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This article is about the mathematical topic. For the book of the same name, see Naive Set Theory (book). See also Naive set theory for the mathematical topic Naive Set Theory is a Mathematics textbook by Paul Halmos

In the discussion of the foundations of mathematics, several set theories have been developed, of which naive set theory[1] is one. Foundations of mathematics is a term sometimes used for certain fields of Mathematics, such as Mathematical logic, Axiomatic set theory, Proof theory 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. 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 Venn diagrams or set diagrams are Diagrams that show all hypothetically possible Logical relations between a finite collection of sets (groups Boolean algebra (or Boolean logic) is a logical calculus of truth values, developed by George Boole in the late 1830s

Sets are of great importance in mathematics; in fact, in modern formal treatments, most mathematical objects (numbers, relations, functions, etc. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. This article sets out the set-theoretic notion of relation For a more elementary point of view see Binary relations and Triadic relations The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function ) 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.

Contents

Requirements

A naive theory is a non-formalized theory, that is to say a theory that uses a natural language to describe sets. In the Philosophy of language, a natural language (or ordinary language) is a Language that is spoken or written in phonemic-alphabetic or phonemically-related 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.

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. 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 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. The 19th century of the Common Era began on January 1, 1801 and ended on December 31, 1900, according to the Gregorian calendar Georg Ferdinand Ludwig Philipp Cantor ( – January 6 1918) was a German Mathematician, born in Russia. 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. A paradox is a true statement or group of statements that leads to a Contradiction or a situation which defies intuition; or inversely Part of the Foundations of mathematics, Russell's paradox (also known as Russell's antinomy) discovered by Bertrand Russell in 1901 showed that the The Berry paradox is a Self-referential Paradox arising from the expression "the smallest possible Integer not definable by a given number of 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). 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 Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common Some believe that Georg Cantor's set theory was not actually implicated in the paradoxes (this is a matter which continues to be discussed). Georg Ferdinand Ludwig Philipp Cantor ( – January 6 1918) was a German Mathematician, born in Russia. A paradox is a true statement or group of statements that leads to a Contradiction or a situation which defies intuition; or inversely 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. Friedrich Ludwig Gottlob Frege ( 8 November 1848, Wismar, Grand Duchy of Mecklenburg-Schwerin  &ndash 26 July 1925 Bertrand Arthur William Russell 3rd Earl Russell, OM, FRS (18 May 1872 – 2 February 1970 was a British Philosopher, Historian

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. Paul Richard Halmos ( March 3 1916 &mdash October 2 2006) was a Hungarian -born Jewish American Mathematician Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common 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.

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. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French Clearly, the set of even numbers is infinitely large; there is no requirement that a set be finite.

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 Greek alphabet (Ελληνικό αλφάβητο is a set of twenty-four letters that has been used to write the Greek language since the late 9th or early Epsilon (uppercase Ε, lowercase ε; Έψιλον is the fifth letter of the Greek alphabet, corresponding phonetically to a Close-mid front unrounded Giuseppe Peano ( August 27, 1858 &ndash April 20, 1932) was an Italian Mathematician, whose work was of exceptional ) The symbol ∉ is sometimes used to write x ∉ A, meaning "x is not in A".

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. Equality is the paradigmatic example of the more general concept of Equivalence relations on a set those binary relations which are reflexive, symmetric (See axiom of extensionality. In Axiomatic set theory and the branches of Logic, Mathematics, and Computer science that use it the axiom of extensionality, or axiom ) 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. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 If the sets A and B are equal, this is denoted symbolically as A = B (as usual).

We also allow for an empty set, often denoted Ø and sometimes {}: a set without any members at all. In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members Since a set is determined completely by its elements, there can only be one empty set. (See axiom of empty set. In Set theory, the axiom of empty set is one of the Axioms of Zermelo–Fraenkel set theory and one of the axioms of Kripke–Platek set theory ) Note that \empty \ne \{0\} \ne \{\empty\}.

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. 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 ) Note the following points:

(These are consequences of the definition of equality in the previous section. )

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 in several different ways

This notation is called set-builder notation (or "set comprehension", particularly in the context of Functional programming). In Set theory and its applications to Logic, Mathematics, and Computer science, set-builder notation (sometimes simply "set notation" In Computer science, functional programming is a Programming paradigm that treats Computation as the evaluation of mathematical functions and Some variants of set builder notation are:

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.

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 "\subsetneq" and "\supsetneq" 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. The President of the United States is the Head of state and Head of government of the United States and is the highest political official in United States by 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.

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. In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true 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). In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members A vacuous truth is a truth that is devoid of content because it asserts something about all members of a class that is empty or because it says "If  A  then 

The set of all subsets of a given set A is called the power set of A and is denoted by 2A or P(A); the "P" is sometimes in a fancy font. In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) If the set A has n elements, then P(A) will have 2n elements.

