Axiomatic Set Theory
Although initially naive set theory, which defines a set merely as any well-defined collection, was well accepted, it soon ran into several obstacles. It was found that this definition spawned several paradoxes, most notably:
- Russell's paradox—It shows that the "set of all sets which do not contain themselves," i.e. the "set" { x : x is a set and x ∉ x } does not exist.
- Cantor's paradox—It shows that "the set of all sets" cannot exist.
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.
For most purposes however, naive set theory is still useful.
Read more about this topic: Set (mathematics)
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