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| {{about|the mathematical topic|the book of the same name|Naive Set Theory (book)}}
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| '''Naive set theory''' is one of several theories of sets used in the discussion of the [[foundations of mathematics]].<ref>Concerning the origin of the term ''naive set theory'', Jeff Miller says, “''Naïve set theory'' (contrasting with axiomatic set theory) was used occasionally in the 1940s and became an established term in the 1950s. It appears in Hermann Weyl's review of P. A. Schilpp (ed) ''The Philosophy of Bertrand Russell'' in the ''American Mathematical Monthly'', 53., No. 4. (1946), p. 210 and Laszlo Kalmar's review of ''The Paradox of Kleene and Rosser'' in ''Journal of Symbolic Logic'', 11, No. 4. (1946), p. 136. (JSTOR).” [http://jeff560.tripod.com/s.html] The term was later popularized by [[Paul Halmos]]' book, ''Naive Set Theory'' (1960).</ref> Unlike [[Set theory#Axiomatic set theory|axiomatic set theories]], which are defined using a formal logic, naive set theory is defined informally, in natural language. It describes the aspects of [[mathematical set]]s familiar in [[discrete mathematics]] (for example [[Venn diagram]]s and symbolic reasoning about their [[Boolean algebra (logic)|Boolean algebra]]), and suffices for the everyday usage of set theory concepts in contemporary mathematics{{citation needed|date=April 2013}}.
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| Sets are of great importance in [[mathematics]]; in fact, in modern formal treatments, most mathematical objects ([[number]]s, [[relation (mathematics)|relations]], [[function (mathematics)|functions]], etc.) are defined in terms of sets. Naive set theory can be seen as a stepping-stone to more formal treatments, and suffices for many purposes.
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| ==Requirements==
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| In the sense of this article, a '''naive theory''' is a non-formalized theory, that is, a theory that uses a [[natural language]] to describe sets. The words '''and''', '''or''', '''if ... then''', '''not''', '''for some''', '''for every''' are not subject to rigorous definition. It is useful to study sets naively at an early stage of mathematics in order to develop facility for working with them. Furthermore, a firm grasp of set theory's concepts from a naive standpoint is a step to understanding the motivation for the formal axioms of set theory.
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| This article develops a naive theory. Sets are defined informally and a few of their properties are investigated. Links in this article to specific axioms of set theory describe some of the relationships between the informal discussion here and the formal [[axiomatization]] of set theory, but no attempt is made to justify every statement on such a basis. The first development of [[set theory]] was a naive set theory. It was created at the end of the 19th century by [[Georg Cantor]] as part of his study of [[infinite set]]s {{citation needed|date=December 2012}} and developed by [[Gottlob Frege]] in his ''[[Begriffsschrift]]''.
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| As it turned out, assuming that one can perform any operation on sets without restriction leads to [[paradox]]es such as [[Russell's paradox]] and [[Berry's paradox]]. Some believe that [[Georg Cantor]]'s set theory was not actually implicated by these [[paradox]]es (see Frápolli 1991); one difficulty in determining this with certainty is that Cantor did not provide an axiomatization of his system. It is undisputed{{citation needed|date=December 2012}} that, by 1900, Cantor was aware of some of the paradoxes and did not believe that they discredited his theory. [[Gottlob Frege]] explicitly axiomatized a theory in which the formalized version of naive set theory can be interpreted, and it is this formal theory which [[Bertrand Russell]] actually addressed when he presented his paradox.
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| Axiomatic set theory was developed in response to these early attempts to understand sets, with the goal of determining precisely what operations were allowed and when. Today, when mathematicians talk about "set theory" as a field, they usually{{citation needed|date=December 2012}} mean axiomatic set theory. Informal applications of set theory in other fields are sometimes referred to as applications of "naive set theory", but usually are understood to be justifiable in terms of an [[axiomatic system]] (normally [[Zermelo–Fraenkel set theory]]).
