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Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It describes the aspects of mathematical sets familiar in
discrete mathematics Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuou ...
(for example
Venn diagram A Venn diagram is a widely used diagram style that shows the logical relation between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationship ...
s and symbolic reasoning about their
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
), and suffices for the everyday use of set theory concepts in contemporary mathematics. Sets are of great importance in mathematics; in modern formal treatments, most mathematical objects (
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual number ...
s, relations, functions, etc.) are defined in terms of sets. Naive set theory suffices for many purposes, while also serving as a stepping-stone towards more formal treatments.


Method

A ''naive theory'' in the sense of "naive set theory" is a non-formalized theory, that is, a theory that uses natural language to describe sets and operations on sets. The words ''and'', ''or'', ''if ... then'', ''not'', ''for some'', ''for every'' are treated as in ordinary mathematics. As a matter of convenience, use of naive set theory and its formalism prevails even in higher mathematics – including in more formal settings of set theory itself. The first development of
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
was a naive set theory. It was created at the end of the 19th century by
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...
as part of his study of infinite sets and developed by
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic p ...
in his ''Grundgesetze der Arithmetik''. Naive set theory may refer to several very distinct notions. It may refer to * Informal presentation of an axiomatic set theory, e.g. as in '' Naive Set Theory'' by Paul Halmos. * Early or later versions of
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...
's theory and other informal systems. * Decidedly inconsistent theories (whether axiomatic or not), such as a theory of
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic p ...
that yielded Russell's paradox, and theories of Giuseppe Peano and Richard Dedekind.


Paradoxes

The assumption that any property may be used to form a set, without restriction, leads to paradoxes. One common example is Russell's paradox: there is no set consisting of "all sets that do not contain themselves". Thus consistent systems of naive set theory must include some limitations on the principles which can be used to form sets.


Cantor's theory

Some believe that
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...
's set theory was not actually implicated in the set-theoretic paradoxes (see Frápolli 1991). One difficulty in determining this with certainty is that Cantor did not provide an axiomatization of his system. By 1899, Cantor was aware of some of the paradoxes following from unrestricted interpretation of his theory, for instance Cantor's paradoxLetter from Cantor to David Hilbert on September 26, 1897, p. 388. and the
Burali-Forti paradox In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction. It is named after C ...
,Letter from Cantor to Richard Dedekind on August 3, 1899, p. 408. and did not believe that they discredited his theory.Letters from Cantor to Richard Dedekind on August 3, 1899 and on August 30, 1899, p. 448 (System aller denkbaren Klassen) and p. 407. (There is no set of all sets.) Cantor's paradox can actually be derived from the above (false) assumption—that any property may be used to form a set—using for " is a
cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. ...
". Frege explicitly axiomatized a theory in which a formalized version of naive set theory can be interpreted, and it is ''this'' formal theory which
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, a ...
actually addressed when he presented his paradox, not necessarily a theory Cantorwho, as mentioned, was aware of several paradoxespresumably had in mind.


Axiomatic theories

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.


Consistency

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 Halmos' ''Naive Set Theory'', 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 does not deal with consistency or completeness of the axiom system. Likewise, an axiomatic set theory is not necessarily consistent: not necessarily free of paradoxes. It follows from Gödel's incompleteness theorems that a sufficiently complicated first order logic system (which includes most common axiomatic set theories) cannot be proved consistent from within the theory itself – even if it actually is consistent. However, the common axiomatic systems are generally believed to be consistent; by their axioms they do exclude ''some'' paradoxes, like Russell's paradox. Based on Gödel's theorem, it is just not known – and never can be – if there are ''no'' paradoxes at all in these theories or in any first-order set theory. The term ''naive set theory'' is still today 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.


Utility

The choice between an axiomatic approach and other approaches is largely a matter of convenience. In everyday mathematics the best choice may be informal use of axiomatic set theory. References to particular axioms typically then occur only when demanded by tradition, e.g. the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
is often mentioned when used. Likewise, formal proofs occur only when warranted by exceptional circumstances. This informal usage of axiomatic set theory can have (depending on notation) precisely the ''appearance'' of naive set theory as outlined below. It is considerably easier to read and write (in the formulation of most statements, proofs, and lines of discussion) and is less error-prone than a strictly formal approach.


