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Based on ancient Greek methods, an axiomatic system is a formal description of a way to establish the mathematical truth that flows from a fixed set of assumptions. Although applicable to any area of mathematics, geometry is the branch of elementary mathematics in which this method has most extensively been successfully applied.
In 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry in the plane, sometimes referred to as Birkhoff's axioms. [1] These postulates are all based on basic geometry that can be confirmed experimentally with a scale and protractor.
He also published The Foundations of Geometry (1940) and The Representations of the Symmetric Groups (1961) as well as Vector Geometry (1962). [1] His last mathematical book was his edition of the collected papers of Alfred Young (1977), and he later wrote short volumes on departmental, local, and family history.
Removing five axioms mentioning "plane" in an essential way, namely I.4–8, and modifying III.4 and IV.1 to omit mention of planes, yields an axiomatization of Euclidean plane geometry. 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.
Moritz Pasch (8 November 1843, Breslau, Prussia (now Wrocław, Poland) – 20 September 1930, Bad Homburg, Germany) was a German mathematician of Jewish ancestry [1] specializing in the foundations of geometry. He completed his Ph.D. at the University of Breslau at only 22 years of age.
Jean George Pierre Nicod (1 June 1893, in France – 16 February 1924, in Geneva, Switzerland) was a French philosopher and logician, best known for his work on propositional logic and induction. Biography
Suitable changes in these axioms yield axiom sets for Euclidean geometry for dimensions 0, 1, and greater than 2 (Tarski and Givant 1999: Axioms 8 (1), 8 (n), 9 (0), 9 (1), 9 (n)). Note that solid geometry requires no new axioms, unlike the case with Hilbert's axioms .
Pasch's axiom — Let A, B, C be three points that do not lie on a line and let a be a line in the plane ABC which does not meet any of the points A, B, C.If the line a passes through a point of the segment AB, it also passes through a point of the segment AC, or through a point of segment BC.