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Encyclopedia > Axiomatic set theory

In mathematics, axiomatic set theory is a rigorous reformulation of set theory in first-order logic created to address paradoxes in naive set theory. The basis of set theory was created principally by the German mathematician Georg Cantor at the end of the 19th century. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Look up Rigour in Wiktionary, the free dictionary. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... First-order logic (FOL) is a formal deductive system used by mathematicians, philosophers, linguists, and computer scientists. ... This article is about the mathematical topic. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845 â€“ January 6, 1918) was a German mathematician. ...

Initially controversial, set theory has come to play the role of a foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects (numbers, functions, etc.,) from all the traditional areas of mathematics (algebra, analysis, topology, etc.) in a single theory, and provides a standard set of axioms to prove or disprove them. At the same time the basic concepts of set theory are used throughout mathematics, the subject is pursued in its own right as a specialty by a comparatively small group of mathematicians and logicians. It should be mentioned that there are also mathematicians using and promoting different approaches to the foundations of mathematics. 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 the branch of mathematics. ... Analysis has its beginnings in the rigorous formulation of calculus. ... A MÃ¶bius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... Logic (from Classical Greek Î»ÏŒÎ³Î¿Ï‚ logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. ...

## The origins of rigorous set theory GA_googleFillSlot("encyclopedia_square");

The important idea of Cantor's, which got set theory going as a new field of study, was to set forth the definition, two sets A and B are said to have the same number of members (the same cardinality) if and only if there is a way of pairing off members of A exhaustively with members of B (i.e. each and every member of A has a corresponding member of B that it is "paired off" with so that A and B are in one-to-one correspondence, and that each and every member of both A and B has a partner). Resulting from this definition, the set $mathbb{N}$ of natural numbers has the same cardinality as the set $mathbb{Q}$ of rational numbers (they are both said to have cardinality $aleph_0$, that is, they are both countably infinite), even though $mathbb{N}$ is a proper subset of $mathbb{Q}$. On the other hand, the set $mathbb{R}$ of real numbers does not have the same cardinality as $mathbb{N}$ or $mathbb{Q}$, but a larger one (it is said to have cardinality $beth_1$, that is, it is not countably infinite). Cantor gave two proofs that $mathbb{R}$ is not countable, and the second of these, using what is known as the diagonal construction, has been extraordinarily influential and has had many applications in logic and mathematics. Image File history File links Broom_icon. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality â€“ one which compares sets directly using bijections and injections, and another which uses cardinal numbers. ... You may be looking for an Injective function, in which (f(a)=f(b)) -> a=b, or a Bijection function, which is both injective and surjective (ie. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In the branch of mathematics known as set theory, aleph usually refers to a series of numbers used to represent the cardinality (or size) of infinite sets. ... In mathematics, a countable set is a set with the same cardinality (i. ... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X... In mathematics, the Hebrew letter (aleph) with various subscripts represents various infinite cardinal numbers (see aleph number). ... Cantors diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. ...

Cantor constructed infinite hierarchies of infinite sets, the ordinal and cardinal numbers. This was controversial in his day, with the opposition led by the finitist Leopold Kronecker, but there is no significant disagreement among mathematicians today that Cantor had the right idea. Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. ... In the philosophy of mathematics, finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps. ... Leopold Kronecker Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the integers; all else is the work of man (Bell 1986, p. ...

## Axioms for set theory

All axiomatic set theories consist of nonlogical axioms added to first order logic and some postulated universe of discourse. An important feature of ZFC is that every object that it deals with (every individual in the universe of discourse of ZFC) is a set. In particular, every element of a set is itself a set. Hence all familiar mathematical objects, such as numbers, must be defined as sets of a certain kind. First-order predicate calculus or first-order logic (FOL) is a theory in symbolic logic that permits the formulation of quantified statements such as there is at least one X such that. ... The term universe of discourse generally refers to the entire set of terms used in a specific discourse, i. ... The term universe of discourse generally refers to the entire set of terms used in a specific discourse, i. ...

The ten axioms of ZFC are listed below. (Strictly speaking, the ZFC axioms are just strings of logical symbols. What follows is only an attempt to express the intended meaning of these axioms in English.) Note that Separation and Replacement are both axiom schemata. Each axiom has its own article giving additional information. In symbolic logic, it is sometimes inconvenient or impossible to express an axiomatic system in a finite number of axioms. ...

