Collected Papers of Gerhard Gentzen (eBook)

Front Cover 1
The Collected Papers of Gerhard Gentzen 4
Copyright Page 5
TABLE OF CONTENTS 10
ACKNOWLEDGMENTS 6
BIBLIOGRAPHY 7
BIOGRAPHICAL SKETCH 8
INTRODUCTION 16
NOTES TO THE INTRODUCTION 39
CHAPTER 1. ON THE EXISTENCE OF INDEPENDENT AXIOM SYSTEMS FOR INFINITE SENTENCE SYSTEMS 44
Section I. Notations and some lemmas 44
Section II. An infinite closed sentence system possessing no independent axiom system 57
Section III. Construction of an independent axiom system for a given denumerably infinite closed linear sentence system 61
CHAPTER 2 . ON THE RELATION BETWEEN INTUITIONIST AND CLASSICAL ARITHMETIC 68
Introduction 68
1. Terminology and notations 68
2. The formal structure of arithmetic 70
3. The intuitionist validity of the law of double negation 73
4. Transformation of proofs of classical arithmetic into proofs of intuitionist arithmetic 75
5. Consequences 80
6. The consistency of arithmetic – the redundancy of negation in intuitionist arithmetic 81
CHAPTER 3 . INVESTIGATIONS INTO LOGICAL DEDUCTION 83
Synopsis 83
Section I. Terminology and notations 84
Section II. The calculus of natural deduction 89
Section III. The deductive calculi LJ,LK and the Hauptsatz 96
Section IV. Some applications of the Hauptsatz 118
Section V. The equivalence of the new calculi NJ, NK, and LJ, LK witha calculus modelled on the formalism of Hilbert 130
CHAPTER 4. THE CONSISTENCY OF ELEMENTARY NUMBER THEORY 147
Section I. Reflections on the purpose and possibility of consistency proofs 147
Section II. The formalization of elementary number theory 154
Section III. Disputable and indisputable forms of inference in elementary number theory 173
Section IV. The consistency proof 185
Section V. Reflections on the consistency proof 208
APPENDIX TO 4 216
CHAPTER 5. THE CONSISTENCY OF THE SIMPLE THEORY OF TYPES 229
1. The formal structure of the simple theory of types 229
2. The consistency proof 232
CHAPTER 6. THE CONCEPT OF INFINITY IN MATHEMATICS 238
CHAPTER 7. THE PRESENT STATE OF RESEARCH INTO THE FOUNDATIONS OF MATHEMATICS 249
1. The different points of view concerning the question of the antinomies and the concept of infinity 249
2. Exact foundational research in mathematics: axiomatics, metalogic, metamathematics, The theorems of Gödel and Skolem 253
3. The continuum 258
4. The possibility of reconciling the different points of view 262
CHAPTER 8. NEW VERSION OF THE CONSISTENCY PROOF FOR ELEMENTARY NUMBER THEORY 267
1. New formalization of number-theoretical proofs 268
2. Outline of the consistency proof 275
3. A reduction step on a contradictive derivation 276
4. The ordinal numbers – concluding remarks 292
CHAPTER 9. PROVABILITY AND NONPROVABILITY OF RESTRICTED TRANSFINITE INDUCTION IN ELEMENTARY NUMBER THEORY 302
1. TJ-derivations 302
2. Characterizations of TJ-derivations 307
3. Demonstrations of nonprovability 312
CHAPTER 10. FUSION OF SEVERAL COMPLETE INDUCTIONS 324
NOTES 327
GLOSSARY 333
INDEX OF SYMBOLS 336
INDEX OF AUTHORS 337
INDEX OF SUBJECTS 339