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.