Axioms for Plane Geometry
The axioms systems of Euclid and Hilbert were intended to provide everything needed for plane geometry without any prior development. The axioms of Hilbert include information about the lines in the plane that implies that each line can be identified with the structure commonly called the “real numbers” and denoted by ℝ. Euclid’s axioms also include information of that kind but the meaning of the “real numbers” may not have been the same in that era as it is now. In both cases, geometry was taken as more fundamental than the real number system. Instead we are going to use ℝ as an ingredient in laying the foundations of hyperbolic plane geometry. (See the Appendix to this chapter for a discussion of the real number system, its properties, its consistency and its uniqueness.) Our axiom system is equivalent to that of Hilbert for the hyperbolic plane, and following the laying of the foundations in this chapter we proceed rigorously with the development of its properties, its consistency and its uniqueness, in later chapters.
KeywordsRational Number Cauchy Sequence Great Circle Plane Geometry Axiom System
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