1. Math
  2. Advanced Math
  3. 2 recall that a model of an axiom system is...

Question: 2 recall that a model of an axiom system is...

Question details

2. Recall that a model of an axiom system is a realization of the undefined terms and relations in some particular context such that the axioms are satisfied. In this set of exercises, students are required to prove that the real Cartesian plane R2 serves as a model for the geometry derived from the Hilberts Axioms I1 13 and B1 - B4 following the steps below a) We start with descriptions of undefined terms. The set of points is considered as the set R2 of odered pairs of real number. The set of lines is considered as the set of linear equations ax +by+ 0, where r, y, a, b,c E R and a, b are not both equal to 0 (1) Interpret Axioms 11. 12 and 13 using the descriptions of points and line in R2 (2) Verify that Axioms I1, I2 and 13 are true statements (b) We start with descriptions of undefined relations. For three dstinct real numbers a, b, c, recall that b is said to be between a and c if either a b<c orc<b< a Apply this idea to state the undefined relation, the betweenness A BC, for three distinct elements A-(a, as), B (h, b), C = (c1-2) of the set R2 as follows. Given A-(al , a2), B-(b1, b2), C-(c1-2 ) E R2, we say A * B * C if one of the following conditions is satisfie d. . ai = bı c1, and b2 lies between a2 and c2, that is, either a2 < b2 < c2 or a2 > b2 > C2 . bi lies between ai and c, that is, either ai < bi < q or ai > bi > C1, and (1) Interpret Axioms B1, B2, B3 and B4 (2) Verify that Axioms B1, B2, B3 and B4 are true statementsGroup I. AXIOMS OF INCIDENCE 11. There is a unique line m incident to any two distinct points A, B. In other words, for any 2. Every line is incident to (that is, contains) at least two points. 13. There exists (at least) three non-collinear points (that is, three distinct points not all two distinct points A, B, there exists a unique line m containing A, B contained in the same line).Group B. AXIOMS OF BETWEENNESS B1. If B is between A and C, (written A*B*C), then A, B, C are three distinct points on a line, and also C B*A. (We may also write A-B-C or C-B-A. B2. For any two distinct points A, B, there exists a point C such that A*B*C 8 B3. Given three distinct points A, B, C are 3 distinct points lying on the same line, then one and only one of them is between the other two. B4. (Pasch) Let A, B, C be three non-collinear points, and let m be a line not incident to any of A, B, C. If m is incident to a point D lying between A and B, then it must also contairn either a point lying between A and Cor a point lying between B and C, but not both.

Solution by an expert tutor
Blurred Solution
This question has been solved
Subscribe to see this solution