Sets, membership and equality

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 An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s. Clearly, the set of even numbers is infinitely large; there is no requirement that a set be finite. The definition of sets goes back to
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( , ;  – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...
. He wrote in his 1915 article
Beiträge zur Begründung der transfiniten Mengenlehre
':
“Unter einer 'Menge' verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objekten unserer Anschauung oder unseres Denkens (welche die 'Elemente' von M genannt werden) zu einem Ganzen.” – Georg Cantor
“A set is a gathering together into a whole of definite, distinct objects of our perception or of our thought—which are called elements of the set.” – Georg Cantor


Note on consistency

It does ''not'' follow from this definition ''how'' sets can be formed, and what operations on sets again will produce a set. The term "well-defined" in "well-defined collection of objects" cannot, by itself, guarantee the consistency and unambiguity of what exactly constitutes and what does not constitute a set. Attempting to achieve this would be the realm of axiomatic set theory or of axiomatic class theory. The problem, in this context, with informally formulated set theories, not derived from (and implying) any particular axiomatic theory, is that there may be several widely differing formalized versions, that have both different sets and different rules for how new sets may be formed, that all conform to the original informal definition. For example, Cantor's verbatim definition allows for considerable freedom in what constitutes a set. On the other hand, it is unlikely that Cantor was particularly interested in sets containing cats and dogs, but rather only in sets containing purely mathematical objects. An example of such a class of sets could be the von Neumann universe. But even when fixing the class of sets under consideration, it is not always clear which rules for set formation are allowed without introducing paradoxes. For the purpose of fixing the discussion below, the term "well-defined" should instead be interpreted as an ''intention'', with either implicit or explicit rules (axioms or definitions), to rule out inconsistencies. The purpose is to keep the often deep and difficult issues of consistency away from the, usually simpler, context at hand. An explicit ruling out of ''all'' conceivable inconsistencies (paradoxes) cannot be achieved for an axiomatic set theory anyway, due to Gödel's second incompleteness theorem, so this does not at all hamper the utility of naive set theory as compared to axiomatic set theory in the simple contexts considered below. It merely simplifies the discussion. Consistency is henceforth taken for granted unless explicitly mentioned.


Membership

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''. This is denoted by ''x'' ∈ ''A''. The symbol ∈ is a derivation from the lowercase Greek letter
epsilon Epsilon (, ; uppercase , lowercase or lunate ; el, έψιλον) is the fifth letter of the Greek alphabet, corresponding phonetically to a mid front unrounded vowel or . In the system of Greek numerals it also has the value five. It was d ...
, "ε", introduced by Giuseppe Peano in 1889 and is the first letter of the wor
ἐστί
(means "is"). The symbol ∉ is often used to write ''x'' ∉ ''A'', meaning "x is not in A".


Equality

Two sets ''A'' and ''B'' are defined to be 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 A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s less than 6. If the sets ''A'' and ''B'' are equal, this is denoted symbolically as ''A'' = ''B'' (as usual).


Empty set

The empty set, denoted as \varnothing and sometimes \, 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 \varnothing\neq\, because the former has no members and the latter has one member. In mathematics, the only sets with which one needs to be concerned can be built up from the empty set alone.


Specifying sets

The simplest way to describe a set is to list its elements between curly braces (known as defining a set ''extensionally''). Thus denotes the set whose only elements are and . (See
axiom of pairing In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo–Fraenkel set theory. It was introduced by as a special case of his axiom of elementary set ...
.) Note the following points: *The order of elements is immaterial; for example, . *Repetition ( multiplicity) of elements is irrelevant; for example, . (These are consequences of the definition of equality in the previous section.) This notation can be informally abused by saying something like to indicate the set of all dogs, but this example would usually be read by mathematicians as "the set containing the single element ''dogs''". An extreme (but correct) example of this notation is , which denotes the empty set. The notation , or sometimes , is used to denote the set containing all objects for which the condition holds (known as defining a set ''intensionally''). For example, denotes the set of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, denotes the set of everything with blonde hair. This notation is called set-builder notation (or "set comprehension", particularly in the context of
Functional programming In computer science, functional programming is a programming paradigm where programs are constructed by applying and composing functions. It is a declarative programming paradigm in which function definitions are trees of expressions tha ...
). Some variants of set builder notation are: * denotes the set of all that are already members of such that the condition holds for . For example, if is the set of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s, then is the set of all
even Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East **Even language, a language spoken by the Evens * Odd and Even, a solitaire game wh ...
integers. (See
axiom of specification In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any ...
.) * denotes the set of all objects obtained by putting members of the set into the formula . For example, is again the set of all even integers. (See axiom of replacement.) * is the most general form of set builder notation. For example, is the set of all dog owners.


