希尔伯特《几何基础》，何处寻？

Posted 2023-03-08 yuanmeng001

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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).
§2. GROUP I: AXIOMS OF CONNECTION.
The axioms of this group establish a connection between the concepts indicated above; namely, points, straight lines, and planes. These axioms are as follows:
I, 1. Two distinct points A and B always completely determine a straight line a. We write AB = a or BA = a.
Insteadof“determine,” wemayalsoemployotherformsofexpression; forexample, we may say A “lies upon” a, A “is a point of” a, a “goes through” A “and through” B, a “joins” A “and” or “with” B, etc. If A lies upon a and at the same time upon another straight line b, we make use also of the expression: “The straight lines” a “and” b “have the point A in common,” etc.
I, 2. Any two distinct points of a straight line completely determine that line; that is, if AB = a and AC = a, where B6= C, then is also BC = a.
3
I, 3. Three points A, B, C not situated in the same straight line always completely determine a plane α. We write ABC = a.
We employ also the expressions: A, B, C, “lie in” α; A, B, C “are points of” α, etc.
I, 4. Anythreepoints A, B, C ofaplane α,whichdonotlieinthesamestraightline,completely determine that plane. I, 5. If two points A, B of a straight line a lie in a plane α, then every point of a lies in α.
In this case we say: “The straight line a lies in the plane α,” etc.
I, 6. If two planes α, β have a point A in common, then they have at least a second point B in common. I, 7. Upon every straight line there exist at least two points, in every plane at least three points not lying in the same straight line, and in space there exist at least four points not lying in a plane.
Axioms I, 1–2 contain statements concerning points and straight lines only; that is, concerningtheelementsofplanegeometry. Wewillcallthem,therefore,theplaneaxioms of group I, in order to distinguish them from the axioms I, 3–7, which we will designate brie y as the space axioms of this group. Of the theorems which follow from the axioms I, 3–7, we shall mention only the following:
Theorem 1. Two straight lines of a plane have either one point or no point in common; two planes have no point in common or a straight line in common; a plane and a straight line not lying in it have no point or one point in common. Theorem 2. Through a straight line and a point not lying in it, or through two distinct straight lines having a common point, one and only one plane may be made to pass.
§3. GROUP II: AXIOMS OF ORDER.2
The axioms of this group de ne 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……