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| A naive set theory is not necessarily inconsistent, if it correctly specifies the sets allowed to be considered. This can be done by the means of definitions, which are implicit axioms. It is possible to state all the axioms explicitly, as in the case of the book ''Naive Set Theory'' by [[Paul Halmos]], which is actually an informal presentation of the usual axiomatic [[Zermelo–Fraenkel set theory]]. It is "naive" in that the language and notations are those of ordinary informal mathematics, and in that it doesn't deal with consistency or completeness of the axiom system. However, the term ''naive set theory'' is{{citation needed|date=December 2012}} also used in some literature to refer to the set theories studied by Frege and Cantor, rather than to the informal counterparts of modern axiomatic set theory.
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| == Sets, membership and equality ==
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| In naive set theory, a '''set''' is described as a well-defined collection of objects. These objects are called the '''elements''' or '''members''' of the set. Objects can be anything: numbers, people, other sets, etc. For instance, 4 is a member of the set of all even [[integer]]s. Clearly, the set of even numbers is infinitely large; there is no requirement that a set be finite.
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| If ''x'' is a member of a set ''A'', then it is also said that ''x'' '''belongs to''' ''A'', or that ''x'' is in ''A''. In this case, we write ''x'' ∈ ''A''.
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| (The symbol ∈ is a derivation from the [[Greek alphabet|Greek letter]] [[epsilon]], "ε", introduced by [[Peano]] in 1888.)
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| The symbol ∉ is sometimes used to write ''x'' ∉ ''A'', meaning "x is not in A".
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| Two sets ''A'' and ''B'' are defined to be '''[[Equality (mathematics)|equal]]''' when they have precisely the same elements, that is, if every element of ''A'' is an element of ''B'' and every element of ''B'' is an element of ''A''. (See [[axiom of extensionality]].) Thus a set is completely determined by its elements; the description is immaterial. For example, the set with elements 2, 3, and 5 is equal to the set of all [[prime number]]s less than 6.
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| If the sets ''A'' and ''B'' are equal, this is denoted symbolically as ''A'' = ''B'' (as usual).
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| The '''[[empty set]]''', often denoted Ø and sometimes <math>\{\}</math>, is a set with no members at all. Because a set is determined completely by its elements, there can be only one empty set. (See [[axiom of empty set]].) Although the empty set has no members, it can be a member of other sets. Thus Ø ≠ {Ø}, because the former has no members and the latter has one member.
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| == Specifying sets ==
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| The simplest way to describe a set is to list its elements between curly braces (known as defining a set ''extensionally''). Thus {1,2} denotes the set whose only elements are 1 and 2.
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| (See [[axiom of pairing]].)
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| Note the following points:
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| *Order of elements is immaterial; for example, {1,2} = {2,1}.
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| *Repetition ([[multiplicity (mathematics)|multiplicity]]) of elements is irrelevant; for example, {1,2,2} = {1,1,1,2} = {1,2}.
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| (These are consequences of the definition of equality in the previous section.)
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| This notation can be informally abused by saying something like {dogs} to indicate the set of all dogs, but this example would usually be read by mathematicians as "the set containing the single element ''dogs''".
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| An extreme (but correct) example of this notation is {}, which denotes the empty set.
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| We can also use the notation {''x'' : ''P''(''x'')}, or sometimes {''x'' | ''P''(''x'')}, to denote the set containing all objects for which the condition ''P'' holds (known as defining a set ''intensionally'').
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| For example, {''x'' : ''x'' is a real number} denotes the set of [[real number]]s, {''x'' : ''x'' has blonde hair} denotes the set of everything with blonde hair, and {''x'' : ''x'' is a dog} denotes the set of all dogs.
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| This notation is called [[set-builder notation]] (or "'''set comprehension'''", particularly in the context of [[Functional programming]]).
