68 found
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  1.  9
    Proof theory.Gaisi Takeuti - 1975 - New York, N.Y., U.S.A.: Sole distributors for the U.S.A. and Canada, Elsevier Science Pub. Co..
    This comprehensive monograph is a cornerstone in the area of mathematical logic and related fields. Focusing on Gentzen-type proof theory, the book presents a detailed overview of creative works by the author and other 20th-century logicians that includes applications of proof theory to logic as well as other areas of mathematics. 1975 edition.
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  2. Proof Theory.Gaisi Takeuti - 1990 - Studia Logica 49 (1):160-161.
     
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  3.  2
    Two applications of logic to mathematics.Gaisi Takeuti - 1978 - [Princeton, N.J.]: Princeton University Press.
    Using set theory in the first part of his book, and proof theory in the second, Gaisi Takeuti gives us two examples of how mathematical logic can be used to obtain results previously derived in less elegant fashion by other mathematical techniques, especially analysis. In Part One, he applies Scott- Solovay's Boolean-valued models of set theory to analysis by means of complete Boolean algebras of projections. In Part Two, he develops classical analysis including complex analysis in Peano's arithmetic, showing that (...)
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  4.  30
    Bounded arithmetic and the polynomial hierarchy.Jan Krajíček, Pavel Pudlák & Gaisi Takeuti - 1991 - Annals of Pure and Applied Logic 52 (1-2):143-153.
    T i 2 = S i +1 2 implies ∑ p i +1 ⊆ Δ p i +1 ⧸poly. S 2 and IΔ 0 ƒ are not finitely axiomatizable. The main tool is a Herbrand-type witnessing theorem for ∃∀∃ П b i -formulas provable in T i 2 where the witnessing functions are □ p i +1.
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  5. Intuitionistic fuzzy logic and intuitionistic fuzzy set theory.Gaisi Takeuti & Satoko Titani - 1984 - Journal of Symbolic Logic 49 (3):851-866.
  6.  15
    Fuzzy logic and fuzzy set theory.Gaisi Takeuti & Satoko Titani - 1992 - Archive for Mathematical Logic 32 (1):1-32.
  7.  13
    Globalization of intui tionistic set theory.Gaisi Takeuti & Satoko Titani - 1987 - Annals of Pure and Applied Logic 33 (C):195-211.
  8.  8
    Bounded arithmetic and truth definition.Gaisi Takeuti - 1988 - Annals of Pure and Applied Logic 39 (1):75-104.
  9.  12
    On a Generalized Logic Calculus.Gaisi Takeuti - 1957 - Journal of Symbolic Logic 22 (4):351-352.
  10. Proof theory and set theory.Gaisi Takeuti - 1985 - Synthese 62 (2):255 - 263.
    The foundations of mathematics are divided into proof theory and set theory. Proof theory tries to justify the world of infinite mind from the standpoint of finite mind. Set theory tries to know more and more of the world of the infinite mind. The development of two subjects are discussed including a new proof of the accessibility of ordinal diagrams. Finally the world of large cardinals appears when we go slightly beyond girard's categorical approach to proof theory.
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  11.  13
    Introduction to axiomatic set theory.Gaisi Takeuti - 1971 - New York,: Springer Verlag. Edited by Wilson M. Zaring.
    In 1963, the first author introduced a course in set theory at the University of Illinois whose main objectives were to cover Godel's work on the con sistency of the Axiom of Choice (AC) and the Generalized Continuum Hypothesis (GCH), and Cohen's work on the independence of the AC and the GCH. Notes taken in 1963 by the second author were taught by him in 1966, revised extensively, and are presented here as an introduction to axiomatic set theory. Texts in (...)
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  12.  50
    A formalization of the theory of ordinal numbers.Gaisi Takeuti - 1965 - Journal of Symbolic Logic 30 (3):295-317.
  13.  21
    S 3 i andV 2 i (BD).Gaisi Takeuti - 1990 - Archive for Mathematical Logic 29 (3):149-169.
    No categories
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  14.  22
    Forcing on bounded arithmetic II.Gaisi Takeuti & Masahiro Yasumoto - 1998 - Journal of Symbolic Logic 63 (3):860-868.
