关于公理系统的无矛盾性                           

回顾历史，欧氏几何的无矛盾性可以归结为算术公理的无矛盾性，希尔伯特首次提出这一论点。

但是，直到1936年，根茨（G.Gentaen，1909-1945）使用超限归纳法证明了算术公理系统的无矛盾性。

    我们投放全国普通高校的连环画微积分电子版教科书在该书结束语中给出了超实数的公理系统。

    希尔伯特《几何学基础》第二章给出了欧氏几何的无矛盾性证明（划归于算术的无矛盾性）。请见本文

袁萌  陈启清   3月8日

附件: 希尔伯特及其《几何学基础》电子版（英文PDF），

The Foundations of Geometry

BY DAVID HILBERT, PH. D.

PROFESSOR OF MATHEMATICS, UNIVERSITY OF GÖTTINGEN

AUTHORIZED TRANSLATION BY E. J. TOWNSEND, PH. D.

UNIVERSITY OF ILLINOIS

REPRINT EDITION

THE OPEN COURT PUBLISHING COMPANY

LA SALLE ILLINOIS

1950

TRANSLATION COPYRIGHTED

BY

The Open Court Publishing Co.

1902.

PREFACE.

The material contained in the following translation was given in substance by Professor Hilbert as a course of lectures on euclidean geometry at the University of Göttingen during the winter semester of 1898–1899. The results of his investigation were re-arranged and put into the form in which they appear here as a memorial address published in connection with the celebration at the unveiling of the Gauss-Weber monument at Göttingen, in June, 1899. In the French edition, which appeared soon after, Professor Hilbert made some additions, particularly in the concluding remarks, where he gave an account of the results of a recent investigation made by Dr. Dehn. These additions have been incorporated in the following translation. As a basis for the analysis of our intuition of space, Professor Hilbert commences his discussion by considering three systems of things which he calls points, straight lines, and planes, and sets up a system of axioms connecting these elements in their mutual relations. The purpose of his investigations is to discuss systematically the relations of these axioms to one another and also the bearing of each upon the logical development of euclidean geometry. Among the important results obtained, the following are worthy of special mention: 1. The mutual independence and also the compatibility of the given system of axioms is fully discussed by the aid of various new systems of geometry which are introduced. 2. The most important propositions of euclidean geometry are demonstrated in such a manner as to show precisely what axioms underlie and make possible the demonstration. 3. Theaxiomsofcongruenceareintroducedandmadethebasisofthedenitionofgeometric displacement. 4. The signicance of several of the most important axioms and theorems in the development of the euclidean geometry is clearly shown; for example, it is shown that the whole of the euclidean geometry may be developed without the use of the axiom of continuity; the signicance of Desargues’s theorem, as a condition that a given plane geometry may be regarded as a part of a geometry of space, is made apparent, etc. 5. Avarietyofalgebrasofsegmentsareintroducedinaccordancewiththelawsofarithmetic. This development and discussion of the foundation principles of geometry is not only of mathematical but of pedagogical importance. Hoping that through an English edition these important results of Professor Hilbert’s investigation may be made more accessible to English speaking students and teachers of geometry, I have undertaken, with his permission, this translation. In its preparation, I have had the assistance of many valuable suggestions from Professor Osgood of Harvard, Professor Moore of Chicago, and Professor Halsted of Texas. I am also under obligations to Mr. Henry Coar and Mr. Arthur Bell for reading the proof.

E. J. Townsend

University of Illinois.

CONTENTS

PAGE Introduction ................................................................ 1

CHAPTER I. THE FIVE GROUPS OF AXIOMS.

§ 1. The elements of geometry and the ve groups of axioms ............. 2 § 2. Group I: Axioms of connection ....................................... 2

§ 3. Group II: Axioms of Order ........................................... 3

§ 4. Consequences of the axioms of connection and order ................. 5

§ 5. Group III: Axiom of Parallels (Euclid’s axiom) ........................ 7

§ 6. Group IV: Axioms of congruence ..................................... 8

§ 7. Consequences of the axioms of congruence ........................... 10

§ 8. Group V: Axiom of Continuity (Archimedes’s axiom) ................ 15

CHAPTER II. THE COMPATIBILITY AND MUTUAL INDEPENDENCE OF THE AXIOMS.

