Singular cardinal ($\mathrm{cf}(\kappa )<\kappa$)

An infinite cardinal number $\kappa$ is singular when $\mathrm{cf}(\kappa )<\kappa$. Thus a short cofinal function reaches arbitrarily high ordinals below $\kappa$. For example, ${\aleph }_{\omega }$ has cofinality $\omega$. The complementary case is a regular cardinal.

Write ${\sigma }_{\alpha }$ for the $\alpha$th uncountable singular cardinal in increasing order. It begins with ${\sigma }_0={\aleph }_{\omega }$ and ${\sigma }_1={\aleph }_{\omega +\omega }$. At a nonzero limit ordinal index it is continuous exactly when the supremum of its earlier values is singular. It can jump when that supremum is a weakly inaccessible cardinal. The first member of uncountable cofinality occurs at index ${\omega }_1$, with value ${\aleph }_{{\omega }_1}$.

There is an uncountable cardinal number $\kappa$ with ${\sigma }_{\kappa }=\kappa$. Iterate ${\kappa }_{n+1}={\sigma }_{{\kappa }_n}$ starting with ${\aleph }_0$. If there is no earlier fixed point, the increasing supremum $\kappa$ has countable cofinality, hence is a singular cardinal. The singular cardinal enumeration is cofinal in $\kappa$ below index $\kappa$, and continuity at this singular supremum gives equality.

The singular cardinals hypothesis asserts that every infinite singular cardinal $\kappa$ satisfiesEquivalently, $\gimel (\kappa )=\max \{{\kappa }^{+},2^{\mathrm{cf}(\kappa )}\}$ for every infinite singular cardinal. Here $\gimel$ is the gimel function. The Generalized continuum hypothesis implies this principle, but the principle only constrains the indicated singular-cardinal exponentiation.