The pieces of mathematics that you are focussing on (Choice, Goedel, etc) are of little importance in the practice of science, and are highly arbitrary.

For instance there is no absolute truth upon which one can say that the Axiom of Choice should be included in our math, and it has been proven that any result that you can prove about integers using ZFC can be proven without choice as well. Since integers can model computing which can model all of our scientific results, right there is a proof that Choice doesn't matter to science.

Math is a fun game. The application of mathematical techniques to science is not a game, and as Ross says, Anyone who as struggled through understanding it will understand (that).

Cheers,
Ben