Graduate Thesis Or Dissertation
 

A critique of the incidence and order axioms of geometry

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  • A set of axioms of incidence and order for geometry was formulated by David Hilbert in 1898. In this paper these axioms are reformulated and particular care is taken with the two relations of order and incidence. Such phrases as " point P lies on line [cursive small letter L]" are defined in terms of the incidence relation. A set of models is developed which illustrates and clarifies the axioms and establishes their independence. A major portion of the paper is devoted to the implications of these axioms. Hilbert gave a list of theorems in his book Foundations of Geometry which he believed were provable on the basis of only these axioms. Some of the proofs were sketched and some were not given. The three major theorems which are a consequence of these axioms are The ordering of a finite number of points on a line.The ordering of angles with a common side. The Jordan Theorem for polygons. A full proof is given of each of these theorems based upon elementary and not metric concepts. Some consequences of the theorems are investigated. Special attention is payed to the significance of Pasch's Axiom in the proof of these theorems.
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