Download Foundations of Euclidean and Non-Euclidean Geometry by Ellerly B. Golos PDF
By Ellerly B. Golos
This ebook is an try and current, at an straightforward point, an method of geometry according to Euclid, and in line with the fashionable advancements in axiomatic arithmetic.
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Extra info for Foundations of Euclidean and Non-Euclidean Geometry
Example text
Axio m Set 2 la. If P and Q are any two points, then there exists at least one line containing both P and Q. 1 b. If P and Q are any two points, then there exists at most one line containing both P and Q. 2. If Z is a line, then there exist at least three points on it. 3. If Z is a line, then there exists a point P not on it. 4. There exists at least one line. 5. If Z and m are any two lines, then there exists at least one point P belonging to both l and m. As is more apparent in this reworded version, only Axiom 4 IS an exist ence statement.
For the system {A 1, A2, ···An} and axioms. morlc,l,;;; for { A1, A 2, • • · not-An}, hence showing that An is independent in the system. If it is now further supposed that the system is categorical, these two models must then be isomorphic; hence, corresponding statements in the two systems are either both true or both false. But this is impossible by the assumption that "A n " is true in one and "not-An" is true in the other. This assumption must therefore be false; hence, if a system is categorical, it is complete.
Help prove some other theorem. Generally, it is called a lemma when its use is more or less restricted to the theorem in question; if it had other uses, one would usually call it a theorem and give it an appropriate number. 1. Q FIGURE 2. Jr,,,,nc, n r - - - Io -- Proof: Suppose l is a line containing exactly n points P1, P2, · · · , Pn . Since by Theorem 8 there exist other lines parallel to l, we may let m be any line parallel to l. Once again by Theorem 8 there exists still another line, parallel to both m and l, and by Axiom 2 it has a point Q on it, which by the definition of parallel cannot be on either m or l.