Universal sets and absolute complements

In certain contexts we may consider all sets under consideration as being subsets of some given universal set. In Mathematical logic, the universe of a structure (or model) is its domain. 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. In Mathematics, the real numbers may be described informally in several different ways 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).

Given a universal set U and a subset A of U, we may define the complement of A (in U) as

AC := {x ∈ U : x ∉ A}. In Discrete mathematics and predominantly in Set theory, a complement is a concept used in comparisons of sets to refer to the unique values of one set in relation

In other words, AC ("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, we may construct their union. In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets This is the set consisting of all objects which are elements of A or of B or of both (see axiom of union). 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 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. 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 It is denoted by A ∩ B.

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. In Discrete mathematics and predominantly in Set theory, a complement is a concept used in comparisons of sets to refer to the unique values of one set in relation It is written as A \ B or A − B. Symbolically, these are respectively

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)}. In Logic and/or Mathematics, logical conjunction or and is a two-place Logical operation that results in a value of true if both of In Logic and Mathematics, negation or not is an operation on Logical values for example the logical value of a Proposition

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.

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 {}. In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members

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 or Boolean lattice is a complemented distributive lattice.

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 distinguishable objects one of which is identified as the first coordinate (or the first entry

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

(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. In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a 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 = {(a,b) : a is in A and b is in B}. Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory.

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. In Mathematics, a tuple is a Sequence (also known as an "ordered list" of values called the components of the tuple It is even possible to define infinite Cartesian products, but to do this we need a more recondite definition of the product. Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory.

Cartesian products were first developed by René Descartes in the context of analytic geometry. Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry If R denotes the set of all real numbers, then R2 := R × R represents the Euclidean plane and R3 := R × R × R represents three-dimensional Euclidean space. In Mathematics, the real numbers may be described informally in several different ways Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria.

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 (also called counting number) can mean either an element of the set (the positive Integers or an In Mathematics, the real numbers may be described informally in several different ways

  1. Natural numbers are used for counting. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an A blackboard bold capital N (\mathbb{N}) often represents this set. Blackboard bold is a Typeface style often used for certain symbols in Mathematics and Physics texts in which certain lines of the symbol (usually vertical
  2. Integers appear as solutions for x in equations like x + a = b. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French A blackboard bold capital Z (\mathbb{Z}) often represents this set (from the German Zahlen, meaning numbers).
  3. Rational numbers appear as solutions to equations like a + bx = c. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions A blackboard bold capital Q (\mathbb{Q}) often represents this set (for quotient, because R is used for the set of real numbers). In Mathematics, a quotient is the result of a division. For example when dividing 6 by 3 the quotient is 2 while 6 is called the dividend, and 3 the
  4. Algebraic numbers appear as solutions to polynomial equations (with integer coefficients) and may involve radicals and certain other irrational numbers. In Mathematics, an algebraic number is a Complex number that is a root of a non-zero Polynomial in one variable with rational (or In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, an n th 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 — that is it is a number which cannot be expressed as a fraction A blackboard bold capital A (\mathbb{A}) or a Q with an overline (\overline{\mathbb{Q}}) often represents this set. The overline denotes the operation of algebraic closure. In Mathematics, particularly Abstract algebra, an algebraic closure of a field K is an Algebraic extension of K that is
  5. Real numbers represent the "real line" and include all numbers that can be approximated by rationals. In Mathematics, the real numbers may be described informally in several different ways These numbers may be rational or algebraic but may also be transcendental numbers, which cannot appear as solutions to polynomial equations with rational coefficients. In Mathematics, a transcendental number is a Complex number that is not algebraic, that is not a solution of a non-zero Polynomial equation A blackboard bold capital R (\mathbb{R}) often represents this set.
  6. Complex numbers are sums of a real and an imaginary number: r + si. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted 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 \mathbb{R} has at least one root in this set. In Mathematics, particularly Abstract algebra, an algebraic closure of a field K is an Algebraic extension of K that is This article is about the zeros of a function which should not be confused with the value at zero. A blackboard bold capital C (\mathbb{C}) often represents this set. Note that since a number r + si 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).

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 Foundations of mathematics, Russell's paradox (also known as Russell's antinomy) discovered by Bertrand Russell in 1901 showed that the

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 Set theory as usually formulated referring to the set of all sets typically leads to a Paradox. 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). In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematical logic, the universe of a structure (or model) is its domain. Alternatively, one can make use of proper classes. In Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously 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. Willard Van Orman Quine (June 25 1908 Akron, Ohio &ndash December 25 2000 (known to intimates as "Van" In Mathematical logic, New Foundations ( NF) is an Axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of The exact resolution employed rarely makes an ultimate difference.

See also

References

External links

Note

  1. ^ 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). Paul Richard Halmos ( March 3 1916 &mdash October 2 2006) was a Hungarian -born Jewish American Mathematician
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