**Naive set theory** is any of several theories of sets used in the discussion of the foundations of mathematics.^{ [3] } 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 (for example Venn diagrams and symbolic reasoning about their Boolean algebra), and suffices for the everyday use of set theory concepts in contemporary mathematics.^{ [4] }

- Method
- Paradoxes
- Cantor's theory
- Axiomatic theories
- Consistency
- Utility
- Sets, membership and equality
- Note on consistency
- Membership
- Equality
- Empty set
- Specifying sets
- Subsets
- Universal sets and absolute complements
- Unions, intersections, and relative complements
- Ordered pairs and Cartesian products
- Some important sets
- Paradoxes in early set theory
- See also
- Notes
- References
- External links

Sets are of great importance in mathematics; in modern formal treatments, most mathematical objects (numbers, 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.

A *naive theory* in the sense of "naive set theory" is a non-formalized theory, that is, a theory that uses a 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 was a naive set theory. It was created at the end of the 19th century by Georg Cantor as part of his study of infinite sets ^{ [5] } and developed by Gottlob Frege 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's theory and other informal systems.
- Decidedly inconsistent theories (whether axiomatic or not), such as a theory of Gottlob Frege
^{ [6] }that yielded Russell's paradox, and theories of Giuseppe Peano^{ [7] }and Richard Dedekind.

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.

Some believe that Georg Cantor'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 paradox ^{ [8] } and the Burali-Forti paradox,^{ [9] } and did not believe that they discredited his theory.^{ [10] } Cantor's paradox can actually be derived from the above (false) assumption—that any property *P*(*x*) may be used to form a set—using for *P*(*x*) "x is a cardinal number". 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 actually addressed when he presented his paradox, not necessarily a theory Cantor—who, as mentioned, was aware of several paradoxes—presumably had in mind.

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.

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^{[ citation needed ]} to refer to the set theories studied by Frege and Cantor, rather than to the informal counterparts of modern axiomatic set theory.

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 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.

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 definition of sets goes back to Georg Cantor. 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

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.

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, "ε", introduced by Giuseppe Peano in 1889 and is the first letter of the word ἐστί (means "is"). The symbol ∉ is often 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*. (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).

The empty set, often denoted Ø 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 Ø ≠ {Ø}, 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.^{ [11] }

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:

- The 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.)

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.

The notation {*x* : *P*(*x*)}, or sometimes {*x* | *P*(*x*)}, is used to denote the set containing all objects for which the condition P holds (known as defining a set *intensionally*). For example, {*x* : *x* ∈ **R**} denotes the set of real numbers, {*x* : *x* has blonde hair} 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). Some variants of set builder notation are:

- {
*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.

Given two sets *A* and *B*, *A* is a ** subset ** 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. 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 or ; the "*P*" is sometimes in a script font. If the set *A* has *n* elements, then will have elements.

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 numbers **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}:= {*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

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 ** 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 := {*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*)}.

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 is a Boolean algebra under the operations of union and intersection.

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.

Formally, an ordered pair with **first coordinate***a*, and **second coordinate***b*, usually denoted by (*a*, *b*), can be 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.

(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).

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*}.

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 products, but this requires a more recondite definition of the product.

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.

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 numbers, and *r* and *s* are real numbers.

- 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 (including ) and certain other irrational numbers. 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: . Here either or (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 has at least one root in this set. A blackboard bold capital
**C**() often represents this set. Note that since a number can be identified with a point in the plane, is basically "the same" as the Cartesian product ("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 ).

The unrestricted formation principle of sets referred to as the axiom schema of unrestricted comprehension,

*If**P**is a property, then there exists a set**Y*= {*x*:*P*(*x*)} (**false**),^{ [12] }

is the source of several early appearing paradoxes:

*Y*= {*x*:*x*is an ordinal} led, in the year 1897, to the Burali-Forti paradox, the first published antinomy.*Y*= {*x*:*x*is a cardinal} produced Cantor's paradox in 1897.^{ [8] }*Y*= {*x*: {} = {}} yielded**Cantor's second antinomy**in the year 1899.^{ [10] }Here the property*P*is true for all*x*, whatever*x*may be, so*Y*would be a universal set, containing everything.*Y*= {*x*:*x*∉*x*}, 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**,

*If**P**is a property, then for any set**X**there exists a set**Y*= {*x*∈*X*:*P*(*x*)},^{ [12] }

then all the above paradoxes disappear.^{ [12] } There is a corollary. With the axiom schema of separation as an axiom of the theory, it follows, as a theorem of the theory:

*The set of all sets does not exist*.

