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In a response I read, someone stated that any uncountable subset if the real numbers had the cardinality of the real numbers. Is this true and if so where can I find a reference to that result?

Howard Cary Morris
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    It's called the continuum hypothesis. – Robert Israel Dec 29 '23 at 15:54
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    This statement is equivalent to Cantor's famous continuum hypothesis.

    But, as proved in steps by Kurt Gödel and Paul Cohen, it is impossible to either prove or disprove this statement using the axioms of ZFC set theory: starting from any model of ZFC set theory, one can construct another model of ZFC set theory in which this statement is true (Gödel), and still another one in which it is false (Cohen).

     – Lee Mosher Dec 29 '23 at 15:55
  • For other links on this site with more information: https://math.stackexchange.com/questions/754268/question-about-the-continuum-hypothesis, https://math.stackexchange.com/questions/2315178/the-continuum-hypothesis – Lee Mosher Dec 29 '23 at 16:03

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This is the famous continuum hypothesis. Famous results by Cohen and Gödel imply that it is impossible to prove/disprove that there exists a cardinal between $\aleph_0$ and $2^{\aleph_0}$ within the framework of ZFC axioms.
There is also a well-known result that any uncountable Borel subset of $\mathbb{R}$ same cardinality as that of continuum. Maybe this was the result they had in mind.

Ajin Shaji Jose
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