Intuitionism and Proof Theory: Proceedings of the Summer Conference at Buffalo N.Y. 1968 (eBook)

Front Cover 1
Intuitionism and Proof Theory 4
Copyright Page 5
Contents 6
Introduction 8
PART A: NEW DIRECTIONS 10
Chapter I. The ultra-intuitionistic criticism and the antitraditional program for foundations of mathematics 12
Chapter II. Computable analysis and differential equations 56
Chapter III. Mathematics as a numerical language 62
Chapter IV. On the notion of randomness 82
Chapter V. Abstract quantification theory 88
PART B: TRADITIONAL INTUITIONISM 102
Chapter VI. Recent progress in intuitionistic analysis 104
Chapter VII. A theory of constructions equivalent to arithmetic 110
Chapter VIII. Church's thesis: a kind of reducibility axiom for constructive mathematics 130
Chapter IX. Formal systems of intuitionistic analysis II: The theory of species 160
Chapter X. Projections of lawless sequences 172
Chapter XI. On subjective mathematical assertions 196
Chapter XII. A palatable substitute for Kripke's schema 206
PART C: CLASSICAL INTERPRETATIONS OF INTUITIONISM 218
Chapter XIII. A characterization of the intuitionistic propositional calculus 220
Chapter XIV. Intuitionistic model theory and the Cohen independence proofs 228
Chapter XV. An abstract notion of realizability for which intuitionistic predicate calculus is complete 236
Chapter XVI. Extending the topological interpretation to intuitionistic analysis, II 244
PART D: PROOF THEORY OF INTUITIONISM 266
Chapter XVII. Some results for intuitionistic logic with second order quantification rules 268
Chapter XVIII. On cut elimination in intuitionistic systems of analysis 280
PART E: PROOF–THEORETIC ORDINALS 296
Chapter XIX. Hereditarily replete functionals over the ordinals 298
Chapter XX. Formal theorles for transfinite iterations of generalized inductive deflnitions and some subsystems of analysis 312
Chapter XXI. Brouwer’s bar theorem and a system of ordinal notations 336
Chapter XXII. Regular ordinals and normal forms 348
Chapter XXIII. Formalization of the theory of ordinal diagrams of infinite order 372
Chapter XXIV. On the relationship between Takeuti’s ordinal diagrams O(n) and schutte's system of ordinal notations S(n) 386
PART F: PROOF THEORY 416
Chapter XXV. On the original Gentzen consistency proof for number theory 418
Chapter XXVI. Herbrand-style consistency proofs 428
Chapter XXVII. Iterated inductive definitions and S1 2-AC* 444
Chapter XXVIII. Assignment of ordinals to terms for primitive recursive functionals of finite type 452
Chapter XXIX. On a number theoretic choice schema and its relation to induction 468
Chapter XXX. Applications of the cut elimination theorem to some subsystems of classical analysis 484
Chapter XXXI. Principles of proof and ordinals implicit in given concepts 498