Subsets

Given two sets ''A'' and ''B'', ''A'' is a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of ''B'' if every element of ''A'' is also an element of ''B''. 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. 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''. 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. 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
U.S. Presidents The president of the United States is the head of state and head of government of the United States, indirectly elected to a four-year term via the Electoral College. The officeholder leads the executive branch of the federal government and ...
. 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''. 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. 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 prove that two sets are equal, one aims to show these two inclusions. 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). The set of all subsets of a given set ''A'' is called the power set of ''A'' and is denoted by 2^A or P(A); the "''P''" is sometimes in a
script Script may refer to: Writing systems * Script, a distinctive writing system, based on a repertoire of specific elements or symbols, or that repertoire * Script (styles of handwriting) ** Script typeface, a typeface with characteristics of ha ...
font. If the set ''A'' has ''n'' elements, then P(A) will have 2^n elements.


Universal sets and absolute complements

In certain contexts, one may consider all sets under consideration as being subsets of some given universal set. For instance, when investigating properties of the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s R (and subsets of R), R may be taken as the universal set. A true universal set is not included in standard set theory (see Paradoxes below), but is included in some non-standard set theories. Given a universal set U and a subset ''A'' of U, the complement of ''A'' (in U) is defined as :''A''C := . In other words, ''A''C ("''A-complement''"; sometimes simply ''A, "''A-prime''" ) is the set of all members of U which are not members of ''A''. Thus with R, Z and ''O'' defined as in the section on subsets, if Z is the universal set, then ''OC'' is the set of even integers, while if R is the universal set, then ''OC'' is the set of all real numbers that are either even integers or not integers at all.


Unions, intersections, and relative complements

Given two sets ''A'' and ''B'', their union 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''. The 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''. Finally, the
relative complement In set theory, the complement of a set , often denoted by (or ), is the set of elements not in . When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set , the absolute complement of is ...
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''. Symbolically, these are respectively :''A'' ∪ B := ; :''A'' ∩ ''B'' :=  =  = ; :''A'' \ ''B'' :=  = . The set ''B'' doesn't have to be a subset of ''A'' for ''A'' \ ''B'' to make sense; this is the difference between the relative complement and the absolute complement (''A''C = ''U'' \ ''A'') from the previous section. 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. 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 over 1000 years old, so ''E'' ∩ ''F'' must be the empty set . For any set ''A'', the power set P(A) is a
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
under the operations of union and intersection.


Ordered pairs and Cartesian products

Intuitively, an
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
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. Formally, an ordered pair with first coordinate ''a'', and second coordinate ''b'', usually denoted by (''a'', ''b''), can be defined as the set . It follows that, two ordered pairs (''a'',''b'') and (''c'',''d'') are equal if and only if ''a'' = ''c'' and ''b'' = ''d''. Alternatively, an ordered pair can be formally thought of as a set with a
total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive ...
. (The notation (''a'', ''b'') is also used to denote an
open interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
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). If ''A'' and ''B'' are sets, then the Cartesian product (or simply product) is defined to be: :''A'' × ''B'' = . 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''. This definition may be extended to a set ''A'' × ''B'' × ''C'' of ordered triples, and more generally to sets of ordered n-tuples for any positive integer ''n''. It is even possible to define infinite
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\t ...
s, but this requires a more recondite definition of the product. Cartesian products were first developed by
René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Ma ...
in the context of
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and enginee ...
. If R denotes the set of all
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, then R2 := R × R represents the
Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...
and R3 := R × R × R represents three-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
.


Some important sets

There are some ubiquitous sets for which the notation is almost universal. Some of these are listed below. In the list, ''a'', ''b'', and ''c'' refer to
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s, and ''r'' and ''s'' are
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s. #
Natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s are used for counting. A blackboard bold capital N (\mathbb) often represents this set. #
Integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s appear as solutions for ''x'' in equations like ''x'' + ''a'' = ''b''. A blackboard bold capital Z (\mathbb) often represents this set (from the German ''Zahlen'', meaning ''numbers''). #
Rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s appear as solutions to equations like ''a'' + ''bx'' = ''c''. A blackboard bold capital Q (\mathbb) often represents this set (for '' quotient'', because R is used for the set of real numbers). #
Algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of th ...
s appear as solutions to
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
equations (with integer coefficients) and may involve
radicals Radical may refer to: Politics and ideology Politics *Radical politics, the political intent of fundamental societal change *Radicalism (historical), the Radical Movement that began in late 18th century Britain and spread to continental Europe and ...
(including i=\sqrt) and certain other irrational numbers. A Q with an overline (\overline) often represents this set. The overline denotes the operation of algebraic closure. #
Real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
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 In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and . Though only a few classes ...
s, which cannot appear as solutions to polynomial equations with rational coefficients. A blackboard bold capital R (\mathbb) often represents this set. #
Complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s are sums of a real and an imaginary number: r+s\,i. Here either r or s (or both) can be zero; thus, the set of real numbers and the set of strictly 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 \mathbb has at least one
root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the su ...
in this set. A blackboard bold capital C (\mathbb) often represents this set. Note that since a number r+s\,i can be identified with a point (r,s) in the plane, \mathbb is basically "the same" as the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\t ...
\R\times\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, as long as multiplication rule is appropriate for \mathbb).