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| Some variants of set builder notation are:
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| *{''x'' ∈ ''A'' : ''P''(''x'')} denotes the set of all ''x'' ''that are already members of A'' such that the condition ''P'' holds for ''x''. For example, if '''Z''' is the set of [[integer]]s, then {''x'' ∈ '''Z''' : ''x'' is even} is the set of all [[even and odd numbers|even]] integers. (See [[axiom of specification]].)
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| *{''F''(''x'') : ''x'' ∈ ''A''} denotes the set of all objects obtained by putting members of the set ''A'' into the formula ''F''. For example, {2''x'' : ''x'' ∈ '''Z'''} is again the set of all even integers. (See [[axiom of replacement]].) | |
| *{''F''(''x'') : ''P''(''x'')} is the most general form of set builder notation. For example, {''x'''s owner : ''x'' is a dog} is the set of all dog owners. | |
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| == Subsets ==
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| Given two sets ''A'' and ''B'' we say that ''A'' is a '''[[subset]]''' of ''B'' if every element of ''A'' is also an element of ''B''.
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| In particular, each set ''B'' is a subset of itself; a subset of ''B'' that is not equal to ''B'' is called a '''proper subset'''.
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| If ''A'' is a subset of ''B'', then one can also say that ''B'' is a '''superset''' of ''A'', that ''A'' is '''contained in''' ''B'', or that ''B'' '''contains''' ''A''. In symbols, ''A'' ⊆ ''B'' means that ''A'' is a subset of ''B'', and ''B'' ⊇ ''A'' means that ''B'' is a superset of ''A''.
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| Some authors use the symbols ⊂ and ⊃ for subsets, and others use these symbols only for ''proper'' subsets. For clarity, one can explicitly use the symbols ⊊ and ⊋ to indicate non-equality.
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| As an illustration, let '''R''' be the set of real numbers, let '''Z''' be the set of integers, let ''O'' be the set of odd integers, and let ''P'' be the set of current or former [[President of the United States|U.S. Presidents]].
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| Then ''O'' is a subset of '''Z''', '''Z''' is a subset of '''R''', and (hence) ''O'' is a subset of '''R''', where in all cases ''subset'' may even be read as ''proper subset''.
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| Note that not all sets are comparable in this way. For example, it is not the case either that '''R''' is a subset of ''P'' nor that ''P'' is a subset of '''R'''.
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| It follows immediately from the definition of equality of sets above that, given two sets ''A'' and ''B'', ''A'' = ''B'' if and only if ''A'' ⊆ ''B'' and ''B'' ⊆ ''A''. In fact this is often given as the definition of equality. Usually when trying to [[mathematical proof|prove]] that two sets are equal, one aims to show these two inclusions. Note that the [[empty set]] is a subset of every set (the statement that all elements of the empty set are also members of any set ''A'' is [[vacuously true]]).
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| The set of all subsets of a given set ''A'' is called the '''[[power set]]''' of ''A'' and is denoted by <math>2^A</math> or <math>P(A)</math>; the "''P''" is sometimes in a [[Script (typefaces)|script]] font. If the set ''A'' has ''n'' elements, then <math>P(A)</math> will have <math>2^n</math> elements.
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| == Universal sets and absolute complements ==
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| In certain contexts we may consider all sets under consideration as being subsets of some given [[universe (mathematics)|universal set]].
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| For instance, if we are investigating properties of the [[real number]]s '''R''' (and subsets of '''R'''), then we may take '''R''' as our universal set. A true universal set is not included in standard set theory (see '''[[#Paradoxes|Paradoxes]]''' below), but is included in some non-standard set theories.
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| Given a universal set '''U''' and a subset ''A'' of '''U''', we may define the '''[[complement (set theory)|complement]]''' of ''A'' (in '''U''') as
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| :''A''<sup>C</sup> := {''x'' {{unicode|∈}} '''U''' : ''x'' {{unicode|∉}} ''A''}. | |
| In other words, ''A''<sup>C</sup> ("''A-complement''"; sometimes simply ''A''', "''A-prime''" ) is the set of all members of '''U''' which are not members of ''A''.
