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1 definition found
 for aleph 0
From The Free On-line Dictionary of Computing (18 March 2015) :

  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.

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