Are all the uncountable infinite always greater than countable infinite?

Question

I read on an article about there are more reals than rationals because reals are uncountable. Here are my questions:

Are there more complex numbers than real numbers? Are all the uncountable infinite always greater than countable infinite?

Thanks in advance.

Answer 1

Depending on your definition of "more".   Mostly we use Cantor's.

  There does not exist an injective function which maps real numbers one-to-one into rational numbers.   So we say $\Bbb R$ has greater cardinality than $\Bbb Q$.

  There does exist a bijective function which maps complex numbers onto real numbers.   So we say $\Bbb C$ has equal cardinality to $\Bbb R$.   They have the cardinality of the continuum.

Although , there do exist sets with greater cardinality than the continuum.   The powerset fir the reals, for instance.