Or, more spectacularly (Halmos' phrasing^{ [13] }): There is no universe. *Proof*: Suppose that it exists and call it *U*. Now apply the axiom schema of separation with *X* = *U* and for *P*(*x*) use *x* ∉ *x*. This leads to Russell's paradox again. Hence *U* can't exist in this theory.^{ [12] }

Related to the above constructions is formation of the set

*Y*= {*x*: (*x*∈*x*) → {} ≠ {}}, where the statement following the implication certainly is false. It follows, from the definition of*Y*, using the usual inference rules (and some afterthought when reading the proof in the linked article below) both that*Y*∈*Y*→ {} ≠ {} and*Y*∈*Y*holds, hence {} ≠ {}. This is Curry's paradox.

It is (perhaps surprisingly) not the possibility of *x* ∈ *x* that is problematic. It is again the axiom schema of unrestricted comprehension allowing (*x* ∈ *x*) → {} ≠ {} for *P*(*x*). With the axiom schema of specification instead of unrestricted comprehension, the conclusion *Y* ∈ *Y* doesn't hold and, hence {} ≠ {} is not a logical consequence.

Nonetheless, the possibility of *x* ∈ *x* is often removed explicitly^{ [14] } or, e.g. in ZFC, implicitly,^{ [15] } by demanding the axiom of regularity to hold.^{ [15] } One consequence of it is

*There is no set**X**for which**X*∈*X*,

or, in other words, no set is an element of itself.^{ [16] }

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.^{ [12] } 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 of ZFC is incompatible with the conceivable **every set of reals is Lebesgue measurable **. The former implies the latter is false.

- ↑ "Earliest Known Uses of Some of the Words of Mathematics (S)". April 14, 2020.
- ↑ Halmos 1960,
*Naive Set Theory*. - ↑ Jeff Miller writes that
*naive set theory*(as opposed to 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. (1946). "The Philosophy of Bertrand Russell".*American Mathematical Monthly*.**53**(4): 210, and in a review by Laszlo Kalmar (Laszlo Kalmar (1946). "The Paradox of Kleene and Rosser".*Journal of Symbolic Logic*.**11**(4): 136.).^{ [1] }The term was later popularized in a book by Paul Halmos.^{ [2] } - ↑ Mac Lane, Saunders (1971), "Categorical algebra and set-theoretic foundations",
*Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967)*, Providence, RI: Amer. Math. Soc., pp. 231–240, MR 0282791 . "The working mathematicians usually thought in terms of a naive set theory (probably one more or less equivalent to ZF) ... a practical requirement [of any new foundational system] could be that this system could be used "naively" by mathematicians not sophisticated in foundational research" (p. 236). - ↑ Cantor 1874.
- ↑ Frege 1893 In Volume 2, Jena 1903. pp. 253-261 Frege discusses the antionomy in the afterword.
- ↑ Peano 1889 Axiom 52. chap. IV produces antinomies.
- 1 2 Letter from Cantor to David Hilbert on September 26, 1897, Meschkowski & Nilson 1991 p. 388.
- ↑ Letter from Cantor to Richard Dedekind on August 3, 1899, Meschkowski & Nilson 1991 p. 408.
- 1 2 Letters from Cantor to Richard Dedekind on August 3, 1899 and on August 30, 1899, Zermelo 1932 p. 448 (System aller denkbaren Klassen) and Meschkowski & Nilson 1991 p. 407. (There is no set of all sets.)
- ↑ Halmos 1974, p.
^{[ page needed ]}. - 1 2 3 4 5 Jech 2002, p. 4.
- ↑ Halmos 1974, Chapter 2.
- ↑ Halmos 1974, See discussion around Russell's paradox.
- 1 2 Jech 2002, Section 1.6.
- ↑ Jech 2002, p. 61.

In mathematics, the **axiom of regularity** 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 axiom reads:

In mathematics, the **cardinality** of a set is a measure of the "number of elements" of the set. For example, the set contains 3 elements, and therefore has a cardinality of 3. Beginning in the late 19th century, this concept was generalized to infinite sets, which allows one to distinguish between the different types of infinity, and to perform arithmetic on them. There are two approaches to cardinality: one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set is also called its **size**, when no confusion with other notions of size is possible.

In mathematics, the **empty set** is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set.

In mathematics, the **natural numbers** are those used for counting and ordering. In common mathematical terminology, words colloquially used for counting are "cardinal numbers", and words used for ordering are "ordinal numbers". The natural numbers can, at times, appear as a convenient set of codes, that is, as what linguists call nominal numbers, forgoing many or all of the properties of being a number in a mathematical sense. The set of natural numbers is often denoted by the symbol .

In mathematics, the **power set** of a set S is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory, the existence of the power set of any set is postulated by the axiom of power set. The powerset of S is variously denoted as P(*S*), 𝒫(*S*), *P*(*S*), , ℘(*S*), or 2^{S}. The notation 2^{S}, meaning the set of all functions from S to a given set of two elements, is used because the powerset of S can be identified with, equivalent to, or bijective to the set of all the functions from S to the given two elements set.

