Set Theory

Set theory is an area of mathematics that focuses on collections of objects, called sets. While any object can be collected into sets, set theory is usually applied to objects with mathematical relevance.

Modern set theory was developed by Richard Dedekind and Georg Cantor in the 1870s. After the discovery of the naive set theory paradoxes, a number of axiom systems were proposed in the early twentieth century, most notably the Zermelo-Fraenkel axioms.

Set theory language can be used in the definitions of just about all mathematical objects, and set theory concepts are widely dispersed in various mathematics curricula at all levels. Simple facts about sets and set membership, including Venn diagrams, Euler diagrams, union, and intersection, can be introduced at the primary grade levels. More advanced concepts like cardinality are a normal component of undergraduate curricula.

Contemporary set theory research covers a wide range of topics, from real number line structure to large cardinal consistency.

A course in set theory will normally cover the following topics:

• Sets
• Relations, Functions, and Orderings
• Natural Numbers
• Finite, Countable, and Uncountable Sets
• Cardinal Numbers
• Ordinal Numbers
• Alephs
• The Axiom of Choice
• Arithmetic of Cardinal Numbers
• Sets of Real Numbers
• Filters and Ultrafilters
• Combinatorial Set Theory
• Large Cardinals
• The Axiom of Foundation
• The Axiomatic Set Theory

Students will be able to find plenty of books on set theory from Google Books and Amazon.com. For a tutorial, Javier R. Movellan's excellent set theory tutorial is a must. A good journal that students should follow is Elsevier's Fuzzy Sets and Systems, their International Journal in Information Science and Engineering. The Math Archives' Logic and Set Theory, made available by the University of Tennessee at Knoxville, offers a terrific compilation of references for this topic.

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