12 found
Order:
  1.  18
    From Hierarchies to Well-Foundedness.Dandolo Flumini & Kentaro Sato - 2014 - Archive for Mathematical Logic 53 (7-8):855-863.
    We highlight that the connection of well-foundedness and recursive definitions is more than just convenience. While the consequences of making well-foundedness a sufficient condition for the existence of hierarchies have been extensively studied, we point out that well-foundedness is a necessary condition for the existence of hierarchies e.g. that even in an intuitionistic setting α⊢wfwhereα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${_\alpha \vdash \mathsf{wf}\, {\rm where}\, _\alpha}$$\end{document} stands for the iteration of Π10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} (...)
    No categories
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   5 citations  
  2.  13
    Ordinal analyses for monotone and cofinal transfinite inductions.Kentaro Sato - 2020 - Archive for Mathematical Logic 59 (3-4):277-291.
    We consider two variants of transfinite induction, one with monotonicity assumption on the predicate and one with the induction hypothesis only for cofinally many below. The latter can be seen as a transfinite analogue of the successor induction, while the usual transfinite induction is that of cumulative induction. We calculate the supremum of ordinals along which these schemata for \ formulae are provable in \. It is shown to be larger than the proof-theoretic ordinal \ by power of base 2. (...)
    Direct download (2 more)  
    Translate
     
     
    Export citation  
     
    Bookmark   1 citation  
  3.  12
    Full and Hat Inductive Definitions Are Equivalent in NBG.Kentaro Sato - 2015 - Archive for Mathematical Logic 54 (1-2):75-112.
    A new research project has, quite recently, been launched to clarify how different, from systems in second order number theory extending ACA0, those in second order set theory extending NBG are. In this article, we establish the equivalence between Δ01-LFP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^1_0\mbox{\bf-LFP}}$$\end{document} and Δ01-FP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^1_0\mbox{\bf-FP}}$$\end{document}, which assert the existence of a least and of a fixed point, respectively, for positive elementary operators. Our proof also shows (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  4.  27
    The Strength of Extensionality II—Weak Weak Set Theories Without Infinity.Kentaro Sato - 2011 - Annals of Pure and Applied Logic 162 (8):579-646.
    By obtaining several new results on Cook-style two-sorted bounded arithmetic, this paper measures the strengths of the axiom of extensionality and of other weak fundamental set-theoretic axioms in the absence of the axiom of infinity, following the author’s previous work [K. Sato, The strength of extensionality I — weak weak set theories with infinity, Annals of Pure and Applied Logic 157 234–268] which measures them in the presence. These investigations provide a uniform framework in which three different kinds of reverse (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   4 citations  
  5.  25
    The Strength of Extensionality I—Weak Weak Set Theories with Infinity.Kentaro Sato - 2009 - Annals of Pure and Applied Logic 157 (2-3):234-268.
    We measure, in the presence of the axiom of infinity, the proof-theoretic strength of the axioms of set theory which make the theory look really like a “theory of sets”, namely, the axiom of extensionality Ext, separation axioms and the axiom of regularity Reg . We first introduce a weak weak set theory as a base over which to clarify the strength of these axioms. We then prove the following results about proof-theoretic ordinals:1. and ,2. and . We also show (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   4 citations  
  6.  29
    A New Model Construction by Making a Detour Via Intuitionistic Theories I: Operational Set Theory Without Choice is Π 1 -Equivalent to KP.Kentaro Sato & Rico Zumbrunnen - 2015 - Annals of Pure and Applied Logic 166 (2):121-186.
  7.  15
    A New Model Construction by Making a Detour Via Intuitionistic Theories II: Interpretability Lower Bound of Feferman's Explicit Mathematics T 0.Kentaro Sato - 2015 - Annals of Pure and Applied Logic 166 (7-8):800-835.
  8.  16
    Double Helix in Large Large Cardinals and Iteration of Elementary Embeddings.Kentaro Sato - 2007 - Annals of Pure and Applied Logic 146 (2):199-236.
    We consider iterations of general elementary embeddings and, using this notion, point out helices of consistency-wise implications between large large cardinals.Up to now, large cardinal properties have been considered as properties which cannot be accessed by any weaker properties and it has been known that, with respect to this relation, they form a proper hierarchy. The helices we point out significantly change this situation: the same sequence of large cardinal properties occurs repeatedly, changing only the parameters.As results of our investigation (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  9.  30
    Semantical Characterizations for Irreflexive and Generalized Modal Languages.Katsuhiko Sano & Kentaro Sato - 2007 - Notre Dame Journal of Formal Logic 48 (2):205-228.
    This paper deals with two main topics: One is a semantical investigation for a bimodal language with a modal operator \blacksquare associated with the intersection of the accessibility relation R and the inequality ≠. The other is a generalization of some of the former results to general extended languages with modal operators. First, for our language L\sb{\square\blacksquare}, we prove that Segerberg's theorem (equivalence between finite frame property and finite model property) fails and establish both van Benthem-style and Goldblatt-Thomason-style characterizations. We (...)
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  10.  14
    Elementary Inductive Dichotomy: Separation of Open and Clopen Determinacies with Infinite Alternatives.Kentaro Sato - 2020 - Annals of Pure and Applied Logic 171 (3):102754.
    We introduce a new axiom called inductive dichotomy, a weak variant of the axiom of inductive definition, and analyze the relationships with other variants of inductive definition and with related axioms, in the general second order framework, including second order arithmetic, second order set theory and higher order arithmetic. By applying these results to the investigations on the determinacy axioms, we show the following. (i) Clopen determinacy is consistency-wise strictly weaker than open determinacy in these frameworks, except second order arithmetic; (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  11.  2
    Bounded inductive dichotomy: separation of open and clopen determinacies with finite alternatives in constructive contexts.Kentaro Sato - forthcoming - Archive for Mathematical Logic:1-37.
    In his previous work, the author has introduced the axiom schema of inductive dichotomy, a weak variant of the axiom schema of inductive definition, and used this schema for elementary ) positive operators to separate open and clopen determinacies for those games in which two players make choices from infinitely many alternatives in various circumstances. Among the studies on variants of inductive definitions for bounded ) positive operators, the present article investigates inductive dichotomy for these operators, and applies it to (...)
    No categories
    Direct download (2 more)  
    Translate
     
     
    Export citation  
     
    Bookmark  
  12.  2
    Forcing for Hat Inductive Definitions in Arithmetic.Kentaro Sato - 2014 - Mathematical Logic Quarterly 60 (4-5):314-318.