Inaccessible Cardinal

An inaccessible cardinal is an infinite cardinal number that cannot be reached, or accessed, from smaller cardinal numbers by the two basic operations set theory otherwise uses to build larger cardinals, taking a union of a smaller collection of smaller cardinals or taking the power set of a smaller cardinal, and which is also uncountable and greater than the smallest infinite cardinal. Weakly inaccessible cardinals, defined using only the regularity condition together with being a limit cardinal, were first studied by Felix Hausdorff in the early twentieth century, and the stronger strongly inaccessible version, which also requires closure under the power set operation, was introduced somewhat later by Waclaw Sierpinski and Alfred Tarski, among others. The existence of an inaccessible cardinal cannot be proved from the standard Zermelo-Fraenkel axioms of set theory, even together with the axiom of choice, since a model containing an inaccessible cardinal can be used to build a model of set theory itself, so its existence is typically treated as a large cardinal axiom, an additional hypothesis considered separately from the standard axioms.