<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
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.
ALEF « ALEPH « Aleph « aleph 0 » alert » Alex