1. Axiom of extensionality: Two sets are the same if and only if they have the same elements.
2. Axiom of empty set: There is a set with no elements. We will use {} to denote this empty set.
3. Axiom of pairing: If x, y are sets, then so is {x,y}, a set containing x and y as its only elements.
4. Axiom of union: Every set has a union. That is, for any set x there is a set y whose elements are precisely the elements of the elements of x.
5. Axiom of infinity: There exists a set x, one of whose members is {}, such that whenever y is in x, so is y U {y}.
6. Axiom schema of separation (or subset axiom): Given any set y and any first order logic formula P(x), where x is a free variable, there is a subset of y containing precisely those elements x for which P(x) comes out true.
7. Axiom schema of replacement: Roughly, if the domain of a function is a set, its range is also a set. More precisely, let w be some set with typical member x, and let P be a binary predicate. Then if P(x,y) and P(x,z) imply y = z (in which case P is a functional predicate and a mapping), there exists a set containing the image of w under P.
8. Axiom of power set: Every set has a power set. That is, for any set x there exists a set y, such that the elements of y are precisely the subsets of x.
9. Axiom of regularity (or axiom of foundation): Every non-empty set x contains some element y such that x and y are disjoint sets.
10. Axiom of choice: (Zermelo's version) Given a set x of mutually disjoint nonempty sets, there is a set y (a choice set for x) containing exactly one element from each member of x.

Extensionality lays down the the fundamental property governing sets and their members. Regularity rules out sets that are not wellfounded. Null Set and Infinity each assert the unconditional existence of a set. Given any set, Union and Power Set each assure the existence of an additional set. Given a set, Separation (Replacement) assures the existence of another set corresponding to each monadic (dyadic functional) formula. Given any set having the requisite structure, Choice assures the existence of a choice set. (Because the existence of a choice set for a finite set is ZF theorem, Choice matters only for infinite sets.) Given any two sets, Pairing assures the existence of a third set. 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 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. ... 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. ... 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. ... 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 infinity is one of the axioms of Zermelo-Fraenkel set theory. ... 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. ... First-order predicate calculus or first-order logic (FOL) is a theory in symbolic logic that permits the formulation of quantified statements such as there is at least one X such that. ... In mathematics and in the sciences, a formula (plural: formulae, formulÃ¦ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ... In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation for a place or places in an expression, into which some definite substitution may take place, or with respect to which some operation (summation or quantification, to give two... â€œSupersetâ€ redirects here. ... 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. ... In mathematics, the domain of a function is the set of all input values to the function. ... This article is about functions in mathematics. ... In mathematics, the range of a function is the set of all output values produced by that function. ... In mathematics, an n-ary relation (or often simply relation) is a generalization of binary relations such as = and < which occur in statements such as 5 < 6 or 2 + 2 = 4. It is the fundamental notion in the relational model for databases. ... In formal logic and related branches of mathematics, a functional predicate, or function symbol, is a logical symbol that may be applied to an object term to produce another object term. ... The word mapping has several senses: In mathematics and related technical fields, it is some kind of function: see map (mathematics). ... In mathematics, the image of an element x in a set X under the function f : X â†’ Y, denoted by f(x), is the unique y in Y that is associated with x. ... In mathematics, the axiom of power set is one of the Zermelo-Fraenkel axioms of axiomatic set theory. ... 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. ... The axiom of regularity (also known as the axiom of foundation) is one of the axioms of Zermelo-Fraenkel set theory. ... In mathematics, two sets are said to be disjoint if they have no element in common. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ... In mathematics, a well-founded (or wellfounded) relation is a set-like binary relation R on a class X where every non-empty subset of X has an R-minimal element; that is, where for every non-empty subset S of X, there is an element m of S such... In formal logic and related branches of mathematics, a functional predicate, or function symbol, is a logical symbol that may be applied to an object term to produce another object term. ... In mathematics and in the sciences, a formula (plural: formulae, formulÃ¦ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ... In set theory, an infinite set is a set that is not a finite set. ...

