Birkhoff's axioms


In 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry in the plane, sometimes referred to as Birkhoff's axioms. These postulates are all based on basic geometry that can be confirmed experimentally with a scale and protractor. Since the postulates build upon the real numbers, the approach is similar to a model-based introduction to Euclidean geometry.
Birkhoff's axiom system was utilized in the secondary-school textbook by Birkhoff and Beatley.
These axioms were also modified by the School Mathematics Study Group to provide a new standard for teaching high school geometry, known as .
A few other textbooks in the foundations of geometry use variants of Birkhoff's axioms.

Postulates

The distance between two points and is denoted by, and the angle formed by three points is denoted by.
Postulate I: Postulate of line measure.
The set of points on any line can be put into a 1:1 correspondence with the real numbers so that for all points and .
Postulate II: Point-line postulate.
There is one and only one line that contains any two given distinct points and .
Postulate III: Postulate of angle measure.
The set of rays through any point can be put into 1:1 correspondence with the real numbers so that if and are points of and, respectively, the difference of the numbers associated with the lines and is. Furthermore, if the point on varies continuously in a line not containing the vertex, the number varies continuously also.
Postulate IV: Postulate of similarity.
Given two triangles and and some constant such that and, then, and.