An uncountable cardinal $\kappa$ is strongly inaccessible if it is regular ($\operatorname{cf}(\kappa) = \kappa$) and a strong limit ($2^\lambda < \kappa$ for all $\lambda < \kappa$). If $\kappa$ is inaccessible, then $V_\kappa \models \text{ZFC}$, so the existence of an inaccessible cardinal implies $\mathrm{Con}(\text{ZFC})$.

A cardinal $\kappa$ is Mahlo if it is inaccessible and the set of inaccessible cardinals below $\kappa$ is stationary in $\kappa$. Mahlo cardinals are strictly stronger than inaccessible cardinals in consistency strength.

An uncountable cardinal $\kappa$ is measurable if there exists a non-principal $\kappa$-complete ultrafilter on $\kappa$. Equivalently, there is a non-trivial elementary embedding $j: V \to M$ with critical point $\kappa$ (the least ordinal moved by $j$).

A cardinal $\delta$ is Woodin if for every function $f: \delta \to \delta$, there exists a cardinal $\kappa < \delta$ with $f[\kappa] \subseteq \kappa$ and an elementary embedding $j: V \to M$ with critical point $\kappa$ such that $V_{j(f)(\kappa)} \subseteq M$. Woodin cardinals play a central role in the theory of determinacy and descriptive set theory.