Topic summary
Large cardinal
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinitecardinal numbers. Cardinals with such properties are, as the name suggests, generally very large, such that the cardinal is inaccesible through standard set theory functions, power sets or successors of smaller cardinals. (for example, bigger than the least α such that α = ωα). The proposition that such cardinals exist cannot be proved in the most common axiomatization of set theory, namely ZFC, and such propositions can be viewed as ways of measuring how "much", beyond ZFC, one needs to assume to be able to prove certain desired results. In other words, they can be seen, in Dana Scott's phrase, as quantifying the fact "that if you want more you have to assume more".
A large cardinal axiom is an axiom stating that there exists a cardinal (or perhaps many of them) with some specified large cardinal property.
Most working set theorists believe that the large cardinal axioms that are currently being considered are consistent with ZFC. These axioms are strong enough to imply the consistency of ZFC. This has the consequence (via Gödel's second incompleteness theorem) that their consistency with ZFC cannot be proven in ZFC (assuming ZFC is consistent).
There is no generally agreed-upon definition of what a large cardinal property is. Rather, "Set theorists often adopt the existence of certain large cardinals as axioms in the foundation of mathematics." For impredicative treatments, properties in the list of large cardinal properties are large cardinal properties (see Motivations and epistemic status for further discussion).