# Foundations of Euclidean and Non-Euclidean Geometry by Ellerly B. Golos PDF

By Ellerly B. Golos

This e-book is an try and current, at an ordinary point, an method of geometry in accordance with Euclid, and according to the trendy advancements in axiomatic arithmetic.

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Extra resources for Foundations of Euclidean and Non-Euclidean Geometry

Sample text

How can one be sure ? This IS a question to which there is no definitive answer. There IS, however, a pragmatic test for consistency that mathematicians have been using for years. To explain precisely how the test works, it will be helpful to introduce a few definitions and then to use the axiomatic systems of the preceding chapter to illustrate the test. , ............. introduced as Axiom Set 1 and Axiom Set 2 are "abstract systems" as long as the terms "point" and "line" are taken as undefined.

Does this mean that independence, as defined in the next definition, is an unnecessary property in an axiom system? Evidently. But there are many mathematicians who, for aesthetic and logical reasons, try to reduce an axiom set to a set of independent axioms. In fact, such an attempt plays a significant role in the history of mathematics, as will be apparent later. Let us consider a new problem. Suppose one wishes to choose an independent set of axioms. How does one go about such a task? We might start with the following definition : Defini tion.

Three points; no line. 6. l = { Pi, P2 } ; m = { P2, P a } ; k = { P a, Pi } . 7. 1 4 1 2 2 1 2 1 3 1 2 5 4 3 4 3 4 3 4 2. 3 6 5 6 5 6 6 5 5 6 8. k = { Pi } ; l = { Pi, P2 } ; m = { Pi, Pa } ; n = { P2, Pa } . 9. k = { P2, P a, P" } ; l = { Pi, P2, Pa } ; m = { Pi, P2, P 4 } . 10. l = { Pi, Pa, P2 } . 11. 1 2 3 4 5 6 7 2 3 4 5 6 7 1. 4 5 6 7 1 2 3 12. l = { P i, P2, Pa } ; m = { P", Ps, P6 } . II. Consider the following axiom system : 1. If l and m are any two distinct lines, they have at least one point in common.