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| Thus with '''R''', '''Z''' and ''O'' defined as in the section on subsets, if '''Z''' is the universal set, then ''O<sup>C</sup>'' is the set of even integers, while if '''R''' is the universal set, then ''O<sup>C</sup>'' is the set of all real numbers that are either even integers or not integers at all.
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| == Unions, intersections, and relative complements ==
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| Given two sets ''A'' and ''B'', we may construct their '''[[union (set theory)|union]]'''. This is the set consisting of all objects which are elements of ''A'' or of ''B'' or of both (see [[axiom of union]]). It is denoted by ''A'' ∪ ''B''.
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| The '''[[intersection (set theory)|intersection]]''' of ''A'' and ''B'' is the set of all objects which are both in ''A'' and in ''B''. It is denoted by ''A'' ∩ ''B''.
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| Finally, the '''[[complement (set theory)|relative complement]]''' of ''B'' relative to ''A'', also known as the '''set theoretic difference''' of ''A'' and ''B'', is the set of all objects that belong to ''A'' but ''not'' to ''B''. It is written as ''A'' \ ''B'' or ''A'' − ''B''.
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| Symbolically, these are respectively
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| :''A'' ∪ B := {''x'' : (''x'' ∈ ''A'') [[logical disjunction|or]] (''x'' ∈ ''B'')};
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| :''A'' ∩ ''B'' := {''x'' : (''x'' ∈ ''A'') [[logical conjunction|and]] (''x'' ∈ ''B'')} = {''x'' ∈ ''A'' : ''x'' ∈ ''B''} = {''x'' ∈ ''B'' : ''x'' ∈ ''A''};
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| :''A'' \ ''B'' := {''x'' : (''x'' ∈ ''A'') and [[negation|not]] (''x'' ∈ ''B'') } = {''x'' ∈ ''A'' : not (''x'' ∈ ''B'')}.
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| Notice that ''A'' doesn't have to be a subset of ''B'' for ''B'' ''A'' to make sense; this is the difference between the relative complement and the absolute complement (''A''<sup>C</sup> = ''U'' ''A'') from the previous section.
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| To illustrate these ideas, let ''A'' be the set of left-handed people, and let ''B'' be the set of people with blond hair. Then ''A'' ∩ ''B'' is the set of all left-handed blond-haired people, while ''A'' ∪ ''B'' is the set of all people who are left-handed or blond-haired or both. ''A'' ''B'', on the other hand, is the set of all people that are left-handed but not blond-haired, while ''B'' ''A'' is the set of all people who have blond hair but aren't left-handed.
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| Now let ''E'' be the set of all human beings, and let ''F'' be the set of all living things over 1000 years old. What is ''E'' ∩ ''F'' in this case? No living human being is [[Oldest people|over 1000 years old]], so ''E'' ∩ ''F'' must be the [[empty set]] {}.
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| For any set ''A'', the power set <math>P(A)</math> is a [[Boolean algebra (structure)|Boolean algebra]] under the operations of union and intersection.
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| == Ordered pairs and Cartesian products ==
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| Intuitively, an '''[[ordered pair]]''' is simply a collection of two objects such that one can be distinguished as the ''first element'' and the other as the ''second element'', and having the fundamental property that, two ordered pairs are equal if and only if their ''first elements'' are equal and their ''second elements'' are equal.
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| Formally, an ordered pair with '''first coordinate''' ''a'', and '''second coordinate''' ''b'', usually denoted by (''a'', ''b''), can be defined as the set <nowiki>{{</nowiki>''a''<nowiki>}</nowiki>, <nowiki>{</nowiki>''a'', ''b''<nowiki>}}</nowiki>.
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| It follows that, two ordered pairs (''a'',''b'') and (''c'',''d'') are equal if and only if ''a'' = ''c'' and ''b'' = ''d''.