The above axioms are not all independent:

• Every instance of Separation can be derived from an instance of Replacement. Hence adding Replacement to Z makes Separation redundant;
• Empty Set follows from Separation when P(x) is always false;
• Pairing can be derived from the other axioms.

Choice and Regularity remain controversial for a minority of mathematicians. Axiomatic set theories closely related to ZFC include Von Neumann-Bernays-Gödel set theory (NBG), the Kripke-Platek set theory (KP), Kripke-Platek set theory with urelements (KPU), and Morse-Kelley set theory. Axiomatic set theories that differ more radically from ZFC include New Foundations, Scott-Potter set theory, and systems of positive set theory. In foundations of mathematics, Von Neumann-Bernays-GÃ¶del set theory (NBG) is an axiom system for set theory designed to yield the same results as Zermelo-Fraenkel set theory, together with the axiom of choice (ZFC), but with only a finite number of axioms, that is without axiom schemas. ... The Kripke-Platek axioms of set theory (KP) are a system of axioms of axiomatic set theory. ... The Kripke-Platek set theory with urelements (KPU) is an axiom system for set theory with urelements that is considerably weaker than the familiar system ZF. // Preliminaries The usual way of stating the axioms presumes a two sorted first order language with a single binary relation symbol . ... Morse-Keylley set theory (MK) is another axiomatization of set theory. ... 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. ... An approach to the foundations of mathematics that is of relatively recent origin, Scottâ€“Potter set theory is a collection of nested axiomatic set theories set out by the philosopher Michael Potter, building on earlier work by the mathematician Dana Scott. ... In mathematical logic, positive set theory is an alternative set theory consisting of the following axioms: The axiom of extensionality: . The axiom of infinity: the von Neumann ordinal exists. ...

## Independence in ZFC

Many important statements are independent of ZFC, see the list of statements undecidable in ZFC. The independence is usually proved by forcing, that is, it is shown that every countable transitive model of ZFC (plus, occasionally, large cardinal axioms) can be expanded to satisfy the statement in question, and (through a different expansion) its negation. An independence proof by forcing automatically proves independence from arithmetical statements, other concrete statements, and large cardinal axioms. Some statements independent of ZFC can be proven to hold in particular inner models, such as in the constructible universe. However, some statements that are true about constructible sets are not consistent with hypothesised large cardinal axioms. In mathematical logic, a statement S is independent of a theory T if it is impossible to prove S from T and it is impossible to prove not S from T. Many interesting statements in set theory are independent of ZF. It is possible for the statement S is independent... The following is a list of mathematical statements that are undecidable in ZFC (the Zermelo-Fraenkel axioms plus the axiom of choice), assuming that ZFC is consistent. ... In axiomatic set theory, forcing is a technique, invented by Paul Cohen, for proving consistency and independence results with respect to the Zermelo-Fraenkel axioms. ... In mathematical logic, suppose T is a theory in the language . If M is a model of describing a set theory and N is a class of M such that is a model of T then we say that N is an inner model of T (in M). ... In mathematics, the constructible universe (or GÃ¶dels constructible universe), denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. ...

Here are some statements whose independence is provable by forcing:

Notes: In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... In mathematics, and particularly in axiomatic set theory, (diamondsuit or diamond) is a certain family of combinatorial principles. ... Suslins problem in mathematics is a question about orders posed by M. Suslin in the early 1920s. ... In axiomatic set theory, Martins axiom, named after Donald A. Martin, is a statement which is independent of the usual axioms of ZFC set theory. ... The axiom of constructibility is a possible axiom for set theory in mathematics. ...

1. Consistency of V=L is not provable by forcing, but is provable through inner models: every model of ZF can be trimmed to be a model of ZFC+V=L.
2. The Diamond Principle implies the Continuum Hypothesis and the negation of the Suslin Hypothesis.
3. Martin's axiom plus the negation of the Continuum Hypothesis implies the Suslin Hypothesis.
4. The constructible universe satisfies the Generalized Continuum Hypothesis, the Diamond Principle, Martin's Axiom and the Kurepa Hypothesis.

