Consider the following axiom set, in which x's, y's, and "on" are the undefined terms:
Axiom 1. There exist exactly five x's.
Axiom 2. Any two distinct x's have exactly one y on both of them.
Axiom 3. Each y is on exactly two x's.
How many y's are there in the system? Prove your result.
Solution (so far):
I think, there the number of y's is undefined because Axiom 2 and Axiom 3 contradict each other.
I should point out that precisely because Plato was able to produce a model that satisfies the axioms (you can easily see that the model works by seeing the edges as y's), the axioms are consistent.
Alternatively we can find a bijection between {1,...,10} and the 2-subsets of {1,...,5} which is what prompted my first response.
In other words, label the x's and call the set X = {1,2,3,4,5}
List the 2-subsets lexicographically as usual
y1 = {1,2}
y2 = {1,3}
...
y10 = {4,5}
See that this satisfies the axioms and that it is unique up to labeling.
Plato's model looks nicer though.