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| Alternatively, an ordered pair can be formally thought of as a set {a,b} with a [[total order]].
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| (The notation (''a'', ''b'') is also used to denote an [[open interval]] on the [[real number line]], but the context should make it clear which meaning is intended. Otherwise, the notation ]''a'', ''b''[ may be used to denote the open interval whereas (''a'', ''b'') is used for the ordered pair).
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| If ''A'' and ''B'' are sets, then the '''[[Cartesian product]]''' (or simply '''product''') is defined to be:
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| :''A'' × ''B'' = {(''a'',''b'') : ''a'' is in ''A'' and ''b'' is in ''B''}.
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| That is, ''A'' × ''B'' is the set of all ordered pairs whose first coordinate is an element of ''A'' and whose second coordinate is an element of ''B''.
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| We can extend this definition to a set ''A'' × ''B'' × ''C'' of ordered triples, and more generally to sets of ordered [[n-tuple]]s for any positive integer ''n''.
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| It is even possible to define infinite [[Cartesian product]]s, but to do this we need a more recondite definition of the product.
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| Cartesian products were first developed by [[René Descartes]] in the context of [[analytic geometry]]. If '''R''' denotes the set of all [[real number]]s, then '''R'''<sup>2</sup> := '''R''' × '''R''' represents the [[Euclidean plane]] and '''R'''<sup>3</sup> := '''R''' × '''R''' × '''R''' represents three-dimensional [[Euclidean space]].
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| == Some important sets ==
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| <small>Note: In this section, ''a'', ''b'', and ''c'' are [[natural number]]s, and r and s are [[real number]]s.</small>
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| # [[Natural number]]s are used for counting. A [[blackboard bold]] capital '''N''' (<math>\mathbb{N}</math>) often represents this set.
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| # [[Integer]]s appear as solutions for ''x'' in equations like ''x'' + ''a'' = ''b''. A blackboard bold capital '''Z''' (<math>\mathbb{Z}</math>) often represents this set (from the German ''Zahlen'', meaning ''numbers'').
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| # [[Rational number]]s appear as solutions to equations like ''a'' + ''bx'' = ''c''. A blackboard bold capital '''Q''' (<math>\mathbb{Q}</math>) often represents this set (for ''[[quotient]]'', because R is used for the set of real numbers).
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| # [[Algebraic number]]s appear as solutions to [[polynomial]] equations (with integer coefficients) and may involve [[Nth root|radicals]] and certain other [[irrational number]]s. A blackboard bold capital '''A''' (<math>\mathbb{A}</math>) or a '''Q''' with an overline (<math>\overline{\mathbb{Q}}</math>) often represents this set. The overline denotes the operation of [[algebraic closure]].
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| # [[Real number]]s represent the "real line" and include all numbers that can be approximated by rationals. These numbers may be rational or algebraic but may also be [[transcendental number]]s, which cannot appear as solutions to polynomial equations with rational coefficients. A blackboard bold capital '''R''' (<math>\mathbb{R}</math>) often represents this set.
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| # [[Complex number]]s are sums of a real and an imaginary number: ''r'' + ''s''i. Here both ''r'' and ''s'' can equal zero; thus, the set of real numbers and the set of imaginary numbers are subsets of the set of complex numbers, which form an [[algebraic closure]] for the set of real numbers, meaning that every polynomial with coefficients in <math>\mathbb{R}</math> has at least one [[Root of a function|root]] in this set. A blackboard bold capital '''C''' (<math>\mathbb{C}</math>) often represents this set. Note that since a number ''r'' + ''s''i can be identified with a point (''r'', ''s'') in the plane, '''C''' is basically "the same" as the Cartesian product '''R'''×'''R''' ("the same" meaning that any point in one determines a unique point in the other and for the result of calculations it doesn't matter which one is used for the calculation).
