The
cardinality of the first
infinite
ordinal,
omega (the number of natural numbers).
Aleph 1 is the cardinality of the smallest
ordinal whose
cardinality is greater than aleph 0, and so on up to aleph
omega and beyond. These are all kinds of
infinity.
The
Axiom of Choice (AC) implies that every set can be
well-ordered, so every
infinite cardinality is an aleph;
but in the absence of AC there may be sets that can't be
well-ordered (don't posses a
bijection with any
ordinal)
and therefore have cardinality which is not an aleph.
These sets don't in some way sit between two alephs; they just
float around in an annoying way, and can't be compared to the
alephs at all. No
ordinal possesses a
surjection onto
such a set, but it doesn't surject onto any sufficiently large
ordinal either.