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MacLane's Postulates

By David A. Smith

In the 1960's Saunders MacLane proposed a new set of axioms that are more like Birkhoff's than Euclid's or Hilbert's axioms. Like Birkhoff's axioms, MacLane's axioms also use the real numbers; and so has fewer axioms than Hilbert's axioms did. MacLane's approach also does not generate any new theorems in Euclidean geometry but rather simplifies previous attempts at erecting the foundations of Euclidean geometry. The main differences are that MacLane uses a function to measure distance between points (called a metric function) and the so called Continuity Axiom, which incorporates the Crossbar Proposition into Euclidean geometry as an axiom. This topic states MacLane's axioms, from which all of Euclidean geometry can be proven.

Comment (Undefined Terms) Point, distance, line, and angle measure.

Axiom (Distance Axioms)
    (i) There are at least two points.
    (ii) If and are points, is a nonnegative number.
    (iii) For points and if and only if
    (iv) If and are points,

Axiom (Line Axioms)
    (i) A line is a set of points containing more than one point.
    (ii) Through two distinct points there is one and only one line.
    (iii) Three distinct points lie on a line if one of them is between the other two.
    (iv) On each ray from a point and to each positive real number there is a point with

Axiom (Angle Axioms)
    (i) If   and are rays from the same point, is a real number modulo
    (ii) If and are three rays from the same point, then
    (iii) If is a ray from and is a real number (modulo 360), then there is a ray from such that
    (iv) If then if and only if

Axiom (Similiarity Axiom) If two ordered triangles and have and (for positive) they are similiar.

Axiom (Continuity Axiom) Let be proper. If is between and then Conversely, if then the ray meets the interval

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