Hilbert's axioms

In 1902, he further showed that one of Hilbert's axioms for geometry was redundant.

These axioms are closely related to Hilbert's axioms of order.

Note that solid geometry requires no new axioms, unlike the case with Hilbert's axioms.

Pythagorean fields can be used to construct models for some of Hilbert's axioms for geometry .

Rosenthal's mathematical research was in geometry, in particular the classification of regular polyhedra and Hilbert's axioms.

Hilbert's axioms also require "ray," "angle," and the notion of a triangle "including" an angle.

In addition to betweenness and congruence, Hilbert's axioms require a primitive binary relation "on," linking a point and a line.

Hilbert's axioms do not constitute a first-order theory because his continuity axioms require second-order logic.

It proposed a formal set, Hilbert's axioms, instead of the traditional axioms of Euclid.

The process of abstract axiomatization as exemplified by Hilbert's axioms reduces geometry to theorem proving or predicate logic.

Other often-used axiomizations of plane geometry are Hilbert's axioms and Tarski's axioms.

An assignment of Halsted's led Moore to prove that one of Hilbert's axioms for geometry was redundant.

In his work on proving the independence of Hilbert's axioms he worked with a three dimensional space in which each line had only three points on it.

The first four groups of axioms of Hilbert's axioms for plane geometry can be proved using Tarski's axioms.

When David Hilbert made his Hilbert's axioms he used Playfair's axiom instead of the original one from Euclid.

He reformulated Hilbert's axioms for geometry so that points were the only primitive notion, thus turning Hilbert's primitive lines and planes into defined notions.

Today's student of geometry has axioms other than Euclid's available: see Hilbert's axioms and Tarski's axioms.

Shortly afterwards he took part in controversy with Gottlob Frege, concerning Hilbert's axioms for the foundations of Euclidean geometry.

He wrote an elementary geometry text, Rational Geometry, based on Hilbert's axioms, which was translated into French, German, and Japanese.

He started with the "betweenness" of Hilbert's axioms to characterize space without coordinatizing it, and then added extra relations between points to do the work formerly done by vector fields.

David Hilbert publishes Grundlagen der Geometrie, proposing a formal set, Hilbert's axioms, to replace Euclid's elements.

Hilbert's axioms, unlike Tarski's axioms, do not constitute a first-order theory because the axioms V.1-2 cannot be expressed in first-order logic.

Hilbert's axioms for plane geometry number 16, and include Transitivity of Congruence and a variant of the Axiom of Pasch.

Hilbert's axioms: Hilbert's axioms had the goal of identifying a simple and complete set of independent axioms from which the most important geometric theorems could be deduced.

The coordinate geometry given by F for F a Pythagorean field satisfies many of Hilbert's axioms, such as the incidence axioms, the congruence axioms and the axioms of parallels.