The first of Hilbert's twenty-three unsolved problems, FTP, name this doubtful set theorem that states that any uncountable set of real numbers has the same cardinality as the entire set of real numbers.
Related to the Baire category, Martin's axiom is a weaker version, but it was G?del that demonstrated that it was compatible with ZFC theory while Paul Cohen proved that it is undecidable in ZFC.  Originating with the diagonal argument of its main advocate, Georg Cantor, it can be stated in both "cardinal numbers" and "sets of reals" versions.  The first of Hilbert's twenty-three unsolved problems, FTP, name this doubtful set theorem that states that any uncountable set of real numbers has the same cardinality as the entire set of real numbers.
Related to the Baire category, Martin's axiom is a weaker version, but it was G?del
continuum hypothesis or CH