I am not a mathematician, but I study it for fun.  I know that Cantor showed that the infinite set of the real numbers is larger than the rationals since the real numbers are uncountable and the rationals are countable.  The real numbers include rational and irrational numbers and the irrational numbers are uncountable.  Is the set of irrational numbers smaller than the set of real numbers, the same size, or does this question not make sense?  Thank you!
I know and I have proved more than once that the set of irrational numbers ($\mathbb{I}$) is uncountable, but now I'm given to solve this problem:   Show that $|\mathbb{I}|=|\mathbb{R}|$,  How can I do that?  Do I need to assume the Continuum Hypothesis in order to that statement to be true?  Thanks
Can someone point me to a proof that the set of irrational numbers is uncountable? I know how to show that the set $\mathbb{Q}$ of rational numbers is countable, but how would you show that the irrationals are uncountable?