  15.  2
    S 3 i andV 2 i.Gaisi Takeuti - 1990 - Archive for Mathematical Logic 29 (3):149-169.
    No categories
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  16. Meetings of the association for symbolic logic.Gaisi Takeuti - 1971 - Journal of Symbolic Logic 36 (2):357-384.
  17.  29
    Formally Self-Referential Propositions for Cut Free Analysis and Related Systems.Georg Kreisel & Gaisi Takeuti - 1985 - Journal of Symbolic Logic 50 (1):244-246.
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  18.  3
    Meetings of the Association for Symbolic Logic, U.S.-Japan Logic Seminar, Tokyo 1969.Gaisi Takeuti - 1971 - Journal of Symbolic Logic 36 (2):357-359.
  19. Boolean simple groups and boolean simple rings.Gaisi Takeuti - 1988 - Journal of Symbolic Logic 53 (1):160-173.
  20.  9
    A Formal System of First-Order Predicate Calculus with Infinitely Long Expressions.Gaisi Takeuti - 1962 - Journal of Symbolic Logic 27 (4):468-468.
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  21.  10
    A Relativization of Axioms of Strong Infinity to ^|^omega;1.Gaisi Takeuti - 1970 - Annals of the Japan Association for Philosophy of Science 3 (5):191-204.
  22.  23
    A transfer principle in harmonic analysis.Gaisi Takeuti - 1979 - Journal of Symbolic Logic 44 (3):417-440.
  23.  4
    Construction of the Set Theory from the Theory of Ordinal Numbers.Gaisi Takeuti - 1959 - Journal of Symbolic Logic 24 (1):66-67.
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  24.  4
    Grzegorcyk's Hierarchy and $IepSigma_1$.Gaisi Takeuti - 1994 - Journal of Symbolic Logic 59 (4):1274-1284.
  25.  21
    Grzegorcyk's hierarchy and IepΣ1.Gaisi Takeuti - 1994 - Journal of Symbolic Logic 59 (4):1274-1284.
  26.  32
    Gödel sentences of bounded arithmetic.Gaisi Takeuti - 2000 - Journal of Symbolic Logic 65 (3):1338-1346.
  27. On the Theory of Ordinal Numbers.Gaisi Takeuti - 1959 - Journal of Symbolic Logic 24 (1):67-67.
  28.  10
    Separations of theories in weak bounded arithmetic.Gaisi Takeuti - 1995 - Annals of Pure and Applied Logic 71 (1):47-67.
  29.  45
    Transcendence of cardinals.Gaisi Takeuti - 1965 - Journal of Symbolic Logic 30 (1):1-7.
  30. Frege proof system and TNC°.Gaisi Takeuti - 1998 - Journal of Symbolic Logic 63 (2):709 - 738.
    A Frege proof systemFis any standard system of prepositional calculus, e.g., a Hilbert style system based on finitely many axiom schemes and inference rules. An Extended Frege systemEFis obtained fromFas follows. AnEF-sequence is a sequence of formulas ψ1, …, ψκsuch that eachψiis either an axiom ofF, inferred from previous ψuand ψv by modus ponens or of the formq↔ φ, whereqis an atom occurring neither in φ nor in any of ψ1,…,ψi−1. Suchq↔ φ, is called an extension axiom andqa new extension (...)
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  31. The critical number of a variable in a function.Gaisi Takeuti - 1994 - Journal of Symbolic Logic 59 (4):1228-1244.
    LetL0be a language on ℕ consisting of 0, 1, +, ∸, ·, ⌊½a⌋, ∣a∣, #, ∧(a,b), ∨(a,b), ¬(a), ≤ (a,b), andμx≤ ∣s∣t(x). Hereμx≤ ∣s∣t(x) is the smallest numberx≤ ∣s∣ satisfyingt(x) > 0 and 0 if there exist no suchxand we stipulate that ifsandt(a) are terms inLo, thenμx≤ ∣s∣t(x) is also a term inLo. The defining axioms of functions ∧(a,b), ∨(a,b), ¬(a), ≤ (a,b) are as follows:LetLa language on ℕ with only predicate constant = andL0⊆L. Letf(b,a1,…,am) be a function for ℕm+1into (...)
