作者：David Hilbert
译者：E. J. Townsend
出版社：Merchant Books
发行时间：May 19th 2007
来源：下载的 pdf 版本
Goodreads：4.14 (36 Ratings)
豆瓣：无
Definitions. A system of segments AB, BC, CD, . . . , KL is called a broken line joining A with L and is designated, briefly, as the broken line ABCDE . . . KL. The points lying within the segments AB, BC, CD, . . . , KL, as also the points A, B, C, D, . . . , K, L, are called the points of the broken line. In particular, if the point A coincides with L, the broken line is called a polygon and is designated as the polygon ABCD . . . K. The segments AB, BC, CD, . . . , KA are called the sides of the polygon and the points A, B, C, D, . . . , K the vertices. Polygons having 3, 4, 5, . . . , n vertices are called, respectively, triangles, quadrangles, pentagons, . . . , n-gons. If the vertices of a polygon are all distinct and none of them lie within the segments composing the sides of the polygon, and, furthermore, if no two sides have a point in common, then the polygon is called a simple polygon.
Definitions. Two angles having the same vertex and one side in common, while the sides not common form a straight line, are called supplementary angles. Two angles having a common vertex and whose sides form straight lines are called vertical angles. An angle which is congruent to its supplementary angle is called a right angle.
Any finite number of points is called a figure. If all of the points lie in a plane, the figure is called a plane figure.
Archimdean Axiom:
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.
Axiom of Completeness.(Vollst¨andigkeit): To a system of points, straight lines, and planes, it is impossible to add other 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.”
