In the discussion of the foundations of mathematics, several set theories have been developed, of which **naive set theory**^{[1]} is one. The informal content of this *naive set theory* supports both the aspects of mathematical sets familiar in discrete mathematics (for example Venn diagrams and symbolic reasoning about their Boolean algebra), and the everyday usage of set theory concepts in most contemporary mathematics. Naive Set Theory is a mathematics textbook by Paul Halmos originally published in 1960. ...
Foundations of mathematics is a term sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory. ...
This article is about sets in mathematics. ...
Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete, in the sense of not supporting or requiring the notion of continuity. ...
A Venn diagram of sets A, B, and C Venn diagrams are illustrations used in the branch of mathematics known as set theory. ...
Boolean algebra is the finitary algebra of two values. ...
Sets are of great importance in mathematics; in fact, in modern formal treatments, most mathematical objects (numbers, relations, 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. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ...
For other uses, see Number (disambiguation). ...
In mathematics, the concept of a relation is a generalization of 2-place relations, such as the relation of equality, denoted by the sign = in a statement like 5 + 7 = 12, or the relation of order, denoted by the sign < in a statement like 5 < 12. Relations that involve two...
Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...
## Requirements
A **naive theory** is a non-formalized theory, that is to say a theory that uses a natural language to talk about 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 theoretical concepts from a naive standpoint is important as a first stage in understanding the motivation for the formal axioms of set theory. The term natural language is used to distinguish languages spoken and signed (by hand signals and facial expressions) by humans for general-purpose communication from constructs such as writing, computer-programming languages or the languages used in the study of formal logic, especially mathematical logic. ...
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 point out 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 in order to allow mathematicians to work with infinite sets consistently. In mathematics, axiomatization is the process of defining the basic axiomatic systems from which mathematical theories can be derived. ...
Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...
Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ...
Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845[1] â€“ January 6, 1918) was a German mathematician. ...
In set theory, an infinite set is a set that is not a finite set. ...
As it turned out, assuming that one could perform any operations on sets without restriction led to paradoxes such as Russell's paradox or Berry's paradox. In response, axiomatic set theory was developed to determine precisely what operations were allowed and when. Today, when mathematicians talk about "set theory" as a field, they usually 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 the Zermelo–Fraenkel set theory). Some believe that Georg Cantor's set theory was not actually implicated in the paradoxes (this is a matter which continues to be discussed).^{[citation needed]} He was aware of some of them and did not appear to believe that they discredited his theory. It is hard to be sure of this because he did not give an axiomatization. Frege did explicitly axiomatize 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. Look up paradox in Wiktionary, the free dictionary. ...
Part of the foundation of mathematics, Russells paradox (also known as Russells antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory of Frege leads to a contradiction. ...
The Berry paradox is the apparent contradiction that arises from expressions such as the following: The smallest positive integer not nameable in under eleven words. ...
This article or section is in need of attention from an expert on the subject. ...
In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. ...
Zermeloâ€“Fraenkel set theory, with the axiom of choice, commonly abbreviated ZFC, is the most common form of axiomatic set theory, and as such is the most common foundation of mathematics. ...
Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845[1] â€“ January 6, 1918) was a German mathematician. ...
Look up paradox in Wiktionary, the free dictionary. ...
Friedrich Ludwig Gottlob Frege Friedrich Ludwig Gottlob Frege (November 8, 1848 - July 26, 1925) was a German mathematician, logician, and philosopher who is regarded as a founder of both modern mathematical logic and analytic philosophy. ...
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS, (18 May 1872 â€“ 2 February 1970), was a British philosopher, logician, mathematician, advocate for social reform, and pacifist. ...
These early attempts therefore led to inconsistency. 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 can be done by systematically making explicit all the axioms, as in the case of the well-known book *Naive Set Theory* by Paul Halmos, which is actually a somewhat (not all that) 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. Paul Halmos Paul Richard Halmos (March 3, 1916 â€” October 2, 2006) was a Hungarian-born American mathematician who wrote on probability theory, statistics, operator theory, ergodic theory, functional analysis (in particular, Hilbert spaces), and mathematical logic. ...
