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  "body": "\n## Summary\n\nHilbert's axioms are the axiom system for [[Euclidean geometry|KRZD96GSVC]] that David Hilbert gave in his book *Grundlagen der Geometrie* of 1899.\nThey speak of points, lines and planes and of the relations of incidence, [[betweenness|CE8GX6ADZS]] and congruence, and they fall into five groups.\nThey fill the gaps of [[Euclid's Elements|83WHDAV68J]].\nThey made it possible to prove that geometry is consistent if the real numbers are.\nThey also made it possible to prove the independence of single axioms, such as the [[parallel postulate|2PC3XY0PYZ]].\n\n## Explanations\n\n### L1 human\n\nEuclid's proofs sometimes use facts that are clear from the picture but not from his axioms.\nHilbert wrote down a complete list of rules, so that every proof can be checked without looking at a picture.\nFor example, his axiom of Pasch says that a line that enters a triangle through one side must leave it through another side.\nHilbert said that one could replace \"points, lines and planes\" by \"tables, chairs and beer mugs\" and the proofs would still work.\n",
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