Birkhoff's axioms

In 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry 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 SMSG axioms. A few other textbooks in the foundations of geometry use variants of Birkhoff's axioms.

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 $a (mod 2 π)$ 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.

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