Large Cardinal

Large cardinal axioms are extension principles in set theory that assert the existence of cardinal numbers with extraordinary structural properties — cardinals so large that their existence cannot be proved from the standard axioms of ZFC, and whose consistency strength towers above that of ZFC alone. Examples include inaccessible cardinals, measurable cardinals, and Woodin cardinals, each imposing stronger closure conditions on the cumulative hierarchy than the last.

The hierarchy of large cardinals forms a linear scale of consistency strength that has become the standard yardstick for measuring the power of mathematical theories. A remarkable empirical fact — the large