This is what mathematicians like to argue about. If you ascribe to the philosophy that the Continuum Hypothesis is false, you can use a technique called forcing, invented by Paul Cohen. You can essentially create a forcing axiom which will make the hypothesis false.
Alternatively, there is a possible axiom called Martin's axiom, proposed in 1970. If you want the Continuum Hypothesis to be true, then you need an inner model axiom. One possibility is called Ultimate-L. W. Hugh Woodin, a professor at Harvard University who is studying the Continuum Hypothesis, originally thought that the Continuum Hypothesis was false, but has since changed his mind and maintains that if the Ultimate-L conjecture is true, not only will it resolve not only the Continuum Hypothesis, but every other problem that has been shown unsolvable using Cohen's methods.
But in the end, the Continuum Hypothesis is the end of the road. Everything from here on out is off-road. We are literally inventing new math – perhaps new math where formal proof is not all-important. And it is the trail-blazing nature of mathematics that requires a great mathemagician to be creative and daring as well as brilliant.
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