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| == Paradoxes ==
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| We referred earlier to the need for a formal, axiomatic approach. What problems arise in the treatment we have given? The problems relate to the formation of sets. One's first intuition might be that we can form any set we want, but this view leads to inconsistencies. For any set ''x'' we can ask whether ''x'' is a member of itself. Define
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| :''Z'' = {''x'' : ''x'' is not a member of ''x''}.
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| Now for the problem: is ''Z'' a member of ''Z''? If yes, then by the defining quality of ''Z'', ''Z'' is not a member of itself, i.e., ''Z'' is not a member of ''Z''. This forces us to declare that ''Z'' is not a member of ''Z''. Then ''Z'' is not a member of itself and so, again by definition of ''Z'', ''Z'' is a member of ''Z''.
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| Thus both options lead us to a contradiction and we have an inconsistent theory. More succinctly, one says that ''Z'' is a member of ''Z'' if and only if ''Z'' is not a member of ''Z''. Axiomatic developments place restrictions on the sort of sets we are allowed to form and thus prevent problems like our set ''Z'' from arising. This particular paradox is [[Russell's paradox]].
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| The penalty is that one must take more care with one's development, as one must in any rigorous mathematical argument.
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| In particular, it is problematic to speak of a set of everything, or to be (possibly) a bit less ambitious, even a [[set of all sets]].
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| In fact, in the standard axiomatisation of set theory, there is no set of all sets.
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| In areas of mathematics that seem to require a set of all sets (such as [[category theory]]), one can sometimes make do with a universal set so large that all of ordinary mathematics can be done within it (see [[universe (mathematics)|universe]]).
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| Alternatively, one can make use of [[class (set theory)|proper classes]].
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| Or, one can use a different axiomatisation of set theory, such as [[W. V. Quine]]'s [[New Foundations]], which allows for a set of all sets and avoids Russell's paradox in another way.
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| ==See also==
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| {{portal|Mathematics}}
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| * [[Algebra of sets]]
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| * [[Axiomatic set theory]]
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| * [[Internal set theory]]
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| * [[Set theory]]
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| * [[Set (mathematics)]]
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| * [[Partially ordered set]]
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| == Notes ==
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| {{Refimprove|date=July 2011}}
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| {{reflist}}
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| ==References==
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| * [[Nicolas Bourbaki|Bourbaki, N.]], ''Elements of the History of Mathematics'', John Meldrum (trans.), Springer-Verlag, Berlin, Germany, 1994.
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| * [[Keith J. Devlin|Devlin, K.J.]], ''The Joy of Sets: Fundamentals of Contemporary Set Theory'', 2nd edition, Springer-Verlag, New York, NY, 1993.
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| * [[María J. Frápolli|Frápolli, María J.]], 1991, "Is Cantorian set theory an iterative conception of set?". ''Modern Logic'', v. 1 n. 4, 1991, 302–318.
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| * [[Paul Halmos|Halmos, Paul]], ''[[Naive Set Theory (book)|Naive Set Theory]]''. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). Reprinted by Martino Fine Books, 2011. ISBN 978-1-61427-131-4 (Paperback edition).
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| * [[John L. Kelley|Kelley, J.L.]], ''General Topology'', Van Nostrand Reinhold, New York, NY, 1955.
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| * [[Jean van Heijenoort|van Heijenoort, J.]], ''From Frege to Gödel, A Source Book in Mathematical Logic, 1879-1931'', Harvard University Press, Cambridge, MA, 1967. Reprinted with corrections, 1977. ISBN 0-674-32449-8.
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| == External links ==
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| * [http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Beginnings_of_set_theory.html Beginnings of set theory] page at St. Andrews
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| * [http://jeff560.tripod.com/s.html Earliest Known Uses of Some of the Words of Mathematics (S)]
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| * [http://www.angelfire.com/az3/nfold/nulltheorem.html Uniqueness of the empty set] Is there only one empty set rather than an infinity of empty subsets?
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| {{Set theory}}
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| {{DEFAULTSORT:Naive Set Theory}}
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| [[Category:Set theory]]
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