Strongly inaccessible cardinal

The existence of a strongly inaccessible cardinal is equivalent to the existence of a Grothendieck universe. If ${\displaystyle \kappa }$ is a strongly inaccessible cardinal then the von Neumann stage${\displaystyle V_{\kappa }}$ is a Grothendieck universe. Conversely, if ${\displaystyle U}$ is a Grothendieck universe then there is a strongly inaccessible cardinal ${\displaystyle \kappa }$ such that ${\displaystyle V_{\kappa }=U}$. As expected from their correspondence with strongly inaccessible cardinals, Grothendieck universes are very well-closed under set-theoretic operations.