Source : Free On-Line Dictionary of Computing

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)