Chapter V – Gödel’s Functional (“Dialectica”) Interpretation

[1]  Michael Beeson,et al.  Recursive models for constructive set theories , 1982, Ann. Math. Log..

[2]  H. Luckhardt Extensional Gödel Functional Interpretation , 1973 .

[3]  H. Weyl,et al.  Das Kontinuum : kritische Untersuchungen über die Grundlagen der Analysis , 1932 .

[4]  Michael Rathjen,et al.  The strength of some Martin-Löf type theories , 1994, Arch. Math. Log..

[5]  John C. Mitchell,et al.  Type Systems for Programming Languages , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[6]  A. Troelstra,et al.  Constructivism in Mathematics: An Introduction , 1988 .

[7]  Harvey M. Friedman Iterated Inductive Definitions and Σ21-AC , 1970 .

[8]  Errett Bishop,et al.  Mathematics as a Numerical Language , 1970 .

[9]  Solomon Feferman,et al.  Proof theoretic equivalences between classical and constructive theories for analysis , 1981 .

[10]  Wilfried Sieg,et al.  Fragments of arithmetic , 1985, Ann. Pure Appl. Log..

[11]  Rance Cleaveland,et al.  Implementing mathematics with the Nuprl proof development system , 1986 .

[12]  T. Coquand,et al.  Metamathematical investigations of a calculus of constructions , 1989 .

[13]  Gerhard Jäger,et al.  Choice Principles, the Bar Rule and Autonomously Iterated Comprehension Schemes in Analysis , 1983, J. Symb. Log..

[14]  Solomon Feferman A MORE PERSPICUOUS FORMAL SYSTEM FOR PREDICATIVITY , 1978 .

[15]  H. Weyl,et al.  The Continuum: A Critical Examination of the Foundation of Analysis , 1987 .

[16]  A. S. Troelstra,et al.  Aspects of Constructive Mathematics , 1977 .

[17]  Solomon Feferman,et al.  Systems of predicative analysis, II: Representations of ordinals , 1968, Journal of Symbolic Logic.

[18]  S. Kleene,et al.  The Foundations of Intuitionistic Mathematics , 1965, The Mathematical Gazette.

[19]  A. Troelstra Metamathematical investigation of intuitionistic arithmetic and analysis , 1973 .

[20]  Jeremy Avigad On the Relationship Between ATR 0 and ID . , 1996 .

[21]  P. Martin-Löf Hauptsatz for the Theory of Species , 1971 .

[22]  S. Shelah,et al.  Annals of Pure and Applied Logic , 1991 .

[23]  J. Fenstad Proceedings of the Second Scandinavian Logic Symposium , 1971 .

[24]  Jeremy Avigad Formalizing Forcing Arguments in Subsystems of Second-Order Arithmetic , 1996, Ann. Pure Appl. Log..

[25]  Peter Aczel etc HANDBOOK OF MATHEMATICAL LOGIC , 1999 .

[26]  J. Gallier On Girard's "Candidats de Reductibilité" , 1989 .

[27]  Jeremy Avigad On the Relationships between $ATR_0$ And $\widehat{ID}_{< \omega}$ , 1996 .

[28]  J. Roger Hindley,et al.  Introduction to combinators and λ-calculus , 1986, Acta Applicandae Mathematicae.

[29]  William W. Tait,et al.  Intensional interpretations of functionals of finite type I , 1967, Journal of Symbolic Logic.

[30]  C. Parsons On a Number Theoretic Choice Schema and its Relation to Induction , 1970 .

[31]  Jon Barwise,et al.  On recursively saturated models of arithmetic , 1975 .

[32]  Stephen A. Cook,et al.  Functional interpretations of feasibly constructive arithmetic , 1989, STOC '89.

[33]  S. C. Kleene,et al.  Recursive functionals and quantifiers of finite types. II , 1959 .

[34]  C. Spector Provably recursive functionals of analysis: a consistency proof of analysis by an extension of princ , 1962 .

[35]  P. Martin-Löf An Intuitionistic Theory of Types: Predicative Part , 1975 .

[36]  Jeremy Avigad,et al.  Predicative Functionals and an Interpretation of ID , 1998, Ann. Pure Appl. Log..

[37]  S. C. Kleene,et al.  The Foundations of Intuitionistic Mathematics. , 1967 .

[38]  Fernando Ferreira,et al.  A feasible theory for analysis , 1994, Journal of Symbolic Logic.

[39]  Jean-Baptiste Lully,et al.  The collected works , 1996 .

[40]  Günter Asser,et al.  Zeitschrift für mathematische Logik und Grundlagen der Mathematik , 1955 .

[41]  D. Hilbert Über das Unendliche , 1926 .

[42]  W. Buchholz Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-theoretical Studies , 1981 .

