A question about a theorem derived from a given set of postulates

Question

Currently, I am reading the book 'Godel's proof' by Ernest Nagel and James Newman, with the forward by Douglas Hofstadter. In that book, on page 15, the authors give an example of an axiomatic system with the following postulates.

Let K & L be two classes i.e. collections of distinguishable objects each.

1. Any two members of K are contained in just one member of L.
2. No member of K is contained in more than two members of L.
3. The members of K are not all contained in a single member of L.
4. Any two members of L contain just one member of K.
5. No member of L contains more than two members of K.

The book says that from the customary rules of inference, it can be proved using the above axioms that K has only 3 elements and then gives examples of 'real-life' sets of objects satisfying the above as follows.

K = set of vertices of a triangle, L = set of lines of a triangle.

So far so good. But does K have to contain only 3 elements? from the postulate list, I think that the classes

K = set of vertices of any n-gon L = set of lines of that n-gon

equally satisfy the above. is this correct or am I missing something very obvious here? thanks in advance.

Answer 1

Your example for $n>3$ fails axiom 1.