<Li> Cantor's paradox--It shows that "the set of all sets" cannot exist . </Li> <P> The reason is that the phrase well - defined is not very well defined . It was important to free set theory of these paradoxes because nearly all of mathematics was being redefined in terms of set theory . In an attempt to avoid these paradoxes, set theory was axiomatized based on first - order logic, and thus axiomatic set theory was born . </P> <P> For most purposes however, naive set theory is still useful . </P> <P> This principle can be used to find the cardinality of the union of sets . </P>

Three ways to describe a set in math