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Hilbert's 20 axioms of the Euclidean geometry

  1. May 2, 2003 #1


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    what are they?
  2. jcsd
  3. May 2, 2003 #2


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    In his book The Foundations of Geometry (Open Court reprint 1965), Hilbert divides the axioms into five groups.

    I Seven Axioms of Connection
    I.1 Two distinct pioints always determine a straight line.
    I.2 Any two points of a line completely determine that line.
    I.3 Three points not situated in the same straight line always completely determine a plane.
    I.4 Any three points of a plane, which do not lie in the same straight line, completely determine that plane.
    I.5 If two points of a straight line lie in a plane a then all the points of that line lie in a.
    I.6 If two planes have a point in common, then they have at least a second point in common.
    I.7 Upon every 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.

    II. Five Axioms of Order
    II.1 If A, B, and C are points of a straight line and B lies between A and C, then B lies also between C and A.
    II.2 If A and C are two points on a straight line thenthese 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.
    II.3 Of three points situated on a straight line, there is always one and only one which lies between the other two.
    II.4 For any four points of a straight line, the names A, b, c, and D can always be assigned to them in such a way 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. The system of two points A and B on a straight line will be called a segment, denoted by AB or BA. The points between A and B are called the points of the segment AB. All other points of the line are referred to as points lying outside 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 in the same plane as A, B, C not passing through any of the points A, B, or C. Then if the straight line a passes through a point of the segment AB, it will also pass through either if the segment BC or a point of the segment AC.

    III One Axiom of Parallels
    In a plane there can be drawn through any point A, lying outside a straight line a, one and only one straight line which does not intersect the line a. This straight line is called the parallel to a through the given point A.

    IV Six Axioms of Congruence
    IV.1 If A, B are two points on a straight line a, and A' is a point on the same or another straight line a', then upon a given side of A' on the stright line a' we can always find one and only one point B' so that the segment AB (or BA) is congruent to the segment A'B'. Every segment is congruent to itself.
    IV.2 If a segment AB is congruent to the segment A'B' and also to the segment A"B" then the segment A'B' is congruent to the segment A"B".
    IV.3. Let AB and BC be two segments of a straight line a which have no points in common except for the point B and furthermore let A'B' and B'C' be two segments of the same or another line a' having likewise no points in common except for the point B'. Then if AB is congruent to A'B' and BC is congruent to B'C' we have AC congruent to A'C'.

    Let a be any arbitrary plane and h,k, any two distinct half rays lying in a and eminationg from the point O so as to form a part of two different straight lines. The system formed by h,k is termed the angle (h,k). From the previous axioms we can prove that the angle divides the plane a into two regions and that one of these, termen the interior of the angle, has the property that any to points in it form a segment which lies entirely within the region.

    IV.4 Let an angle (h,k) be given in the plane a and let a straight line a' be given in the plane a'. Suppose also that in the plane a' a defineite side of the straight line a' is assigned. Denote by k' a half ray of this line eminating from a point O' on this line. Then in the plane a' there is one and only one half ray k' such that the angle (h,k) is congruent to the angle (h',k') and at the same time all the interior points lie in the assiged side if the straight line a'. This defines the congruence of angled. Every angle is congruent to itself.
    IV.5 If angle (h,k) is congruent to angle (h'k') and to the angle (h",k") then angel (h',k') is congruent to angle (h",k").
    IV.6 If in the two triangles ABC and A'B'C' the congruences AB congruent to A'B' AC congruent to A'C' angle BAC congruent to angle B'A'C' hold, then
    angle ABC will be congruent to angle A'B'C' and angle ACB will be congruent to angle A'C'B'.

    V. Axiom of Continuity
    Let A1 be any point on a straight line between two 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, etc. be all 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.

    That's the lot. I eliminated a few definitions like ray and triangle.
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