Inaccessible cardinal

Summary

An inaccessible cardinal is an uncountable cardinal $\kappa$ that cannot be reached from smaller cardinals. Power sets of smaller sets, and unions of fewer than $\kappa$ smaller sets, stay smaller than $\kappa$. It is the smallest kind of large cardinal. If $\kappa$ is inaccessible, the sets of rank less than $\kappa$ form a model of ZFC. So ZFC cannot prove that inaccessible cardinals exist, if ZFC is consistent. In category theory, inaccessible cardinals give Grothendieck universes.