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By Ellerly B. Golos

This ebook is an try to current, at an user-friendly point, an method of geometry according to Euclid, and in response to the trendy advancements in axiomatic arithmetic.

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Additional info for Foundations of Euclidean and Non-Euclidean Geometry

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3. It is possible for a theorem to be implied by one axiomatic system and its contradictory by another axiomatic system. 4. In an axiomatic system completeness is always desirable. 5. In an axiomatic system independence is always desirable. 6. An inconsistent axiomatic system might imply a statement and its contradictory. 7. If an axiomatic system is satisfiable its model must be finite. 8. The test for independence involves consistency. 9. Completeness, independence, and consistency-as determined by the tests-all depend on the concept of satisfiability.

If there exists a correspondence between two sets S1 and S2 such that every statement which 1s true when made about elements of S1 is also true when made about the corresponding elements of S2, the correspond­ ence is said to preserve relations. 4 Completeness 49 Definition. Two models of an axiomatic system are said to be isomorphic with respect to that system if there exists at least one one-one correspondence between the element~ of the system which preserves relations. This concept 1s important enough to cover in more detail, rather than digress now, however, w·e shall postpone it until the next section, ,vhere we shall attempt to clarify it with several illustrations.

By Axiom la every pair of points must have a line containing them. Point A now lies on a line with B, C, D, and E and hence must lie on a line 26 Finite Geometries with F; but this line must have at least one more point, and by Axiom lb it cannot be B, C, D, or E, so there must exist another point G such that AFG. Thus, there exist at, least seven points. Continuing in this manner, we have ABC, A DE, BDF, and AFG and at least seven points. By Axiom la the following possible lines must be considered: BE .