19 found
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  1. Multiverse Conceptions in Set Theory.Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo - 2015 - Synthese 192 (8):2463-2488.
    We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’. Furthermore, we propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and potentialism with regard to the (...)
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  2.  3
    Multiverse Conceptions in Set Theory.Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo - 2018 - In Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo (eds.), The Hyperuniverse Project and Maximality. Birkhäuser. pp. 47-73.
    We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’. Furthermore, we propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and potentialism with regard to the (...)
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  3.  4
    On Strong Forms of Reflection in Set Theory.Sy-David Friedman & Radek Honzik - 2018 - In Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo (eds.), The Hyperuniverse Project and Maximality. Birkhäuser. pp. 125-134.
    In this paper we review the most common forms of reflection and introduce a new form which we call sharp-generated reflection. We argue that sharp-generated reflection is the strongest form of reflection which can be regarded as a natural generalization of the Lévy reflection theorem. As an application we formulate the principle sharp-maximality with the corresponding hypothesis IMH#. IMH# is an analogue of the IMH :591–600, 2006)) which is compatible with the existence of large cardinals.
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  4.  4
    Indestructibility of the Tree Property.Radek Honzik & Šárka Stejskalová - 2020 - Journal of Symbolic Logic 85 (1):467-485.
    In the first part of the article, we show that if $\omega \le \kappa < \lambda$ are cardinals, ${\kappa ^{ < \kappa }} = \kappa$, and λ is weakly compact, then in $V\left[M {\left} \right]$ the tree property at $$\lambda = \left^{V\left[ {\left} \right]} $$ is indestructible under all ${\kappa ^ + }$-cc forcing notions which live in $V\left[ {{\rm{Add}}\left} \right]$, where ${\rm{Add}}\left$ is the Cohen forcing for adding λ-many subsets of κ and $\left$ is the standard Mitchell forcing for (...)
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  5.  36
    Fusion and Large Cardinal Preservation.Sy-David Friedman, Radek Honzik & Lyubomyr Zdomskyy - 2013 - Annals of Pure and Applied Logic 164 (12):1247-1273.
    In this paper we introduce some fusion properties of forcing notions which guarantee that an iteration with supports of size ⩽κ not only does not collapse κ+ but also preserves the strength of κ. This provides a general theory covering the known cases of tree iterations which preserve large cardinals [3], Friedman and Halilović [5], Friedman and Honzik [6], Friedman and Magidor [8], Friedman and Zdomskyy [10], Honzik [12]).
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  6.  12
    On Strong Forms of Reflection in Set Theory.Sy-David Friedman & Radek Honzik - 2016 - Mathematical Logic Quarterly 62 (1-2):52-58.
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  7.  11
    The Tree Property at the Double Successor of a Singular Cardinal with a Larger Gap.Sy-David Friedman, Radek Honzik & Šárka Stejskalová - 2018 - Annals of Pure and Applied Logic 169 (6):548-564.
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  8.  9
    Easton’s Theorem and Large Cardinals.Sy-David Friedman & Radek Honzik - 2008 - Annals of Pure and Applied Logic 154 (3):191-208.
    The continuum function αmaps to2α on regular cardinals is known to have great freedom. Let us say that F is an Easton function iff for regular cardinals α and β, image and α<β→F≤F. The classic example of an Easton function is the continuum function αmaps to2α on regular cardinals. If GCH holds then any Easton function is the continuum function on regular cardinals of some cofinality-preserving extension V[G]; we say that F is realised in V[G]. However if we also wish (...)
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  9.  6
    The Hyperuniverse Project and Maximality.Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo (eds.) - 2018 - Basel, Switzerland: Birkhäuser.
    This collection documents the work of the Hyperuniverse Project which is a new approach to set-theoretic truth based on justifiable principles and which leads to the resolution of many questions independent from ZFC. The contributions give an overview of the program, illustrate its mathematical content and implications, and also discuss its philosophical assumptions. It will thus be of wide appeal among mathematicians and philosophers with an interest in the foundations of set theory. The Hyperuniverse Project was supported by the John (...)
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  10. Global Singularization and the Failure of SCH.Radek Honzik - 2010 - Annals of Pure and Applied Logic 161 (7):895-915.
    We say that κ is μ-hypermeasurable for a cardinal μ≥κ+ if there is an embedding j:V→M with critical point κ such that HV is included in M and j>μ. Such a j is called a witnessing embedding.Building on the results in [7], we will show that if V satisfies GCH and F is an Easton function from the regular cardinals into cardinals satisfying some mild restrictions, then there exists a cardinal-preserving forcing extension V* where F is realised on all V-regular (...)