Zermeloâ€“Fraenkel set theory, with the axiom of choice, commonly abbreviated ZFC, is the most common form of axiomatic set theory, and as such is the most common foundation of mathematics. ...
## 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 integers. Clearly, the set of even numbers is infinitely large; there is no requirement that a set be finite. The integers are commonly denoted by the above symbol. ...
If *x* is a member of *A*, then it is also said that *x* **belongs to** *A*, or that *x* is in *A*. In this case, we write *x* ∈ *A*. (The symbol ∈ is a derivation from the Greek letter epsilon, "ε", introduced by Peano in 1888.) The symbol ∉ is sometimes used to write *x* ∉ *A*, meaning "x is not in A". The Greek alphabet (Greek: ) is an alphabet consisting of 24 letters that has been used to write the Greek language since the late 8th or early 8th century BC. It was the first alphabet in the narrow sense, that is, a writing system using a separate symbol for each vowel...
Look up Î•, Îµ in Wiktionary, the free dictionary. ...
Giuseppe Peano (August 27, 1858 – April 20, 1932) was an Italian mathematician and philosopher. ...
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 numbers less than 6. If the sets *A* and *B* are equal, this is denoted symbolically as *A* = *B* (as usual). In mathematics, two mathematical objects are considered equal if they are precisely the same in every way. ...
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo-Fraenkel set theory. ...
In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ...
We also allow for an **empty set**, often denoted Ø and sometimes {}: a set without any members at all. Since a set is determined completely by its elements, there can only be one empty set. (See axiom of empty set.) The empty set is the set containing no elements. ...
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of empty set is one of the axioms of Zermelo_Fraenkel set theory. ...
## Specifying sets 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. (See axiom of pairing.) Note the following points: 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. ...
- Order of elements is immaterial; for example, {1,2} = {2,1}.
- Repetition (multiplicity) of elements is irrelevant; for example, {1,2,2} = {1,1,1,2} = {1,2}.
(These are consequences of the definition of equality in the previous section.) In mathematics, the multiplicity of a member of a multiset is how many memberships in the multiset it has. ...
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*". An extreme (but correct) example of this notation is {}, which denotes the empty set. 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*). For example, {*x* : *x* is a real number} denotes the set of real numbers, {*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. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
This notation is called set-builder notation (or "**set comprehension**", particularly in the context of Functional programming). Some variants of set builder notation are: In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by indicating the properties that its members must satisfy. ...
Functional programming is a programming paradigm that treats computation as the evaluation of mathematical functions and avoids state and mutable data. ...
- {
*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 integers, then {*x* ∈ **Z** : *x* is even} is the set of all even integers. (See axiom of specification.) - {
*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. The integers are commonly denoted by the above symbol. ...
Look up een, even in Wiktionary, the free dictionary. ...
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of specification, or axiom schema of separation, or axiom schema of restricted comprehension, is a schema of axioms in Zermelo-Fraenkel set theory. ...
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom schema of replacement is a schema of axioms in Zermelo-Fraenkel set theory. ...
## Subsets 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*. Notice that in particular, *B* is a subset of itself; a subset of *B* that isn't equal to *B* is called a **proper subset**. â€œSupersetâ€ redirects here. ...
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. In this encyclopedia, "⊆" and "⊇" are used for subsets while "⊂" and "⊃" are reserved for proper subsets. 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. 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*. 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**. Federal courts Supreme Court Circuit Courts of Appeal District Courts Elections Presidential elections Midterm elections Political Parties Democratic Republican Third parties State & Local government Governors Legislatures (List) State Courts Local Government Other countries Atlas US Government Portal For other uses, see President of the United States (disambiguation). ...
It follows immediately from the definition of equality of sets above, that given two sets *A* and *B*, *A* = *B* iff *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. 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). IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation...
In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ...
The empty set is the set containing no elements. ...
Informally, a logical statement is vacuously true if it is true but doesnt say anything; examples are statements of the form everything with property A also has property B, where there is nothing with property A. It is tempting to dismiss this concept as vacuous or silly. ...
