Is there a limit to building large cardinals in set theory?

If one could define a cardinality for classes - wouldn't this, in some sense be a 'limit' of all cardinals in Set... Sure, if you modify the definitions to allow this then the cardinalilty of any proper class would be considered the largest cardinal. read more

Many examples of large cardinal axioms take properties of the class of cardinals, e.g. infinitude, regularity (closure under limits of short sequences,) and closure under power sets, and posit that some (set-sized) cardinal already has these property. read more

With increasing frequency, forcing, large cardinals, axioms of determinacy, descriptive set theory, infinitary combinatorics, etc. find applications in the resolution of problems that had been open for many years in areas such as Topology, Algebra, Real and Complex Analysis, Functional Analysis, Measure Theory, etc. read more

In set theory, there are many kinds of large cardinal axioms. Their existence cannot be proved in Zermelo–Fraenkel set theory. But for many large cardinal properties, if we don't need them be uncountable infinitely, "they are "reflect" to smaller cardinals" [unclear what this means]. read more