Euclid redux

In 1882, Moritz Pasch published a book revealing unstated assumptions in Euclid's Elements and calling for a new examination of the foundations of geometry.

Here are three axiomatizations of plane geometry. I'm posting this because they're three surprisingly creative, different approaches.

• Hilbert's axioms (1899) are simply a more rigorous version of Euclid's axioms. Whereas Euclid states five postulates and five “common notions”, Hilbert requires 20 axioms—plus set theory.

• Birkhoff's axioms (1932) take a totally different approach. Birkhoff starts with the real numbers. The distance between points on a line is defined by analogy to the reals, and something similar defines angle measures. The astonishing result is a complete axiomatization of plane geometry in just four postulates.

• Tarski's axioms (1983) are built directly on top of first-order logic, so they don't require set theory. As a result, Tarski was able to prove (posthumously, no less) that the system was consistent, complete, and decidable.