 Just as \(\aleph_1\) is a cardinality that can’t be reached by the operations of ordinal addition, multiplication, or exponentiation, even in the limit, one might conjecture that even before one reaches the Cantorian absolute infinity, there are further cardinalities that can still be grasped in some sense, but can’t be reached even by the stronger operation of cardinal exponentiation, even in the limit. Such conjectures have turned out to be surprisingly fruitful for the study of set theory and mathematical logic generally. (See the entry on  large cardinals and determinacy.) 