Use the axiomatic system of Example 1

a. What is the smallest positive number of points on a model of the system? Repeat for lines and explain your answers. Find a model with the minimum number of points and lines.

b. Find a model with six points not isomorphic to any models presented in the text. Explain why it is not isomorphic to any of them.

c. Two nonisomorphic models with nine points. Explain why they are not isomorphic.

d. Find and prove a theorem of this system.  

terms point, line, and on and the axioms Example I. Figures 2.6 and 2.7 give different models of the axiomatic system with the undefined 23 Models and Metamathematics 83 (1) Every point is on exactly two lines. Gi) Every line is on exactly three points. tese and later models on has the obvious interpretation. While Figure 2.6 interprets the edefined terms in conventional ways, the choice of undefined terms in Figure 2.7 may seem disconcerting, but our only requirement is that the axioms are true in the model with the interpretation we give. And indeed, each edge is on two faces and each face is on three edges. ) d. b C C Figure 2.6 Interpret point by a dot, line by a line segment. Figure 2.7 Interpret point by an edge, line by a face of the pyramid.