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 fancy font. If the set *A* has *n* elements, then *P*(*A*) will have 2^{n} elements. In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ...
## Universal sets and absolute complements In certain contexts we may consider all sets under consideration as being subsets of some given universal set. For instance, if we are investigating properties of the real numbers **R** (and subsets of **R**), then we may take **R** as our universal set. A universal set is only temporarily defined by the context; there is no such thing as a "universal" universal set, "the set of everything" (see **Paradoxes** below). In mathematics, and particularly in applications to set theory and the foundations of mathematics, a universe or universal class (or if a set, universal set) is, roughly speaking, a class that is large enough to contain (in some sense) all of the sets that one may wish to use. ...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
Given a universal set **U** and a subset *A* of **U**, we may define the **complement** of *A* (in **U**) as In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ...
*A*^{C} := {*x* ∈ **U** : *x* ∉ *A*}. 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 *O*^{C} is the set of even integers, while if **R** is the universal set, then *O*^{C} 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*, we may construct their **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*. In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of union is one of the axioms of Zermelo_Fraenkel set theory, stating that, for any two sets, there is a set that contains exactly the elements of both. ...
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*. In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ...
Finally, the **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*. Symbolically, these are respectively In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ...
*A* ∪ B := {*x* : (*x* ∈ *A*) or (*x* ∈ *B*)}; *A* ∩ *B* := {*x* : (*x* ∈ *A*) and (*x* ∈ *B*)} = {*x* ∈ *A* : *x* ∈ *B*} = {*x* ∈ *B* : *x* ∈ *A*}; *A* *B* := {*x* : (*x* ∈ *A*) and not (*x* ∈ *B*) } = {*x* ∈ *A* : not (*x* ∈ *B*)}. 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 from the previous section. OR logic gate. ...
AND Logic Gate In logic and mathematics, logical conjunction (usual symbol and) is a two-place logical operation that results in a value of true if both of its operands are true, otherwise a value of false. ...
Negation (i. ...
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 human being is over 1000 years old, so *E* ∩ *F* must be the empty set {}. The empty set is the set containing no elements. ...
For any set *A*, the power set *P*(*A*) is a Boolean algebra under the operations of union and intersection. In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. ...
## Ordered pairs and Cartesian products 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. In mathematics, an ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element (the first and second elements are also known as left and right projections). ...
Formally, an ordered pair with **first coordinate** *a*, and **second coordinate** *b*, usually denoted by (*a*, *b*), is defined as the set {{*a*}, {*a*, *b*}}. 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 {a,b} with a total order. In mathematics and set theory, a total order, linear order, simple order, or (non-strict) ordering is a binary relation (here denoted by infix â‰¤) on some set X. The relation is transitive, antisymmetric, and total. ...
(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). In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ...
In mathematics, the real line is simply the set of real numbers. ...
If *A* and *B* are sets, then the **Cartesian product** (or simply **product**) is defined to be: In mathematics, the Cartesian product is a direct product of sets. ...
*A* × *B* = {(*a*,*b*) : *a* is in *A* and *b* is in *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*. We can extend this definition 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 products, but to do this we need a more recondite definition of the product. In mathematics, a tuple is a finite sequence of objects (a list of a limited number of objects). ...
In mathematics, the Cartesian product is a direct product of sets. ...
Cartesian products were first developed by René Descartes in the context of analytic geometry. If **R** denotes the set of all real numbers, then **R**^{2} := **R** × **R** represents the Euclidean plane and **R**^{3} := **R** × **R** × **R** represents three-dimensional Euclidean space. Descartes redirects here. ...
Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra. ...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...
Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
## Some important sets Note: In this section, *a*, *b*, and *c* are natural numbers, and r and s are real numbers. In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
- Natural numbers are used for counting. A blackboard bold capital
**N** () often represents this set. - Integers appear as solutions for
*x* in equations like *x* + *a* = *b*. A blackboard bold capital **Z** () often represents this set (from the German *Zahlen*, meaning *numbers*). - Rational numbers appear as solutions to equations like
*a* + *bx* = *c*. A blackboard bold capital **Q** () often represents this set (for *quotient*, because R is used for the set of real numbers). - Algebraic numbers appear as solutions to polynomial equations (with integer coefficients) and may involve radicals and certain other irrational numbers. A blackboard bold capital
**A** () or a **Q** with an overline () often represents this set. The overline denotes the operation of algebraic closure. - Real numbers 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 numbers, which cannot appear as solutions to polynomial equations with rational coefficients. A blackboard bold capital
**R** () often represents this set. - Complex numbers 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 has at least one root in this set. A blackboard bold capital **C** () 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). In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...
