aleph 0

    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)