Hilbert's axioms [SEP] named after
Hilbert's axioms
Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book "Grundlagen der Geometrie" (tr. "The Foundations of Geometry") as the foundation for a modern treatment of Euclidean geometry. Other well-known modern axiomatizations of Euclidean geometry are those of Alfred Tarski and of George Birkhoff.
The axioms.
Hilbert's axiom system is constructed with six primitive notions: three primitive terms:
- point;
- line;
- plane;
Huntington, Veblen and Heyting. Their objective was the axiomatisation of branches of mathematics like geometry, arithmetic, analysis and set theory. Most notable was Hilbert's Program, which sought to ground all of mathematics to a finite set of axioms, proving its consistency by "finitistic" means and providing a procedure which would decide the truth or falsity of any mathematical statement. The standard axiomatization of the natural numbers is named the Peano axioms in his honor. Peano maintained a clear distinction between mathematical and logical symbols. While unaware of