You are using

in an unconventional manner

usually denote the cardinality of the naturals and

that of the continuum (the reals). The continuum hypothesis is that

(that is that there is no set with cadinality between that of naturals and that of the reals).

The algebraic numbers are in fact countable and so have cardinality

.

The trancendentals have cardinality

as they are those reals which are not algebraic.

The set of all functions on the reals is equivalent to the power set of the reals and we know that the cardinality of the power set of a set is strictly larger than that of the set itself.

CB