This article explains why the cardinality of a set must be either Aleph0 or |ℝ|.

1. Rational numbers are discrete
2. Real numbers are continuous
3. Collectively exhaustive and mutually exclusive events
4. Continuum hypothesis
5. Cardinality of discontinuous subsets of real numbers

Please read the article at
PDF Continuous set and continuum hypothesis
http://pengkuanonmaths.blogspot.com/2015/12/continuous-set-and-continuum-hypothesis.html
or
Word https://www.academia.edu/19589645/Continuous_set_and_continuum_hypothesis