Difference between revisions of "Set theory"
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'''Set theory''' is a branch of [[mathematics]] dealing with collections of objects, called [[set]]s. It revolutionized mathematics and made possible enormous new insights. | '''Set theory''' is a branch of [[mathematics]] dealing with collections of objects, called [[set]]s. It revolutionized mathematics and made possible enormous new insights. | ||
| − | *The language of set theory is based on a single fundamental relation, called membership. We say that | + | *The language of set theory is based on a single fundamental relation, called membership. We say that <math>x</math> is a member of <math>A</math> (in symbols <math>x \in A</math>), or that the set <math>A</math> contains <math>x</math> as an element.<ref>The mathematician Giuseppe Peano created the symbol ∈ in 1889 to mean "is an element of," from the Greek epsilon that is the first letter of εἰμί, which means "I am."</ref> 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> or, equivalently, if each is a [[subset]] of the other. |
==History of set theory== | ==History of set theory== | ||
Revision as of 02:56, July 20, 2014
Set theory is a branch of mathematics dealing with collections of objects, called sets. It revolutionized mathematics and made possible enormous new insights.
- The language of set theory is based on a single fundamental relation, called membership. We say that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is a member of (in symbols Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in A} ), or that the set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} contains Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} as an element.[1] 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,[2] or, equivalently, if each is a subset of the other.
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. The most commonly used such axiomatization is Zermelo-Fraenkel set theory.
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.
The Empty Set
The empty set is the set with no members. Because sets are uniquely defined by membership, the empty set is unique. The empty set is usually denoted by {} or ∅.
References
- ↑ The mathematician Giuseppe Peano created the symbol ∈ in 1889 to mean "is an element of," from the Greek epsilon that is the first letter of εἰμί, which means "I am."
- ↑ http://plato.stanford.edu/entries/set-theory/