I think they do have some ideas. You might look at Believing The Axioms (http://cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf http://cs.umd.edu/~gasarch/BLOGPA

I think they do have some ideas. You might look at Believing The Axioms (http://cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf http://cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf, but it's two parts) which studies set theorists who are investigating new axioms for set theory, especially large cardinal axioms, iirc. Honestly, that work is a ways beyond what I can understand, but it's clear that it's an active area of research. You might also read about Hugh Woodin has a very developed program arguing for axioms which would imply the negation of the Continuum Hypothesis. (This doesn't necessarily apply to this specific question, but in general, research into axiomatic foundations of math is ongoing).