In reading https://mathoverflow.net/questions/1924/what-are-some-reasonable-sounding-statements-that-are-independent-of-zfc?page=2&tab=votes#tab-top I began to wonder: if our axioms don’t describe everything then why don’t we just add new ones?
Why are we no longer looking for axioms that could help us solve polynomials degree 5?
Why have we (mostly) given up on proving the continuum hypothesis only because it is independent of our axioms?
I’m sure there are many other problems like these. I’m puzzled because a solution to them would be extremely useful, and the purpose of mathematics is ultimately to solve problems (unless you are a game formalist, of course ;)).
Addition I know no axioms are complete, but certainly some systems are better than others.
This is a soft question.