希尔伯特《几何基础》，何处寻？

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袁萌   陈启清  3月18日

附件：希尔伯特《几何基础》第一章第1节的相关内容如下:

THE FIVE GROUPS OF AXIOMS.

§1. THE ELEMENTS OF GEOMETRY AND THE FIVE GROUPS OF AXIOMS.

Let us consider

three distinct systems things. The things composing the rst system, we will call points and designate them by the letters A, B, C,...; those of the second, we will call straight lines and designate them by the letters a, b, c,...; and those of the third system, we will call planes and designate them by the Greek letters α, β, γ,...The points are called the elements of linear geometry; the points and straight lines, the elements of plane geometry; and the points, lines, and planes, the elements of the geometry of space or the elements of space. We think of these points, straight lines, and planes as having certain mutual relations, which we indicate by means of such words as “are situated,” “between,” “parallel,” “congruent,” “continuous,” etc. The complete and exact description of these relations follows as a consequence of the axioms of geometry. These axioms may be arranged in ve groups. Each of these groups expresses, by itself, certain related fundamental facts of our intuition. We will name these groups as follows: I, 1–7. Axioms of connection. II, 1–5. Axioms of order. III. Axiom of parallels (Euclid’s axiom). IV, 1–6. Axioms of congruence. V. Axiom of continuity (Archimedes’s axiom).