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  11.  10
    The Tree Property at the ℵ 2 N 's and the Failure of SCH at ℵ Ω.Sy-David Friedman & Radek Honzik - 2015 - Annals of Pure and Applied Logic 166 (4):526-552.
  12.  9
    A Laver-Like Indestructibility for Hypermeasurable Cardinals.Radek Honzik - 2019 - Archive for Mathematical Logic 58 (3-4):275-287.
    We show that if \ is \\)-hypermeasurable for some cardinal \ with \ \le \mu \) and GCH holds, then we can extend the universe by a cofinality-preserving forcing to obtain a model \ in which the \\)-hypermeasurability of \ is indestructible by the Cohen forcing at \ of any length up to \ is \\)-hypermeasurable in \). The preservation of hypermeasurability is useful for subsequent arguments. The construction of \ is based on the ideas of Woodin and Cummings :1–39, (...)
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  13.  6
    Definability of Satisfaction in Outer Models.Sy-David Friedman & Radek Honzik - 2016 - Journal of Symbolic Logic 81 (3):1047-1068.
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  14.  5
    The Tree Property and the Continuum Function Below ℵω.Radek Honzik & Šárka Stejskalová - 2018 - Mathematical Logic Quarterly 64 (1-2):89-102.
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  15.  2
    Large Cardinals and the Continuum Hypothesis.Radek Honzik - 2018 - In Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo (eds.), The Hyperuniverse Project and Maximality. Birkhäuser. pp. 205-226.
    This is a survey paper which discusses the impact of large cardinals on provability of the Continuum Hypothesis. It was Gödel who first suggested that perhaps “strong axioms of infinity” could decide interesting set-theoretical statements independent over ZFC, such as CH. This hope proved largely unfounded for CH—one can show that virtually all large cardinals defined so far do not affect the status of CH. It seems to be an inherent feature of large cardinals that they do not determine properties (...)
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  16.  12
    A Lifting Argument for the Generalized Grigorieff Forcing.Radek Honzík & Jonathan Verner - 2016 - Notre Dame Journal of Formal Logic 57 (2):221-231.
    In this short paper, we describe another class of forcing notions which preserve measurability of a large cardinal $\kappa$ from the optimal hypothesis, while adding new unbounded subsets to $\kappa$. In some ways these forcings are closer to the Cohen-type forcings—we show that they are not minimal—but, they share some properties with treelike forcings. We show that they admit fusion-type arguments which allow for a uniform lifting argument.
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  17.  6
    Small $$\mathfrak {u}(\kappa )$$ u ( κ ) at singular $$\kappa $$ κ with compactness at $$\kappa ^{++}$$ κ + +.Radek Honzik & Šárka Stejskalová - 2022 - Archive for Mathematical Logic 61 (1):33-54.
    We show that the tree property, stationary reflection and the failure of approachability at \ are consistent with \= \kappa ^+ < 2^\kappa \), where \ is a singular strong limit cardinal with the countable or uncountable cofinality. As a by-product, we show that if \ is a regular cardinal, then stationary reflection at \ is indestructible under all \-cc forcings.
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  18.  26
    Eastonʼs Theorem and Large Cardinals From the Optimal Hypothesis.Sy-David Friedman & Radek Honzik - 2012 - Annals of Pure and Applied Logic 163 (12):1738-1747.
    The equiconsistency of a measurable cardinal with Mitchell order o=κ++ with a measurable cardinal such that 2κ=κ++ follows from the results by W. Mitchell [13] and M. Gitik [7]. These results were later generalized to measurable cardinals with 2κ larger than κ++ .In Friedman and Honzik [5], we formulated and proved Eastonʼs theorem [4] in a large cardinal setting, using slightly stronger hypotheses than the lower bounds identified by Mitchell and Gitik , for a suitable μ, instead of the cardinals (...)
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  19.  1
    Definability of Satisfaction in Outer Models.Sy-David Friedman & Radek Honzik - 2018 - In Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo (eds.), The Hyperuniverse Project and Maximality. Birkhäuser. pp. 135-160.
    Let M be a transitive model of ZFC. We say that a transitive model of ZFC, N, is an outer model of M if M ⊆ N and ORD ∩ M = ORD ∩ N. The outer model theory of M is the collection of all formulas with parameters from M which hold in all outer models of M. Satisfaction defined with respect to outer models can be seen as a useful strengthening of first-order logic. Starting from an inaccessible cardinal (...)
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