---
tag: KM1FB8XZ2R
kind: concept
name: Hilbert's axioms
msc: [51M05, 51F05, 03B30]
sources:
  - url: https://en.wikipedia.org/wiki/Hilbert%27s_axioms
same_as:
  wikidata: Q840243
  wikipedia: Hilbert's axioms
---

## Summary

Hilbert's axioms are the axiom system for [[Euclidean geometry|KRZD96GSVC]] that David Hilbert gave in his book *Grundlagen der Geometrie* of 1899.
They speak of points, lines and planes and of the relations of incidence, [[betweenness|CE8GX6ADZS]] and congruence, and they fall into five groups.
They fill the gaps of [[Euclid's Elements|83WHDAV68J]].
They made it possible to prove that geometry is consistent if the real numbers are.
They also made it possible to prove the independence of single axioms, such as the [[parallel postulate|2PC3XY0PYZ]].

## Explanations

### L1 human

Euclid's proofs sometimes use facts that are clear from the picture but not from his axioms.
Hilbert wrote down a complete list of rules, so that every proof can be checked without looking at a picture.
For example, his axiom of Pasch says that a line that enters a triangle through one side must leave it through another side.
Hilbert said that one could replace "points, lines and planes" by "tables, chairs and beer mugs" and the proofs would still work.