The twentieth-century choice for the axioms of set theory are the Zermelo-Frankel axioms together with the axiom of choice; these are the ZFC axioms. This particular choice has led to a twenty-first-century problem:

Perhaps the most famous example is given by the problem of the continuum hypothesis: suppose X is an infinite set of real numbers; must it be the case that either X is countable or that the set X has cardinality equal to the cardinality of the set of all real numbers?

One interpretation of this development is as follows:

Here and in what follows, the “Skeptic” simply refers to the metamathematical position that denies any genuine meaning to a conception of uncountable sets. The counterview is that of the “Set Theorist”:

Elaborating further, as a consequence of this calibration, it has been discovered that in many cases, very different lines of investigation have led to problems whose degree of unsolvability is the same.