在今年“两会”期间，我们国推出希尔伯特《几何基础》电子版，目的是：把现代几何学的5组几何公理通过国内互联网直接摆在广大读者面前，增进思考。
今天，希尔伯特《几何基础》的5组现代几何公理已经全部摆出来了。
这5组几何公理有什么用？简而言之，它能够确保直线上的点与实数的一一对挺关系。这种关系的重要性就无需多言。
袁萌 陈启清 3月20日
附件：现代几何学的连续性公理
§8. GROUP V. AXIOM OF CONTINUITY. (ARCHIMEDEAN AXIOM.)
This axiom makes possible the introduction into geometry of the idea of continuity. In order to state this axiom, we must first establish a convention concerning the equality of two segments. For this purpose, we can either base our idea of equality upon the axioms relating o the congruence of segments and define as“equal” thecorres pondingly
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congruent segments, or upon the basis of groups I and II, we may determine how, by suitable constructions (see Chap. V, § 24), a segment is to be laid off from a point of a given straight tlines ot hatanew,definite segmeis obtained“equal” toit . Inconformity with such a convention, the axiom of Archimedes may be stated as follows:
V. Let A1 be any point upon a straight line between the arbitrarily chosen points A and B. Take the points A2, A3, A4,... so that A1 lies between A and A2, A2 between A1 and A3, A3 between A2 and A4 etc. Moreover, let the segments AA1, A1A2, A2A3, A3A4, ...
be equal to one another. Then, among this series of points, there always exists a certain point An such that B lies between A and An.
The axiom of Archimedes is a linear axiom. Remark.3 To the preceeding five groups of axioms, we may add the following one, which, although not of a purely geometrical nature, merits particular attention from a theoretical point of view. It may be expressed in the following form:
Axiom of Completeness.4 (Vollständigkeit): Toasystem o fpoints,straight lines, and planes, it is impossible to add ot elements in such a manner that the system thus generalized shall form a new geometry obeying all of the five groups of axioms. In other words, the elements of geometry form a system which is not susceptible of extension, if we regard the five groups of axioms as valid.
This axiom gives us nothing directly concerning the existence of limiting points, or of the idea of convergence. Nevertheless, it enables us to demonstrate Bolzano’s theorem by virtue of which, for all sets of points situated upon a straight line between two definite points of the same line, there exists necessarily a point of condensation, that is to say, a limiting point. From a theoretical point of view, the value of this axiom is that it leads indirectly to the introduction of limiting points, and, hence, renders it possible to establish a one-to-one correspondence between the points of a segment and the system of real numbers. However, in what is to follow, no use will be made of the “axiom of completeness.”
3Added by Professor Hilbert in the French translation.—Tr. 4See Hilbert, “Ueber den Zahlenbegriff,” Berichte der deutschen Mathematiker-Vereinigung, 1900.
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COMPATIBILITY AND MUTUAL INDEPENDENCE OF THE AXIOMS.