does not hold, but the axiom of choice (AC) does, then , as the smallest uncountable cardinal, is strictly less than . If AC also does not hold then may be incomparable with , but never larger than .

The existence of does not depend on AC, as it can be constructed explicitly as the Hartogs number of . More concretely, the set of all well-orderings on can be constructed as a subset of all binary relations on , and