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|Dave Rusin; The Mathematical Atlas|
|A short article designed to provide an introduction to set theory. Naive set theory considers elementary properties of the union and intersection operators - Venn diagrams, the DeMorgan laws, elementary counting techniques such as the inclusion-exclusion principle, partially ordered sets, and so on. This is perhaps as much of set theory as the typical mathematician uses. Indeed, one may "construct" the natural numbers, real numbers, and so on in this framework. However, situations such as Russell's paradox show that some care must be taken to define what, precisely, is a set. Axiomatic Set Theory studies the axioms used to describe sets. While alternatives have been proposed (for example the von Neumann-Bernays-Godel and Morse-Kelley formulations), most sets of axioms for Set Theory include the Zermelo-Frankl axioms (ZF). Again, within this formulation one may define the natural numbers, the real numbers, and so on, and thus in principle carry out most ordinary mathematics. This formulation is rich enough to prove, for example, the Schoeder-Bernstein theorem (If X and Y are isomorphic to subsets of each other, then they are isomorphic). History; applications and related fields and subfields; textbooks, reference works, and tutorials; software and tables; other web sites with this focus.|
|Math Topics:||Axiomatic Systems, Set Theory|
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