In the mathematical field of
set theory, a
large cardinal property is a certain kind of property of
transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least α such that α=ω
α). 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".