Paradoxes in early set theory

The unrestricted formation principle of sets referred to as the axiom schema of unrestricted comprehension, is the source of several early appearing paradoxes: * led, in the year 1897, to the
Burali-Forti paradox In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction. It is named after C ...
, the first published
antinomy Antinomy (Greek ἀντί, ''antí'', "against, in opposition to", and νόμος, ''nómos'', "law") refers to a real or apparent mutual incompatibility of two laws. It is a term used in logic and epistemology, particularly in the philosophy of I ...
. * produced Cantor's paradox in 1897. * yielded Cantor's second antinomy in the year 1899. Here the property is true for all , whatever may be, so would be a universal set, containing everything. *, i.e. the set of all sets that do not contain themselves as elements, gave Russell's paradox in 1902. If the axiom schema of unrestricted comprehension is weakened to the axiom schema of specification or axiom schema of separation, then all the above paradoxes disappear. There is a corollary. With the axiom schema of separation as an axiom of the theory, it follows, as a theorem of the theory: Or, more spectacularly (Halmos' phrasing): There is no
universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the univers ...
. ''Proof'': Suppose that it exists and call it . Now apply the axiom schema of separation with and for use . This leads to Russell's paradox again. Hence cannot exist in this theory. Related to the above constructions is formation of the set *, where the statement following the implication certainly is false. It follows, from the definition of , using the usual inference rules (and some afterthought when reading the proof in the linked article below) both that and holds, hence . This is Curry's paradox. It is (perhaps surprisingly) not the possibility of that is problematic. It is again the axiom schema of unrestricted comprehension allowing for . With the axiom schema of specification instead of unrestricted comprehension, the conclusion does not hold and hence is not a logical consequence. Nonetheless, the possibility of is often removed explicitly or, e.g. in ZFC, implicitly, by demanding the
axiom of regularity In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set ''A'' contains an element that is disjoint from ''A''. In first-order logic, the a ...
to hold. One consequence of it is or, in other words, no set is an element of itself. The axiom schema of separation is simply too weak (while unrestricted comprehension is a very strong axiom—too strong for set theory) to develop set theory with its usual operations and constructions outlined above. The axiom of regularity is of a restrictive nature as well. Therefore, one is led to the formulation of other axioms to guarantee the existence of enough sets to form a set theory. Some of these have been described informally above and many others are possible. Not all conceivable axioms can be combined freely into consistent theories. For example, the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
of ZFC is incompatible with the conceivable "every set of reals is Lebesgue measurable". The former implies the latter is false.


See also

*
Algebra of sets In mathematics, the algebra of sets, not to be confused with the mathematical structure of ''an'' algebra of sets, defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the ...
*
Axiomatic set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
*
Internal set theory Internal set theory (IST) is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the nonstandard analysis introduced by Abraham Robinson. Instead of adding new elements to the real numbers, ...
* List of set identities and relations *
Set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
*
Set (mathematics) A set is the mathematical model for a collection of different things; a set contains '' elements'' or ''members'', which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, o ...
*
Partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...


Notes


References

* Bourbaki, N., ''Elements of the History of Mathematics'', John Meldrum (trans.), Springer-Verlag, Berlin, Germany, 1994. * * Devlin, K.J., ''The Joy of Sets: Fundamentals of Contemporary Set Theory'', 2nd edition, Springer-Verlag, New York, NY, 1993. * 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. * * ** ** * * Kelley, J.L., ''General Topology'', Van Nostrand Reinhold, New York, NY, 1955. * 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. . * * *


External links


Beginnings of set theory
page at St. Andrews

{{Mathematical logic Set theory Systems of set theory