information:classification
Tarski's axioms are an axiom system for Euclidean geometry, specifically for that portion of Euclidean geometry that is formulable in first-order logic with identity. As such, it does not require an underlying set theory. The only primitive objects of the system are "points" and the only primitive predicates are "betweenness" and "congruence". The system contains infinitely many axioms. The axiom system is due to Alfred Tarski who first presented it in 1926. Other modern axiomizations of Euclidean geometry are Hilbert's axioms (1899) and Birkhoff's axioms (1932). Using his axiom system, Tarski was able to show that the first-order theory of Euclidean geometry is consistent, complete and decidable: every sentence in its language is either provable or disprovable from the axioms, and we have an algorithm which decides for any given sentence whether it is provable or not.