I'm sorry for this uneducated question, but I've been thinking of this for a few hours and I couldn't find anything on the topic. Perhaps it is just a failure on my part and a limitation of my experience, but assuming not:
Could we take something like Goldbach's conjecture as an axiom, and go from there to prove new theorems? Does this ever happen, or do mathematicians build up structures from both assumptions (one structure assumes the conjecture is true, and one false, and then go on to build up theorems and additional structures from there).
I use Goldbach's conjecture as an example because it has a lot of "evidence" (not really but maybe empirically, even though math has nothing to do with empiricism, I'm just using Goldbach but we could use any conjecture) to err on the side of it being true, so perhaps assuming it's true, then going on to use it to prove other things, may result in some pretty results. If it turns out that the conjecture is false, we just throw out the results and the system. But assuming a system that includes "Goldbachs conjecture is true" as an axiom is consistent, then why wouldn't we be able to use the system? Do mathematicians ever do this, or is consider bad form (I can see why this would be completely banned).
Basically, could I say "Let us assume Goldbach's conjecture is true. Then..." and go one to build up books of math and results and theorems using this?