Inaccessible and Mahlo cardinals

Work in ZFC; The Axiom of Choice is the ambient axiom used for arbitrary cardinality comparisons. Cardinals are initial ordinals as in Cardinal (initial ordinal) and cardinality, regular means $\mathrm{cf}(\kappa )=\kappa$ as in Cofinality $\mathrm{cf}(\alpha )$, and regular and singular cardinals, and $2^{\mu }$ is the cardinality of the power set in Cardinal sum $\kappa \oplus \lambda$, product $\kappa \otimes \lambda$ and exponentiation ${\kappa }^{\lambda }$, and why they are written apart from the ordinal operations.

An inaccessible cardinal is an uncountable regular strong limit cardinal: $2^{\mu }<\kappa$ for every cardinal $\mu <\kappa$. A weakly inaccessible cardinal is an uncountable regular limit cardinal. Strong limit and limit cardinal are different conditions.

For a regular uncountable kappa, a subset S is stationary if it meets every club subset of kappa, using Closed unbounded subsets of ordinals. An inaccessible kappa is Mahlo if $\{\alpha <\kappa :\alpha \text{ is an uncountable regular cardinal}\}$ is stationary. None of these definitions asserts existence.