This paper presents a proof that the classical geometry as stated by Karol Borsuk [1] follows from the join geometry of Walter Prenowitz [2].
The approach taken is to assume the axioms of Prenowitz. Using these as the foundation, the theory of join geometry is then developed to include such ideas as 'convex set', 'linear set', the important concept of 'dimension', and finally the relation of 'betweenness'. The development is in the form of definitions with the important extensions given in the form of theorems.
With a firm foundation of theorems in the join geometry, the axioms of classical geometry are examined, and then they are proved as theorems or modified and proved as theorems.
The basic notation to be used is that of set theory. No distinction is made between the set consisting of a single element and the element itself. Thus the notation for set containment is ⊂, and is used to denote element containment also. The set containing no elements, or the empty set, is denoted by Ø, The set of points belonging to at least one of the sets under consideration is called union, denoted ∪. The set of points belonging to each of the sets under consideration is called the intersection and denoted by ∩. Any other notation used will be defined at the first usage.
| Identifer | oai:union.ndltd.org:UTAHS/oai:digitalcommons.usu.edu:etd-7850 |
| Date | 01 May 1963 |
| Creators | Giegerich, Louis John, Jr. |
| Publisher | DigitalCommons@USU |
| Source Sets | Utah State University |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | All Graduate Theses and Dissertations |
| Rights | Copyright for this work is held by the author. Transmission or reproduction of materials protected by copyright beyond that allowed by fair use requires the written permission of the copyright owners. Works not in the public domain cannot be commercially exploited without permission of the copyright owner. Responsibility for any use rests exclusively with the user. For more information contact digitalcommons@usu.edu. |
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