Hilbert's axioms

Hilbert's axioms are a set of 20 basic ideas that help us understand and work with Euclidean geometry, the kind of geometry most people learn in school. They were proposed by a famous mathematician named David Hilbert in 1899. Hilbert wrote about these ideas in his important book called The Foundations of Geometry.

These axioms provide a clear and modern way to build up all the rules and facts of geometry from simple starting points. Before Hilbert, people used the ideas of the ancient Greek mathematician Euclid, but his work had some gaps that Hilbert wanted to fix.

Other mathematicians, like Alfred Tarski and George Birkhoff, also created their own sets of axioms for Euclidean geometry later on. Hilbert's work remains very important for both learning geometry and for advanced mathematical studies.

The axioms

Hilbert's axioms are a set of rules that help us understand basic geometry. David Hilbert created these rules in 1899 to make geometry clearer and easier to study. They use simple ideas like points, lines, and planes to explain how shapes work.

The axioms start with six basic ideas:

• Point: A spot in space, like a dot you might draw on paper.
• Line: An imaginary straight path that goes on forever in both directions.
• Plane: A flat surface that stretches out in all directions.

These ideas connect in three important ways:

• Incidence: This tells us when a point is on a line or a plane.
• Betweenness: This helps us understand the order of points on a line — which points come between others.
• Congruence: This lets us compare lengths and angles to see if they are the same.

These rules help build the foundation for studying shapes, sizes, and spaces in geometry. Other famous sets of rules for geometry were also created by mathematicians like Alfred Tarski and George Birkhoff.

Hilbert's discarded axiom

David Hilbert originally included a 21st axiom in his work on geometry. This axiom, known as Pasch's theorem, described how points are arranged on a straight line. Later, mathematicians E. H. Moore and R. L. Moore showed that this axiom wasn't necessary because the other axioms already covered its ideas. Because of this, Pasch's axiom was moved to a different number in the list.

Main article: Pasch's theorem

Editions and translations of Grundlagen der Geometrie

David Hilbert wrote Grundlagen der Geometrie in 1899 for a special talk, and it was soon translated into French and English. The French version added an important rule called the Completeness Axiom. In 1902, an English version by E.J. Townsend included these changes and more.

Over the years, many new editions of the book came out in German, with the seventh edition being the last one Hilbert himself saw. Later editions had updates mostly in extra sections at the end. A new English version was made in 1971 by Leo Unger, based on a later German edition and including more changes suggested by Paul Bernays. These updates included renaming some rules and changing how the axioms were organized.

Application

Hilbert's axioms provide a way to describe Euclidean geometry, which is the study of shapes and spaces. By changing some of these rules, we can focus just on flat, two-dimensional geometry.

These axioms helped shape modern mathematics by showing new ways to think about rules and proofs. They inspired many other mathematicians to develop their own systems for understanding geometry. Later efforts to use computers to check Hilbert's work found that some of his proofs needed clearer explanations.

Main articles: Euclidean plane geometry, Tarski's axioms, first-order logic, metamathematical, formal systems