I have some other examples: tha cardinality of the family of dense non denumerable subsets of $\mathbb{R}^{n}$ (n=1,2...) is $2^{2^{\aleph_{0}}}$, the dimension of $\mathbb{R}$ like a $\mathbb{Q}$-vector space is $2^{2^{\aleph_{0}}}$, the cardinalitiy of the family of lebesgue non mesurable (and mesurables, mesurables with measure r (r a given real number), with infinite measure) sets is $2^{2^{\aleph_{0}}}$, the cardinality of the family of riemann integrable functions (in $\mathbb{R}^{n}$, n=1,2...)is $2^{2^{\aleph_{0}}}$, and so on...

I have some other examples: the cardinality of the family of dense non denumerable subsets of $\mathbb{R}^{n}$ ($n=1,2...$) is $2^{2^{\aleph_{0}}}$, the dimension of $\mathbb{R}$ like a $\mathbb{Q}$-vector space is $2^{2^{\aleph_{0}}}$, the cardinality of the family of Lebesgue non mesurable (and mesurables, mesurables with measure r (r a given real number), with infinite measure) sets is $2^{2^{\aleph_{0}}}$, the cardinality of the family of riemann integrable functions (in $\mathbb{R}^{n}$, n=1,2...)is $2^{2^{\aleph_{0}}}$, and so on...