Difference between revisions of "Set theory"
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| − | Set theory a branch of mathematics dealing with collections of objects. | + | '''Set theory''' a branch of [[mathematics]] dealing with collections of objects. |
| − | *The language of set theory is based on a single fundamental relation, called membership. We say that A is a member of B (in symbols A ∈ B), or that the set B contains A as its element. The understanding is that a set is determined by its elements; in other words, two sets are deemed equal if they have exactly the same elements. | + | *The language of set theory is based on a single fundamental relation, called membership. We say that A is a member of B (in symbols A ∈ B), or that the set B contains A as its element. The understanding is that a set is determined by its elements; in other words, two sets are deemed equal if they have exactly the same elements. <ref>http://plato.stanford.edu/entries/set-theory/</ref> |
==History of set theory== | ==History of set theory== | ||
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One paradox in naive set theory was announced by [[Bertrand Russell]] in 1901, and is known as [[Russell's Paradox]]. | One paradox in naive set theory was announced by [[Bertrand Russell]] in 1901, and is known as [[Russell's Paradox]]. | ||
| − | Like all sufficiently strong mathematical theories, set theory is incomplete, as shown by [[Kurt Godel]]. However, set theory is the received axiomatization of mathematics today, with subjects like analysis, algebra, topology, and geometry using set theory and its language for their own foundation. | + | Like all sufficiently strong mathematical theories, set theory is incomplete, as shown by [[Kurt Godel]]. However, set theory is the received axiomatization of mathematics today, with subjects like analysis, [[algebra]], topology, and [[geometry]] using set theory and its language for their own foundation. |
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| + | ==References== | ||
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| + | <References/> | ||
[[Category: set theory]] | [[Category: set theory]] | ||
Revision as of 10:04, December 29, 2007
Set theory a branch of mathematics dealing with collections of objects.
- The language of set theory is based on a single fundamental relation, called membership. We say that A is a member of B (in symbols A ∈ B), or that the set B contains A as its element. The understanding is that a set is determined by its elements; in other words, two sets are deemed equal if they have exactly the same elements. [1]
History of set theory
It was developed in the late 1800s, primarly by the German mathematician Georg Cantor. This initial attempt became known as "naive set theory" because mathematicians found flaws in it. It was replaced by "axiomatic set theory" in the early 1900s.
One paradox in naive set theory was announced by Bertrand Russell in 1901, and is known as Russell's Paradox.
Like all sufficiently strong mathematical theories, set theory is incomplete, as shown by Kurt Godel. However, set theory is the received axiomatization of mathematics today, with subjects like analysis, algebra, topology, and geometry using set theory and its language for their own foundation.
