"The central result of logicians in the 20th century was that, in the end, it will always be necessary to extend your axioms—things you just assume to be true without proving them—if you are to extend your idea of truth."
I'm in total agreement with this.
On a philosophical note, I even hold that it's actually possible to create a sound mathematical formalism that is based directly on this principle.
I personally hold that our current mathematical formalism is fundamentally "wrong". And this mainly has to do with Cantor's "empty set" theory.
IMHO, that was a very wrong turn for mathematics to take.
Henri Poincare, once stated, "Cantor's Set Theory is a disease from which the mathematical community will someday recover from".
I agree with Poincare. Cantor's Set Theory based on an empty set is a bad idea, and we should have never gone down that road. It's an unnecessary path to take. In fact, there are far better paths to take. We will not advance properly in our mathematics until we go back and correct this wrong turn.