Birkhoff's Axioms - Postulates

Postulates

Postulate I: Postulate of Line Measure. A set of points {A, B, ...} on any line can be put into a 1:1 correspondence with the real numbers {a, b, ...} so that |b − a| = d(A, B) for all points A and B.

Postulate II: Point-Line Postulate. There is one and only one line, ℓ, that contains any two given distinct points P and Q.

Postulate III: Postulate of Angle Measure. A set of rays {ℓ, m, n, ...} through any point O can be put into 1:1 correspondence with the real numbers a (mod 2π) so that if A and B are points (not equal to O) of ℓ and m, respectively, the difference a_m − a_ℓ (mod 2π) of the numbers associated with the lines ℓ and m is AOB. Furthermore, if the point B on m varies continuously in a line r not containing the vertex O, the number a_m varies continuously also.

Postulate IV: Postulate of Similarity. Given two triangles ABC and A'B'C' and some constant k > 0, d(A', B' ) = kd(A, B), d(A', C' ) = kd(A, C) and B'A'C' = ±BAC, then d(B', C' ) = kd(B, C), C'B'A' = ±CBA, and A'C'B' = ±ACB.