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.
but in the absence of AC there may be sets that can't be
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
such a set, but it doesn't surject onto any sufficiently large
ordinal either.
(1995-03-29)