The alephs enumerate the infinite initial ordinals: , successor indices select the next initial ordinal, and limit indices select the least initial ordinal above all earlier values.
Assuming the axiom of choice, the well-ordering theorem makes every set equipotent to an initial ordinal. The infinite initial ordinals are exactly the , so every infinite cardinal is for a unique ordinal .
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Aleph numbers are a family of cardinal numbers used to represent the sizes of infinite sets in set theory. The first Aleph number, denoted as \( \aleph_0 \) (aleph-null or aleph-zero), represents the cardinality of the set of natural numbers, which is the smallest infinite cardinal number.