18 found
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  1.  75
    A Note on Boolos' Proof of the Incompleteness Theorem.Makoto Kikuchi - 1994 - Mathematical Logic Quarterly 40 (4):528-532.
    We give a proof of Gödel's first incompleteness theorem based on Berry's paradox, and from it we also derive the second incompleteness theorem model-theoretically.
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  2.  66
    On proofs of the incompleteness theorems based on Berry's paradox by Vopěnka, Chaitin, and Boolos.Makoto Kikuchi, Taishi Kurahashi & Hiroshi Sakai - 2012 - Mathematical Logic Quarterly 58 (4-5):307-316.
    By formalizing Berry's paradox, Vopěnka, Chaitin, Boolos and others proved the incompleteness theorems without using the diagonal argument. In this paper, we shall examine these proofs closely and show their relationships. Firstly, we shall show that we can use the diagonal argument for proofs of the incompleteness theorems based on Berry's paradox. Then, we shall show that an extension of Boolos' proof can be considered as a special case of Chaitin's proof by defining a suitable Kolmogorov complexity. We shall show (...)
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  3.  30
    On Formalization of Model-Theoretic Proofs of Gödel's Theorems.Makoto Kikuchi & Kazuyuki Tanaka - 1994 - Notre Dame Journal of Formal Logic 35 (3):403-412.
    Within a weak subsystem of second-order arithmetic , that is -conservative over , we reformulate Kreisel's proof of the Second Incompleteness Theorem and Boolos' proof of the First Incompleteness Theorem.
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  4.  27
    Generalizations of gödel’s incompleteness theorems for ∑n-definable theories of arithmetic.Makoto Kikuchi & Taishi Kurahashi - 2017 - Review of Symbolic Logic 10 (4):603-616.
    It is well known that Gödel’s incompleteness theorems hold for ∑1-definable theories containing Peano arithmetic. We generalize Gödel’s incompleteness theorems for arithmetically definable theories. First, we prove that every ∑n+1-definable ∑n-sound theory is incomplete. Secondly, we generalize and improve Jeroslow and Hájek’s results. That is, we prove that every consistent theory having ∏n+1set of theorems has a true but unprovable ∏nsentence. Lastly, we prove that no ∑n+1-definable ∑n-sound theory can prove its own ∑n-soundness. These three results are generalizations of Rosser’s (...)
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  5.  43
    Kolmogorov complexity and the second incompleteness theorem.Makoto Kikuchi - 1997 - Archive for Mathematical Logic 36 (6):437-443.
    We shall prove the second incompleteness theorem via Kolmogorov complexity.
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  6.  26
    Universal Rosser predicates.Makoto Kikuchi & Taishi Kurahashi - 2017 - Journal of Symbolic Logic 82 (1):292-302.
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  7.  50
    Set-theoretic mereology.Joel David Hamkins & Makoto Kikuchi - 2016 - Logic and Logical Philosophy 25 (3):285-308.
    We consider a set-theoretic version of mereology based on the inclusion relation ⊆ and analyze how well it might serve as a foundation of mathematics. After establishing the non-definability of ∈ from ⊆, we identify the natural axioms for ⊆-based mereology, which constitute a finitely axiomatizable, complete, decidable theory. Ultimately, for these reasons, we conclude that this form of set-theoretic mereology cannot by itself serve as a foundation of mathematics. Meanwhile, augmented forms of set-theoretic mereology, such as that obtained by (...)
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  8.  18
    Illusory models of peano arithmetic.Makoto Kikuchi & Taishi Kurahashi - 2016 - Journal of Symbolic Logic 81 (3):1163-1175.
    By using a provability predicate of PA, we define ThmPA(M) as the set of theorems of PA in a modelMof PA. We say a modelMof PA is (1) illusory if ThmPA(M) ⊈ ThmPA(ℕ), (2) heterodox if ThmPA(M) ⊈ TA, (3) sane ifM⊨ ConPA, and insane if it is not sane, (4) maximally sane if it is sane and ThmPA(M) ⊆ ThmPA(N) implies ThmPA(M) = ThmPA(N) for every sane modelNof PA. We firstly show thatMis heterodox if and only if it is (...)
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  9.  65
    Liar-type Paradoxes and the Incompleteness Phenomena.Makoto Kikuchi & Taishi Kurahashi - 2016 - Journal of Philosophical Logic 45 (4):381-398.
    We define a liar-type paradox as a consistent proposition in propositional modal logic which is obtained by attaching boxes to several subformulas of an inconsistent proposition in classical propositional logic, and show several famous paradoxes are liar-type. Then we show that we can generate a liar-type paradox from any inconsistent proposition in classical propositional logic and that undecidable sentences in arithmetic can be obtained from the existence of a liar-type paradox. We extend these results to predicate logic and discuss Yablo’s (...)
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  10.  39
    Three Short Stories around Gödel's Incompleteness Theorems.Makoto Kikuchi & Taishi Kurahashi - 2011 - Journal of the Japan Association for Philosophy of Science 38 (2):75-80.
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  11.  16
    Abstract design theory.Yuzuru Kakuda & Makoto Kikuchi - 2001 - Annals of the Japan Association for Philosophy of Science 10 (3):109-125.
  12.  21
    Kolmogorov complexity and characteristic constants of formal theories of arithmetic.Shingo Ibuka, Makoto Kikuchi & Hirotaka Kikyo - 2011 - Mathematical Logic Quarterly 57 (5):470-473.
    We investigate two constants cT and rT, introduced by Chaitin and Raatikainen respectively, defined for each recursively axiomatizable consistent theory T and universal Turing machine used to determine Kolmogorov complexity. Raatikainen argued that cT does not represent the complexity of T and found that for two theories S and T, one can always find a universal Turing machine such that equation image. We prove the following are equivalent: equation image for some universal Turing machine, equation image for some universal Turing (...)
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  13.  52
    Hyperformulas and Classifications.Yuzuru Kakuda, Makoto Kikuchi & Hirofumi Miki - 2000 - Annals of the Japan Association for Philosophy of Science 10 (1):33-52.
  14.  3
    特集「人工知能の哲学」趣旨.Makoto Kikuchi - 2017 - Kagaku Tetsugaku 50:33-34.
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  15.  20
    Analysis and Design from a Viewpoint of Information Flow.Makoto Kikuchi - 2003 - In Benedikt Löwe, Thoralf Räsch & Wolfgang Malzkorn (eds.), Foundations of the Formal Sciences II. Kluwer Academic Publishers. pp. 119--122.
  16.  15
    A Mathematical Model of Deductive and Non-Deductive Inferences.Makoto Kikuchi - 2009 - Annals of the Japan Association for Philosophy of Science 17:1-11.
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  17.  4
    Fukanzensei teiri =.Makoto Kikuchi - 2014 - Tōkyō-to Bunkyō-ku: Kyōritsu Shuppan.
    専門的な予備知識は仮定せずに完全性定理や計算可能性から論じ、第一および第二不完全性定理、Rosserの定理、Hilbertのプログラム、G ̈odelの加速定理、算術の超準モデル、Kolmogorov複雑性などを紹介して、不完全性定理の数学的意義と、その根源にある哲学的問題を説く。.
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  18.  16
    On Mathematical Aspects of Varieties of Cognitive Structures of Class-Inclusion Reasoning.Mizuho Mishima & Makoto Kikuchi - 2009 - Journal of the Japan Association for Philosophy of Science 36 (2):53-57.
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