In mathematics, a **set** is a collection of elements. The elements that make up a set can be any kind of mathematical objects: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if and only if they have precisely the same elements.

**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 concerned with those that are relevant to mathematics as a whole.

In mathematical logic, **Russell's paradox**, is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains an unrestricted comprehension principle leads to contradictions. The paradox had already been discovered independently in 1899 by the German mathematician Ernst Zermelo. However, Zermelo did not publish the idea, which remained known only to David Hilbert, Edmund Husserl, and other academics at the University of Göttingen. At the end of the 1890s, Georg Cantor – considered the founder of modern set theory – had already realized that his theory would lead to a contradiction, which he told Hilbert and Richard Dedekind by letter.

In set theory, **Cantor's diagonal argument**, also called the **diagonalisation argument**, the **diagonal slash argument**, the **anti-diagonal argument**, the **diagonal method**, and **Cantor's diagonalization proof**, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. Such sets are now known as uncountable sets, and the size of infinite sets is now treated by the theory of cardinal numbers which Cantor began.

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 definable subclass of a set is a set.

In set theory, the **complement** of a set A, often denoted by *A*^{c}, are the elements not in A.

In set theory, **Zermelo–Fraenkel set theory**, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated **ZFC**, where C stands for "choice", and **ZF** refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded.

In set theory and its applications to logic, mathematics, and computer science, **set-builder notation** is a mathematical notation for describing a set by enumerating its elements, or stating the properties that its members must satisfy.

In mathematics, and particularly in set theory, category theory, type theory, and the foundations of mathematics, a **universe** is a collection that contains all the entities one wishes to consider in a given situation.

In elementary set theory, **Cantor's theorem** is a fundamental result which states that, for any set , the set of all subsets of has a strictly greater cardinality than itself. For finite sets, Cantor's theorem can be seen to be true by simple enumeration of the number of subsets. Counting the empty set as a subset, a set with members has a total of subsets, so that if then , and the theorem holds because for all non-negative integers.

In mathematics, an **element** of a set is any one of the distinct objects that belong to that set.

Constructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. The same first-order language with "" and "" of classical set theory is usually used, so this is not to be confused with a constructive types approach. On the other hand, some constructive theories are indeed motivated by their interpretability in type theories.

**General set theory** (**GST**) is George Boolos's (1998) name for a fragment of the axiomatic set theory Z. GST is sufficient for all mathematics not requiring infinite sets, and is the weakest known set theory whose theorems include the Peano axioms.

In mathematics, a **real number** is a value of a continuous quantity that can represent a distance along a line. The adjective *real* in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the real transcendental numbers, such as π (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more. The set of real numbers is denoted using the symbol **R** or and is sometimes called "the reals".

**S** is an axiomatic set theory set out by George Boolos in his 1989 article, "Iteration Again". **S**, a first-order theory, is two-sorted because its ontology includes “stages” as well as sets. Boolos designed **S** to embody his understanding of the “iterative conception of set“ and the associated iterative hierarchy. **S** has the important property that all axioms of Zermelo set theory *Z*, except the axiom of extensionality and the axiom of choice, are theorems of **S** or a slight modification thereof.

- Bourbaki, N.,
*Elements of the History of Mathematics*, John Meldrum (trans.), Springer-Verlag, Berlin, Germany, 1994. - Cantor, Georg (1874), "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen",
*J. Reine Angew. Math.*,**77**: 258–262, doi:10.1515/crll.1874.77.258, See also pdf version CS1 maint: postscript (link) - 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. - Frege, Gottlob (1893),
*Grundgesetze der Arithmetik*,**1**, Jena - Halmos, Paul (1960).
*Naive Set Theory*. Princeton, NJ: D. Van Nostrand Company.- Halmos, Paul (1974).
*Naive Set Theory*(Reprint ed.). New York: Springer-Verlag. ISBN 0-387-90092-6. - Halmos, Paul (2011).
*Naive Set Theory*(Paperback ed.). Mansfield Centre, CN: D. Van Nostrand Company. ISBN 978-1-61427-131-4.

- Halmos, Paul (1974).
- Jech, Thomas (2002).
*Set theory, third millennium edition (revised and expanded)*. Springer. ISBN 3-540-44085-2. - 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. ISBN 0-674-32449-8. - Meschkowski, Herbert; Nilson, Winfried (1991),
*Georg Cantor: Briefe. Edited by the authors.*, Berlin: Springer, ISBN 3-540-50621-7 - Peano, Giuseppe (1889),
*Arithmetices Principies nova Methoda exposita*, Turin - Zermelo, Ernst (1932),
*Georg Cantor: Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. Mit erläuternden Anmerkungen sowie mit Ergänzungen aus dem Briefwechsel Cantor-Dedekind. Edited by the author.*, Berlin: Springer

- Beginnings of set theory page at St. Andrews
- Earliest Known Uses of Some of the Words of Mathematics (S)

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