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Birkoff's Postulates

    Birkhoff was the first to build the real number system into the foundations of Euclidean geometry. In doing so; he only needed four axioms instead of Hilbert's sixteen axioms. Birkhoff's approach has gained acceptance even though it does not produce new theorems in Euclidean geometry. His axioms do produce a logical equivalent version of Euclidean geometry with fewer needed axioms and so his approach is appealing. One postulate is the usual assumption that two points determine a unique line. His postulates of Line Measure and Angle Measure is where the real numbers enter; he assumes that the points on a line (and angles through a point) can be put into a correspondence with real numbers and real numbers mod 360 in a way that is compatible with distance measurement and angle measurement, respectively. In particular, no mention of betweenness is needed in Birkhoff's axioms because a point being between two others can be defined in terms of distance and real numbers. This topic states Birkhoff's axioms as published in the Annals of Mathematics in 1932.

Comment (Undefined Elements and Relations)
    (a) points,
    (b) sets of points called lines,
    (c) distance between any two points: a non-negative number with
    (d) angle formed by three ordered distinct points a real number The point is called the vertex of the angle.  

Postulate (Postulate of Line Measure) The points of any line can be point into correspondence with the real numbers so that for all points

Definition (Between) A point is between A and if

Definition (Segment) The points and together with all points between and form segment

Definition (Half-Line) The half-line with endpoint is defined by two points in line   as the set of all points of such that is not betwen and

Definition (Triangle) If are three distinct points, then the three segments and are said to form a triangle with sides and are vertices and If are in the same line, is said to be degenerate.

Postulate (Point-line Postulate) One and only one line contains two given points and

Definition (Parallel) If two distinct lines have no points in common they are parallel. A line is always regarded as parallel to itself.

Postulate (Angle Measure) The half-lines thru any point can be put into correspondence with the real numbers so that if and with and are points of and respectively, the difference is

Definition (Half-Lines) Two half-lines thru are said to form a straight angle if Two half-lines thru are said to form a right angle if in which case we also say that is perpendicular to

Postulate (Similarity) If in two triangles and and for some constant and also then also and

Definition (Similiar) Any two geometric figures are similiar if there exists a correspondence between the points of the two figures such that all corresponding distances are in proportion and corresponding angles are either equal or all negatives of each other. Any two geometric figures are congruent if they are similiar with in the Similarity Postulate.

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