Seonglae ChoDefinition: There is no set whose cardinality is strictly between that of the natural numbers and the real numbers.
This is an important problem in set theory that has had a significant impact on the foundations of modern mathematics. The result is that it has been proven to be neither provable nor disprovable within the ZFC axiom system.
Some things are too large to be sets. In other words, some concepts are too large to be elements of a set. Therefore, Russell's paradox is treated as a forbidden definition accordingly. Thus, sets cannot be defined arbitrarily but must be well-defined. This simultaneously means that non-symbolized language cannot be the basis for mathematical definitions.
