<mathematics> The cardinality
of the first infinite ordinal
(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.
possesses a surjection
onto such a set, but it doesn't surject onto any sufficiently large ordinal either.