[1] : The Free On-line Dictionary of Computing (27 SEP 03)

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
     
     
see also:
cardinality infinite ordinal omega natural numbers infinity 
axiom of choice well-ordered bijection surjection 

Aleph >> A Words
Aleph, definition of term: Aleph
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