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aleph 0

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

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

alephs at all. No ordinal possesses a surjection onto

such a set, but it doesn't surject onto any sufficiently large

ordinal either.