Definition
Aleph
The aleph numbers
enumerate the infinite cardinalities of well-orderable sets in increasing order: The symbol
is the Hebrew letter aleph. The index is an ordinal: it specifies a position in this hierarchy, which continues beyond the finite indices.
(aleph-null) is the smallest infinite cardinality, shared by all countably infinite sets. is the smallest uncountable cardinality. is the smallest cardinality strictly greater than . Assuming the axiom of choice, every infinite set has cardinality
for some ordinal .
Comparing with Aleph-Null
Thus an at most countable set need not have cardinality