number of statements on the existence of certain hierarchies on the universe, and that under global choice, $$MOC$$ is in fact equivalent to the $${{\,\mathrm{GCH}\,}}$$. Our main results then show that $$MOC$$ corresponds to a weak version of global choice in models of the $${{\,\mathrm{GCH}\,}}$$ : it can fail in models of the (...)
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  9.  21
    Large cardinals and lightface definable well-orders, without the gch.Sy-David Friedman, Peter Holy & Philipp Lücke - 2015 - Journal of Symbolic Logic 80 (1):251-284.
  10.  5
    Asymmetric Cut and Choose Games.Christopher Henney-Turner, Peter Holy, Philipp Schlicht & Philip Welch - forthcoming - Bulletin of Symbolic Logic:1-31.
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  11.  6
    Ideal topologies in higher descriptive set theory.Peter Holy, Marlene Koelbing, Philipp Schlicht & Wolfgang Wohofsky - 2022 - Annals of Pure and Applied Logic 173 (4):103061.
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  12.  15
    Local club condensation and l-likeness.Peter Holy, Philip Welch & Liuzhen Wu - 2015 - Journal of Symbolic Logic 80 (4):1361-1378.
  13.  10
    Σ1-wellorders without collapsing.Peter Holy - 2015 - Archive for Mathematical Logic 54 (3-4):453-462.
    Given an uncountable cardinal κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa}$$\end{document} that satisfies κκ=κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa^{\kappa}=\kappa}$$\end{document}, we provide a forcing that is <κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${<\kappa}$$\end{document} -closed, has size 2κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${2^\kappa}$$\end{document} and is κ+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa^+}$$\end{document} -cc, to introduce a Σ1-definable wellorder of H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} (...)
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