Consider the axiomatic system with the undefined terms point , line , and on and
ID: 3282067 • Letter: C
Question
Consider the axiomatic system with the undefined terms point, line, and on and the following axioms.
(i) There are a line and a point not on the line.
(ii) Every two distinct points have a unique line on them both.
(iii) Every two distinct lines have at least one point on them both.
(iv) Every line has at least three points on it.
(a) Given two distinct lines prove that they have exactly one point on them.
(b) Prove that there are at least seven points.
(c) Given a point prove that it has at least three lines on it. Hint: First consider a
point not on the line of axiom (i).
(d) Prove that there are at least seven lines.
Explanation / Answer
No, it wouldn't. This set of axioms is for a general geometry- don't restrict your model to a circular space.
Points a,b,c are supposed to be on the first line which is guaranteed to exist.
Circular models are commonly used to illustrate how you can form a consistent model when you *don't* include some axioms... so even though you see circular models in class, don't try to limit yourself to them. Your model should just be the whole infinite plane.