Chapter: Hilbert's Axiomatic System

As introduced in Historical Development of Geometry, the work Elements of Euclid (about 325 BC to 265 BC) was a milestone in the development of the axiomatic method and mathematics as a whole, but Euclid's axiomatic system was very vague with respect to the definitions of the mathematical objects “points”, “lines” and “planes”. It depended on the intuitive understanding of these objects.

David Hilbert (1862 - 1943) developed a more modern axiomatic system, and his key idea was to detach "points", "lines" and "planes" from real-world objects. He did so at the cost of the relatively high number of $21$ axioms as compared to only $5$ axioms in the "Elements".

In this chapter, we will introduce these axioms and derive theorems from them. These theorems and conclusions are today the standard way to establish the theory of Euclidean geometry. Hilbert grouped his axioms into the following groups, and we will follow his approach:

• axioms of connection,
• axioms of order,
• axioms of congruence,
• axiom of parallels,
• axioms of continuity.

Chapters: 1
Parts: 2

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References

Bibliography

1. Hilbert, David: "Grundlagen der Geometrie", Leipzig, B.G. Teubner, 1903