Following Venema’s book, we examine a “toy” collection of incidence axioms: Our primitive (undefined) terms are point, line, and the relation “to lie on” (between a point and a line)
- For every pair of distinct points and there exists exactly one line such that both and lie on .
- For every line there exist at least two distinct points and such that both and lie on .
- There exist three points that do not all lie on any one line.
Axiom 3 gives us in particular that there are at least three points. However, as shown in lecture (or see example 2.2.2 in Venema’s book), we cannot prove from these axioms that there are more than three, because there is a model of these axioms with precisely three points, and three lines, each line containing exactly two of the points.
For a fixed positive integer , replace axiom 2 with axiom :
For every line there exist at least distinct points, all of which lie on .
(In particular, axiom is the same as axiom 2.)
Now consider the theory consisting of axioms 1, , and 3. The comment above indicates that the smallest possible number of points that a model of can have is three.
- Try to find the smallest possible number of points that a model of can have. That is, describe a model of with as few points as you can manage. (To show that the number, say , you find is optimal, that is, it cannot be reduced, one would need to prove from the axioms of the theorem that says that there exist at least points. That would be great, but I am not requiring that. The number you identify does not need to be optimal, but try to make it as small as possible. Of course, for all we know at this point, it could well be that any model of is infinite.)
- Do the same for and .
- If possible, can you say something in general about the number of points of the smallest model of (as a function of )?
The implicit suggestion here is to play with these theories, trying to understand what can and cannot be deduced from them. Ryan suggested to look at a variant of axiom 3, namely: Given any points, there is another point such that the resulting points do not all lie on the same line. Is this a theorem of ? Are there other interesting variants we can consider?
Feel free to include in your homework any results you find about these theories or their models, even if not directly related to the three questions above.
(In principle, this set is due February 2 at the beginning of lecture, but if needed, I am fine with extending the deadline so you have time to further explore the axioms. Let me know.)