In set theory, a large cardinal is a type of cardinal number that possesses certain strong and often large-scale properties, which typically extend beyond the standard axioms of set theory (like Zermelo-Fraenkel set theory with the Axiom of Choice, ZFC). Large cardinals are significant in the study of the foundations of mathematics because they often have implications for the consistency and structure of set theory. There are various kinds of large cardinals, each with different defining properties.
Articles by others on the same topic
A large cardinal is an infinite cardinal number satisfying a strong additional property, such as inaccessibility, weak compactness, or carrying a highly complete measure. The phrase names a family of axioms rather than one numerical size threshold.
For example, a strongly inaccessible cardinal is an uncountable regular strong limit cardinal, whereas a measurable cardinal carries a nonprincipal highly complete ultrafilter. Their significance is structural: they support strong reflection and compactness principles and organize consistency strengths in set theory. Merely taking a larger successor cardinal does not ensure any of these properties.