现代几何学的次序公理
大家知道，微积分建立在实数系统之上，而实数系统是一个“有序结构”，离开次序公理，谈何微积分？
本文附件列出了现代几何学次序公理组的5条公理。
袁萌 陈启清 3月18日
附件：希尔伯特《几何基础》电子版次序公理（II. 1-5）
§3. GROUP II: AXIOMS OF ORDER.2
The axioms of this group define the idea expressed by the word “between,” and make possible, upon the basis of this idea, an order of sequence of the points upon a straight line, in a plane, and in space. The points of a straight line have a certain relation to one another which the word “between” serves to describe. The axioms of this group are as follows:
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Fig. 1.
II, 1. If A, B,C are points of a straightl ine and B lies between A and C,then B lies also between C and A.
II, 2. If A and C are two points of a straight line, then there exists at least one point B lying between A and C and at least one point D so situated that C lies between A and D.
Fig. 2.
II, 3. Of any three points situated on a straight line, there is always one and only one which lies between the other two. II, 4. Any four points A, B, C, D of a straight line can always be so arranged that B shall lie between A and C and also between A and D, and, furthermore, that C shall lie between A and D and also between B and D.
Definition. We will call the system of two points A and B, lying upon a straight line, a segment and denote it by AB or BA. The points lying between A and B are called the points of the segment AB or the points lying within the segment AB. All other points of the straight line are referred to as the points lying outside the segment AB. The points A and B are called the extremities of the segment AB.
II, 5. Let A, B, C be three points not lying in the same straight line and let a be a straight line lying in the plane ABC and not passing through any of the points A, B, C. Then, if the straight line a passes through a point of the segment AB, it will also pass through either a point of the segment BC or a point of the segment AC. Axioms II,
I1–4 contain statements concerning the points of a straight line only, and, hence, we will call them the linear axioms of group II. Axiom
II, 5 relates to the elements of plane geometry and, consequently, shall be called the plane axiom of group II. 2These axioms were first studied in detail by M. Pasch in his Vorlesungen über neuere Geometrie, Leipsic, 1882. Axiom
II, 5 is in particular due to him.
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Fig. 3.