§ 9. Compatibility of the axioms .......................................... 17

§10. Independence of the axioms of parallels. Non-euclidean geometry ... 19

§11. Independence of the axioms of congruence ........................... 20

§12. Independence of the axiom of continuity. Non-archimedean geometry 21

CHAPTER III. THE THEORY OF PROPORTION.

§13. Complex number-syst...... 23

§14. Demonstration of Pascal’s theorem ............................ 25

§15. An algebra of segments, based upon Pascal’s theorem ................ 30

§16. Proportion and the theorems of similitude ............................ 33

§17. Equations of straight lines and of planes ............................. 35

CHAPTER IV. THE THEORY OF PLANE AREAS.

§18. Equal area and equal content of polygons ............................ 38

§19. Parallelograms and triangles having equal bases and equal altitudes . 40

§20. The measure of area of triangles and polygons ................ 41

§21. Equality of content and the measure of area ...................... 44

CHAPTER V. DESARGUES’S THEOREM.

§22. Desargues’s theorem and its demonstration for plane geometry by aid of the axioms of congruence ...................................... 48

§23. The impossibility of demonstrating Desargues’s theorem for the plane without the help of the axioms of congruence ............. 50§24. Introduction of an algebra of segments based upon Desargues’s theorem and independent of the axioms of congruence ............. 53

§25. The commutative and the associative law of addition for our new algebra of segments ...................................................... 55

§26. The associative law of multiplication and the two distributive laws for the new algebra of segments ......................................... 56

§27. Equation of the straight line, based upon the new algebra of segments ... 61

§28. The totality of segments, regarded as a complex number system ......... 64 §29. Construction of a geometry of space by aid of a desarguesian number system ............................................ 65

§30. Signicance of Desargues’s theorem ............................... 67

CHAPTER VI. PASCAL’S THEOREM.

§31. Two theorems concerning the possibility of proving Pascal’s theorem .... 68

§32. The commutative law of multiplication for an archimedean number system ............................................. 68

§33. The commutative law of multiplication for a non-archimedean number system ........................................ 70

§34. Proof of the two propositions concerning Pascal’s theorem. Non-pascalian geometry. ................................................. 72

§35. The demonstration, by means of the theorems of Pascal and Desargues, of any theorem relating to points of intersection ......................... 73

CHAPTER VII. GEOMETRICAL CONSTRUCTIONS BASED UPON THE AXIOMS I–V.

§36. Geometrical constructions by means of a straight-edge and a transferer of segments ................................................... 74

§37. Analytical representation of the co-ordinates of points which can be so constructed ............................................. 76

§38. The representation of algebraic numbers and of integral rational functions as sums of squares ....................................................... 78

§39. Criterion for the possibility of a geometrical construction by means of a straight-edge and a transferer of segments .. 80 Conclusion ............ 83

“All human knowledge begins with intuitions, thence passes to concepts and ends with ideas.” Kant, Kritik der reinen Vernunft, Elementariehre, Part 2, Sec. 2.

INTRODUCTION.

Geometry, like arithmetic, requires for its logical development only a small number of simple, fundamental principles. These fundamental principles are called the axioms of geometry. The choice of the axioms and the investigation of their relations to one another is a problem which, since the time of Euclid, has been discussed in numerous excellent memoirs to be found in the mathematical literature.1 This problem is tantamount to the logical analysis of our intuition of space. The following investigation is a new attempt to choose for geometry a simple and complete set of independent axioms and to deduce from these the most important geometrical theorems in such a manner as to bring out as clearly as possible the signicance of the different groups of axioms and the scope of the conclusions to be derived from the individual axioms.

1Compare the comprehensive and explanatory report of G. Veronese, Grundzüge der Geometrie, German translation by A. Schepp, Leipzig, 1894 (Appendix). See also F. Klein, “Zur ersten Verteilung des Lobatschefskiy-Preises,” Math. Ann., Vol. 50