You can simply dream up whatever axioms, undefined terms, and rules of logic you want. However, one runs the risk of having an inconsistent system or a system that is not interesting to others. Godel's Incompleteness Theorem does not say that this can't be done. Furthermore the "underlying contraints" imposed by Godel's Incompleteness Theorem is not at all what most mathematicians study. Unless I'm misinterpreting your meaning here.

There are knowledgeable people who do not believe that mathematics is independent of our minds. It's not too far fetched of an idea. While I do not personally agree with this, I won't downplay such beliefs.