In this respect, Martin's axiom put forward by D.A. Martin and others in 1970 is the best choice. It complements and develops with the forced method, and finally obtains a Martin maximum principle, which has a wide range of applications and is still under further study.
The most important problem in mathematical foundation is how to deal with the relationship between natural number and real number, that is, how to construct a continuum by discrete method. Since the ancient Greeks discovered this problem, it has not been completely solved. Using Dedeking division, we can't get all real numbers, but only all algebraic numbers at most, and most transcendental numbers can't be constructed.
Even in the case of unknowability, the intuitive mathematician Heiding once commented on Dydykin's division. This algorithm still doesn't provide us with any method to determine whether a rational number A is on the left or right of C or just equal to C, so we can't guarantee that Euler constant C is a real number, and nonstandard analysis also provides us with such an impression.
In fact, the points on a straight line are infinitely separable, which shows that we know very little about continuum. Mathematical induction is the basic method of mathematics, which is based on natural numbers. The negative proof of the continuum hypothesis just shows that people can approach the continuum infinitely in this way, but they will never reach its end.
As Pascal said, man is just a boat floating between infinity and nothingness. We always want to pursue some certainty, but we can never catch it. If we are not careful, our whole foundation will fall apart, and this is the bottomless abyss. For nature, we humans will always be explorers, not terminators.