An example of blackboard bold letters. ...
The integers are commonly denoted by the above symbol. ...
In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...
In mathematics, a quotient is the end result of a division problem. ...
In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ...
In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...
In mathematics, an nth root of a number a is a number b, such that bn=a. ...
In mathematics, an irrational number is any real number that is not a rational number â€” that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. ...
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
In mathematics, a transcendental number is any complex number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ...
In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ...
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ...
In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ...
## Paradoxes 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 sets we want, but this view leads to inconsistencies. For any set *x* we can ask whether *x* is a member of itself. Define *Z* = {*x* : *x* is not a member of *x*}. 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*. 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. Part of the foundation of mathematics, Russells paradox (also known as Russells antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory of Frege leads to a contradiction. ...
The penalty is that one must take more care with one's development, as one must in any rigorous mathematical argument. 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. In fact, in the standard axiomatisation of set theory, there is no set of all sets. 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). Alternatively, one can make use of proper classes. 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. The exact resolution employed rarely makes an ultimate difference. In set theory, referring to the set of all sets typically leads to a paradox. ...
In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...
In mathematics, and particularly in applications to set theory and the foundations of mathematics, a universe or universal class (or if a set, universal set) is, roughly speaking, a class that is large enough to contain (in some sense) all of the sets that one may wish to use. ...
In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ...
W. V. Quine Willard Van Orman Quine (June 25, 1908 - December 25, 2000) was one of the most influential American philosophers and logicians of the 20th century. ...
In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica. ...
## See also The algebra of sets develops and describes the basic properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality (mathematics) and set inclusion. ...
This article or section is in need of attention from an expert on the subject. ...
Internal set theory (IST) is a mathematical theory of sets developed by Edward Nelson which provides an axiomatic basis for a portion of the non-standard analysis introduced by Abraham Robinson. ...
Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...
In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...
In mathematics, especially order theory, a partially ordered set (or poset) is a set equipped with a partial order relation. ...
## References - Halmos, P.R.,
*Naive Set Theory*, D. Van Nostrand Company, Princeton, NJ, 1960. Reprinted, Springer-Verlag, New York, NY, 1974, ISBN 0-387-90092-6. - Bourbaki, N.,
*Elements of the History of Mathematics*, John Meldrum (trans.), Springer-Velag, Berlin, Germany, 1994. - Devlin, K.J.,
*The Joy of Sets: Fundamentals of Contemporary Set Theory*, 2nd edition, Springer-Verlag, New York, NY, 1993. - 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. - Kelley, J.L.,
*General Topology*, Van Nostrand Reinhold, New York, NY, 1955. Paul Halmos Paul Richard Halmos (born March 3, 1916) is a Hungarian-born American mathematician who has done research in the fields of probability theory, statistics, operator theory, ergodic theory, and functional analysis (in particular Hilbert spaces). ...
Naive Set Theory is a mathematics textbook by Paul Halmos originally published in 1960. ...
This article is about the group of mathematicians named Nicolas Bourbaki. ...
Keith J. Devlin is an English mathematician and writer. ...
Jean van Heijenoort (prounounced highenort) (July 23, 1912, Creil France - March 29, 1986, Mexico City) was a pioneer historian of mathematical logic. ...
John Leroy Kelley (December 6, 1916 â€“ November 26, 1999) was an American mathematician at University of California, Berkeley who worked in general topology and functional analysis. ...
## External links ## Note **^** Concerning the origin of the term *naive set theory*, Jeff Miller has this to say: “*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).” [1] The term was later popularized by Paul Halmos' book, *Naive Set Theory* (1960). |