In model theory there is a notion of Grothendieck ring defined here http://math.berkeley.edu/~scanlon/papers/greu12jun00.pdf. Do we know anything about the cardinality of these rings ? Can they be finite ?
If a structure M had a Grothendieck ring of cardinality say 2 this means that:
there is a definable function on $M$ injective and whose image is $M$ minus two points.
There is a definable function between $M$ and $M^2$.
(3. $M$ is infinite)
If we take a structure satisfying this, then its Grothendieck ring is either of cardinality 2 or trivial.
So in order to prove that there exists Grothendieck ring of cardinality say 2, we have to prove that there exists such a $M$ whose the Grothendieck ring is non trivial which is equivalent to the fact that there is no definable set in bijection with itself minus a point.
Could you have any suggestion of how to see that ?
Thank you in advance !