Large cardinal axioms

Large cardinal axioms are extensions of ZFC that assert the existence of cardinal numbers so large that their existence cannot be proved from ZFC alone. They form a hierarchy of increasing strength: a measurable cardinal implies the existence of a certain kind of ultrafilter, a supercompact cardinal implies the existence of inner models with rich structure, and each step up the ladder settles questions that lower rungs leave open. Large cardinal axioms are not mere mathematical curiosities; they are the primary tool for calibrating the consistency strength of mathematical theories. If a statement implies the existence of a large cardinal, and that cardinal is known to be consistent with ZFC, then the statement itself is consistent. The hierarchy of large cardinals is a map of the landscape beyond ZFC.

The large cardinal hierarchy is the closest thing mathematics has to a telescope aimed at infinity. Each stronger axiom sees further — and each is a bet that the universe of sets is larger than we have yet dared to imagine.