Assignment:
Consider the following axiom system. The undefined terms are point, line, and on. Definition: Two lines are parallel if there is no point on both of them.
Axioms:
I. Given any two distinct points, there exactly one line on both of them.
II. Every line has at least one point on it.
III. Given any line, there is at least one other line that is parallel to it.
IV. There exists at least one line.
[The book defines each axiom an "undefined term" called a "interpretation of a system."]
[If, for a given interpretation of a system, all the axioms are "correct" statements, we call the interpretation a "MODEL"]
Questions: (a) Show that this axiom system is consistent by finding a model for it.
(b) Using models, show that Axiom III is independent of the other axioms.
(c) Show that this axiom system is not complete.
Provide complete and step by step solution for the question and show calculations and use formulas.