I would like to see other examples of (ideas of) proofs and results in the same vein as the proof of the Ax-Grothendieck theorem. To explain what I mean by "in the same vein", I will quote from the ...
The collection of prime powers can be characterized in the following way: There is a field with $q$ elements if and only if $q$ is a prime power. Furthermore if it exists then this field is unique ...
I read on Ax's article that the elementary theory of finite fields is decidable if one assumes the continuum hypothesis to be true. What about if one assumes the hypothesis to be false?