[43]  D. Prawitz Ideas and Results in Proof Theory , 1971 .

[44]  J. Diller Eine Variante zur Dialectica-Interpretation der Heyting-Arithmetik endlicher Typen , 1974 .

[45]  Solomon Feferman Monotone Inductive Definitions , 1982 .

[46]  Jeremy Avigad,et al.  Godel's functional interpretation , 1998 .

[47]  J. Girard,et al.  Proofs and types , 1989 .

[48]  J. Y. Girard,et al.  Interpretation fonctionelle et elimination des coupures dans l'aritmetique d'ordre superieur , 1972 .

[49]  Ulrich Kohlenbach,et al.  Effective bounds from ineffective proofs in analysis: An application of functional interpretation and majorization , 1992, Journal of Symbolic Logic.

[50]  Jean-Pierre Jouannaud,et al.  Rewrite Systems , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[51]  W. A. Howard,et al.  Functional interpretation of bar induction by bar recursion , 1968 .

[52]  Wilfrid Hodges,et al.  Logic: from foundations to applications: European logic colloquium , 1996 .

[53]  Solomon Feferman,et al.  Iterated Inductive Fixed-Point Theories: Application to Hancock's Conjecture , 1982 .

[54]  Charles D. Parsons,et al.  On n-quantifier induction , 1972, Journal of Symbolic Logic.

[55]  Ulrich Kohlenbach,et al.  Mathematically strong subsystems of analysis with low rate of growth of provably recursive functionals , 1996, Arch. Math. Log..

[56]  William A. Howard,et al.  A system of abstract constructive ordinals , 1972, Journal of Symbolic Logic.

[57]  W. Tait Infinitely Long Terms of Transfinite Type , 1965 .

[58]  Wolfgang Maaß,et al.  Eine Funktionalinterpretation der prädikativen Analysis , 1977, Arch. Math. Log..

[59]  Von Kurt Gödel,et al.  ÜBER EINE BISHER NOCH NICHT BENÜTZTE ERWEITERUNG DES FINITEN STANDPUNKTES , 1958 .

[60]  Ulrich Kohlenbach,et al.  Effective Moduli from Ineffective Uniqueness Proofs. An Unwinding of de La Vallée Poussin's Proof for Chebycheff Approximation , 1993, Ann. Pure Appl. Log..

[61]  Stephen G. Simpson,et al.  A Finite Combinatorial Principle Which is Equivalent to the 1-Consistency of Predicative Analysis , 1982 .

[62]  H. Schwichtenberg Proof Theory: Some Applications of Cut-Elimination , 1977 .

[63]  H.C.M. de Swart Logic and Computer Science , 1994 .

[64]  J. Girard Une Extension De ĽInterpretation De Gödel a ĽAnalyse, Et Son Application a ĽElimination Des Coupures Dans ĽAnalyse Et La Theorie Des Types , 1971 .

[65]  U. Kohlenbach Analysing proofs in analysis , 1996 .

[66]  W. Tait Normal Form Theorem for Bar Recursive Functions of Finite Type , 1971 .

[67]  Michael Beeson,et al.  A type-free Gödel interpretation , 1978, Journal of Symbolic Logic.

[68]  Solomon Feferman,et al.  Gödel's Dialectica Interpretation and Its Two-Way Stretch , 1993, Kurt Gödel Colloquium.

[69]  William W. Tait,et al.  Normal derivability in classical logic , 1968 .

[70]  Wolfgang Friedrich Gödelsche Funktionalinterpretation für Eine Erweiterung der Klassischen Analysis , 1985, Math. Log. Q..

[71]  Harvey Gerber,et al.  Brouwer's Bar Theorem and a System of Ordinal Notations , 1970 .

[72]  Stephen G. Simpson,et al.  Polynomial time computable arithmetic and conservative extensions , 1988 .

[73]  Carl A. Gunter,et al.  In handbook of theoretical computer science , 1990 .

[74]  John C. Reynolds,et al.  Towards a theory of type structure , 1974, Symposium on Programming.

[75]  Solomon Feferman,et al.  Systems of predicative analysis , 1964, Journal of Symbolic Logic.

[76]  W. A. Howard Assignment of Ordinals to Terms for Primitive Recursive Functionals of Finite Type , 1970 .

[77]  Georg Kreisel,et al.  On the interpretation of non-finitist proofs—Part I , 1951, Journal of Symbolic Logic.

[78]  Akiko Kino,et al.  Intuitionism and Proof Theory , 1970 .

[79]  Andrea Cantini,et al.  Asymmetric Interpretations for Bounded Theories , 1996, Math. Log. Q..

[80]  Solomon Feferman,et al.  Theories of Finite Type Related to Mathematical Practice , 1977 .

[81]  J. Heijenoort From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 , 1967 .