aleph 0
<mathematics> 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.
(1995-03-29)