Large cardinal

Summary

A large cardinal is an infinite cardinal number with a property that makes it very large. If ZFC is consistent, ZFC cannot prove that such cardinals exist. Examples are inaccessible, Mahlo, weakly compact, measurable, Woodin and supercompact cardinals. There is no precise definition of the term; it names a family of axioms that is, as far as is known, ordered by consistency strength. Large cardinal axioms measure the strength of other statements, and they decide many questions about sets of reals, such as projective determinacy.