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  32. Godel Sentences of Bounded Arithmetic.Arnold Beckmann & Gaisi Takeuti - 2002 - Bulletin of Symbolic Logic 8 (3):433.
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  33.  12
    Incompleteness Theorems and S i 2 versus S i+1 2Godel Sentences of Bounded Arithmetic.Arnold Beckmann & Gaisi Takeuti - 2002 - Bulletin of Symbolic Logic 8 (3):433.
  34. Weaves.Barry Burd & Gaisi Takeuti - 1977 - Annals of the Japan Association for Philosophy of Science 5 (2):47-56.
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  35.  21
    Meeting of the Association for Symbolic Logic, Chicago, 1977.Carl G. Jockusch, Robert I. Soare, William Tait & Gaisi Takeuti - 1978 - Journal of Symbolic Logic 43 (3):614 - 619.
  36.  33
    Two interpolation theorems for a π11 predicate calculus.Shoji Maehara & Gaisi Takeuti - 1971 - Journal of Symbolic Logic 36 (2):262 - 270.
  37.  5
    A Metamathematical Theorem on the Theory of Ordinal Numbers.Gaisi Takeuti - 1959 - Journal of Symbolic Logic 24 (1):62-62.
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  38. A Metamathematical Theorem on Functions.Gaisi Takeuti - 1959 - Journal of Symbolic Logic 24 (1):65-66.
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  39.  6
    Axioms of Infinity of Set Theory.Gaisi Takeuti - 1962 - Journal of Symbolic Logic 27 (3):354-355.
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  40.  6
    A Remark on Gentzen's paper "Beweisbarkeit und Unbeweisbarkeit von Anfangsfallen der Transfiniten Induktion in der Reinen Zahlentheorie." I-II.Gaisi Takeuti - 1964 - Journal of Symbolic Logic 29 (1):58-59.
  41. A Second Order Version of S i 2 and U 1 2.Gaisi Takeuti - 1991 - Journal of Symbolic Logic 56 (3):1038.
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  42.  6
    A second order version of S2i and U21.Gaisi Takeuti - 1991 - Journal of Symbolic Logic 56 (3):1038-1063.
  43.  1
    Construction of Ramified Real Numbers.Gaisi Takeuti - 1957 - Journal of Symbolic Logic 22 (4):352-352.
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  44.  3
    Construction of Ramified Real Numbers.Gaisi Takeuti - 1956 - Annals of the Japan Association for Philosophy of Science 1 (1):41-61.
  45. Forcing on Bounded Arithmetic II.Gaisi Takeuti & Masahiro Yasumoto - 1998 - Journal of Symbolic Logic 63 (3):860-868.
     
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  46.  21
    Foundations of Mathematics.Gaisi Takeuti & Setsuya Seki - 1959 - Journal of Symbolic Logic 24 (1):73.
  47.  9
    Frege Proof System and TNC$^circ$.Gaisi Takeuti - 1998 - Journal of Symbolic Logic 63 (2):709-738.
    A Frege proof systemFis any standard system of prepositional calculus, e.g., a Hilbert style system based on finitely many axiom schemes and inference rules. An Extended Frege systemEFis obtained fromFas follows. AnEF-sequence is a sequence of formulas ψ1, …, ψκsuch that eachψiis either an axiom ofF, inferred from previous ψuand ψv by modus ponens or of the formq↔ φ, whereqis an atom occurring neither in φ nor in any of ψ1,…,ψi−1. Suchq↔ φ, is called an extension axiom andqa new extension (...)
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  48.  19
    Global intuitionistic analysis.Gaisi Takeuti & Satoko Titani - 1986 - Annals of Pure and Applied Logic 31:307-339.
  49. Godel Sentences of Bounded Arithmetic.Gaisi Takeuti - 2000 - Journal of Symbolic Logic 65 (3):1338-1346.
     
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  50.  7
    Incompleteness Theorems and S i 2 versus S i+1 2.Gaisi Takeuti - 2002 - Bulletin of Symbolic Logic 8 (3):433-435.
1 — 50 / 68