We can see that we step directly from $\aleph_0$ to c. Is there anything in between? See the page about the Continuum Hypothesis.
The sizes of various infinite sets are represented by transfinite numbers first developed by Georg Cantor.

$\aleph_0$ , c , $2^c$ , $2^{2^c}$ , $2^{2^{2^c}}$ ...

CantorSet
ContinuumHypothesis
RationalNumber
ZornsLemma
(none) (none)
CountableSet
GeorgCantor
UncountableSet
(none)

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TransfiniteNumbers
ContinuumHypothesis
CantorSet
Denominator
FamousPeople
Numerator
SquareNumber
Axiom
CountingNumber
Integer
NaturalNumber
RationalNumber
RealNumber
(none)