10 found
Order:
  1.  17
    Guessing Models and Generalized Laver Diamond.Matteo Viale - 2012 - Annals of Pure and Applied Logic 163 (11):1660-1678.
    We analyze the notion of guessing model, a way to assign combinatorial properties to arbitrary regular cardinals. Guessing models can be used, in combination with inaccessibility, to characterize various large cardinal axioms, ranging from supercompactness to rank-to-rank embeddings. The majority of these large cardinal properties can be defined in terms of suitable elementary embeddings j:Vγ→Vλ. One key observation is that such embeddings are uniquely determined by the image structures j[Vγ]≺Vλ. These structures will be the prototypes guessing models. We shall show, (...)
    Direct download (6 more)  
     
    Export citation  
     
    Bookmark   9 citations  
  2.  40
    Absoluteness Via Resurrection.Giorgio Audrito & Matteo Viale - 2017 - Journal of Mathematical Logic 17 (2):1750005.
    The resurrection axioms are forcing axioms introduced recently by Hamkins and Johnstone, developing on ideas of Chalons and Veličković. We introduce a stronger form of resurrection axioms for a class of forcings Γ and a given ordinal α), and show that RAω implies generic absoluteness for the first-order theory of Hγ+ with respect to forcings in Γ preserving the axiom, where γ = γΓ is a cardinal which depends on Γ. We also prove that the consistency strength of these axioms (...)
    Direct download (6 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  3.  14
    Martin’s Maximum Revisited.Matteo Viale - 2016 - Archive for Mathematical Logic 55 (1-2):295-317.
    We present several results relating the general theory of the stationary tower forcing developed by Woodin with forcing axioms. In particular we show that, in combination with class many Woodin cardinals, the forcing axiom MM++ makes the Π2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi_2}$$\end{document}-fragment of the theory of Hℵ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H_{\aleph_2}}$$\end{document} invariant with respect to stationary set preserving forcings that preserve BMM. We argue that this is a promising generalization to (...)
    No categories
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   5 citations  
  4. On the Notion of Guessing Model.Matteo Viale - forthcoming - Annals of Pure and Applied Logic.
  5.  15
    The Proper Forcing Axiom and the Singular Cardinal Hypothesis.Matteo Viale - 2006 - Journal of Symbolic Logic 71 (2):473 - 479.
    We show that the Proper Forcing Axiom implies the Singular Cardinal Hypothesis. The proof uses the reflection principle MRP introduced by Moore in [11].
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   8 citations  
  6. The Cumulative Hierarchy and the Constructible Universe of ZFA.Matteo Viale - 2004 - Mathematical Logic Quarterly 50 (1):99.
    We present two results which shed some more light on the deep connection between ZFA and the standard ZF set theory: First of all we refine a result of Forti and Honsell in order to prove that the universe of ZFA can also be obtained as the least fixed point of a continuous operator and not only as the greatest fixed point of the powerset operator. Next we show that it is possible to define a new absolute Gödel operation in (...)
    Direct download  
     
    Export citation  
     
    Bookmark   2 citations  
  7.  4
    Second Order Arithmetic as the Model Companion of Set Theory.Giorgio Venturi & Matteo Viale - forthcoming - Archive for Mathematical Logic:1-25.
    This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically we develop here a general framework linking Woodin’s generic absoluteness results for second order number theory and the theory of universally Baire sets to model companionship and show that a \-property formalized in an appropriate language for second order number theory is forcible from some \large cardinals if and only if (...)
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  8. J. T. Moore. Set Mapping Reflection. Journal of Mathematical Logic, Vol. 5 (2005), Pp. 87–97. - J. T. Moore. A Five Element Basis for the Uncountable Linear Orders. Annals of Mathematics, Vol. 163 (2006), Pp. 669–688. [REVIEW]Matteo Viale - 2009 - Bulletin of Symbolic Logic 15 (3):322-325.
  9.  2
    Incompatible Bounded Category Forcing Axioms.David Asperó & Matteo Viale - 2022 - Journal of Mathematical Logic 22 (2).
    Journal of Mathematical Logic, Volume 22, Issue 02, August 2022. We introduce bounded category forcing axioms for well-behaved classes [math]. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe [math] modulo forcing in [math], for some cardinal [math] naturally associated to [math]. These axioms naturally extend projective absoluteness for arbitrary set-forcing — in this situation [math] — to classes [math] with [math]. Unlike projective absoluteness, these higher bounded category forcing (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  10.  25
    Forcing Axioms, Supercompact Cardinals, Singular Cardinal Combinatorics.Matteo Viale - 2008 - Bulletin of Symbolic Logic 14 (1):99-113.
    The purpose of this communication is to present some recent advances on the consequences that forcing axioms and large cardinals have on the combinatorics of singular cardinals. I will introduce a few examples of problems in singular cardinal combinatorics which can be fruitfully attacked using ideas and techniques coming from the theory of forcing axioms and then translate the results so obtained in suitable large cardinals properties.The first example I will treat is the proof that the proper forcing axiom PFA (...)
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark