< mathematics > 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)