Dictionary.net

The cardinality of the first infiniteordinal, omega (the number of natural numbers).Aleph 1 is the cardinality of the smallest ordinal whosecardinality is greater than aleph 0, and so on up to alephomega and beyond. These are all kinds of infinity.The Axiom of Choice (AC) implies that every set can bewell-ordered, so every infinite cardinality is an aleph;but in the absence of AC there may be sets that can't bewell-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 justfloat around in an annoying way, and can't be compared to thealephs at all. No ordinal possesses a surjection ontosuch a set, but it doesn't surject onto any sufficiently largeordinal either.