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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 approach of set theory also offers a powerful way to think about life, handle [[anxiety]], escape [[addiction]], and understand the miracles in the [[New Testament]]. Problems, which otherwise seem impossible to overcome, can be more easily solved or managed with a set-theory approach. |
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| − | *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.
| + | The devoutly [[Christian]] founder of set theory, [[Georg Cantor]], viewed it as bringing freedom to mathematics. He summarized set theory as follows: "A set is a Many that allows itself to be thought of as a One."<ref>http://thinkexist.com/quotes/georg_cantor/</ref> [[David Hilbert]], a leading mathematician in the early 20th century, described set theory as follows: |
| | + | {{cquote|No one shall drive us from the paradise which Cantor has created for us.<ref>https://plus.maths.org/content/glimpse-cantors-paradise</ref>}} |
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| | + | [[John von Neumann]], considered by many to have been the most brilliant man of the 20th century (and a convert to [[Roman Catholic Church|Catholicism]], did his best work in set theory. He developed a definition of an ordinal number as the set of all smaller ordinal numbers, the definition that is commonly accepted today.<ref>https://www.britannica.com/biography/John-von-Neumann</ref> That approach helps circumvent issues caused by the transfinite numbers of Georg Cantor. |
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| | + | 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 name=stanford>[https://plato.stanford.edu/entries/set-theory/ ''Set Theory''], retrieved from [[Stanford Encyclopedia of Philosophy]] on Apr 15, 2022.</ref> or, equivalently, if each is a [[subset]] of the other. |
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| | ==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]]. |
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| − | 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 Gödel]]. 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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| − | ==The Empty Set== | + | ==The empty set== |
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| | 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 ∅. | | 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 ∅. |
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| | + | In the axiomatic set theory, '''any''' other set is build over the empty set. An example of a set with one element is the set { ∅ } - the set whose single element is the empty set. Von Neumann called the empty set ''0'', this set { ∅ } = ''1'', and proceeded in defining each ''natural number'' ''n'' as the set with the ''n'' elements from ''0'' to ''n-1''. In this way, it's possible to define the (infinite) set of natural numbers. Ordered pairs, functions, relations, etc, are all defined as sets with some properties.<ref>[https://www.britannica.com/science/set-theory/Schema-for-transfinite-induction-and-ordinal-arithmetic ''Axiomatic set theory''], retrieved from [[Encyclopedia Britannica]] on Apr 15, 2022.</ref><ref name=stanford /> |
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| | == See also == | | == See also == |
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| | *[http://math.stackexchange.com/questions/24854/what-are-the-issues-in-modern-set-theory Forum explaining the status of set theory in mathematics and philosophy today] | | *[http://math.stackexchange.com/questions/24854/what-are-the-issues-in-modern-set-theory Forum explaining the status of set theory in mathematics and philosophy today] |
| | + | *[http://castle.eiu.edu/~mathcs/mat2120/index/set03-2x3.pdf good overview of set theory] |
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| | ==References== | | ==References== |
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| | <References/> | | <References/> |
| − | | + | {{Greatest Thinkers}} |
| | [[Category:Set Theory]] | | [[Category:Set Theory]] |