A variation on the method of forcing can also be used to demonstrate the consistency and unprovability of the axiom of choice, i.e., that the axiom of choice is independent of ZF. The consistency of choice can be (relatively) easily verified by proving the inner model L satisfies choice (thus every model of ZF contains a submodel of ZFC hence Con(ZF) implies Con(ZFC)). Since forcing preserves choice we cannot directly produce a model contradicting choice from a model satisfying choice. However, we can use forcing to create a model which contains a suitable submodel, namely one which satisfies ZF but not C. In axiomatic set theory, forcing is a technique, invented by Paul Cohen, for proving consistency and independence results with respect to the Zermelo-Fraenkel axioms. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...

Forcing is perhaps the most useful method of proving independence results but not the only method. In particular Godel's 2nd incompleteness theorem which asserts that no sufficiently complex recursively axiomatizable system can prove its own consistency can be used to prove independence results. In this approach it is demonstrated that a particular statement in set theory can be used to prove the existence of a set model of ZFC and thereby demonstrate the consistency of ZFC. Since we know that Con(ZFC) (the sentence asserting the consistency of ZFC in the language of set theory) is unprovable in ZFC no statement allowing such a proof can itself be provable in ZFC. For instance this method can be used to demonstrate the existence of large cardinals is not provable in ZFC (but it is essentially impossible to show they are consistent). In mathematical logic, GÃ¶dels incompleteness theorems are two celebrated theorems proven by Kurt GÃ¶del in 1931. ... In mathematics, a cardinal is called a large cardinal if it belongs to a class of cardinals, the existence of which provably cannot be proved within the standard axiomatic set theory ZFC, if one assumes ZFC itself is consistent. ...

## Set theory (ZFC) foundations for mathematics

Main article: Set-theoretic definition of natural numbers

From these initial axioms for sets one can construct all other mathematical concepts and objects: number (discrete and continuous), order, relation, function , etc. Several ways have been proposed to define the natural numbers using set theory. ... For other uses, see Number (disambiguation). ... 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. ... In mathematics, the word continuum sometimes denotes the real line. ... Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ... In mathematics, an n-ary relation (or often simply relation) is a generalization of binary relations such as = and < which occur in statements such as 5 < 6 or 2 + 2 = 4. It is the fundamental notion in the relational model for databases. ... This article is about functions in mathematics. ...

For example, while the elements of a set have no intrinsic ordering it is possible to construct models of ordered lists. The essential step is to be able to model the ordered pair (a, b) which represents the pairing of two objects in this order. The defining property of an ordered pair is that (a, b) = (c, d) if and only if a = c and b = d. The approach is basically to specify the two elements and additionally note which one is the first using the construction: In mathematics, a tuple is a finite sequence of objects (a list of a limited number of objects). ... In mathematics, an ordered pair is a collection of two not necessarily distinct objects, one of which is distinguished as the first coordinate (or first entry or left projection) and the other as the second coordinate (second entry, right projection). ... $( a, b ) = lbrace lbrace a, b rbrace, lbrace a rbrace rbrace,$

Ordered lists of greater length can be constructed inductively: Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ... begin{align} (a, b, c) & = ( (a, b), c ) (a, b, c, d) & = ( (a, b, c), d ) vdots end{align}

As another example, there is a construction for the natural numbers, due to von Neumann. We require to produce an infinite sequence of distinct sets with a successor relation as a model for the Peano axioms. This provides a canonical representation for the number N as being a particular choice of set containing precisely N distinct elements. We proceed inductively: For other persons named John Neumann, see John Neumann (disambiguation). ... In mathematics, the Peano axioms (or Peano postulates) are a set of second-order axioms proposed by Giuseppe Peano which determine the theory of the natural numbers. ... Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ... begin{align} 0 & = lbrace rbrace 1 & = lbrace 0 rbrace & =,, & lbrace lbrace rbrace rbrace 2 & = lbrace 0, 1 rbrace & =,, & lbrace lbrace rbrace, lbrace lbrace rbrace rbrace rbrace 3 & = lbrace 0, 1, 2 rbrace & =,, & lbrace lbrace rbrace, lbrace lbrace rbrace rbrace, lbrace lbrace rbrace, lbrace lbrace rbrace rbrace rbrace rbrace vdots end{align}

At each stage we construct a new set with N elements as being the set containing the (already defined) elements 0, 1, 2, ..., N-1. More formally, at each step the successor of N is N  ∪  { N }. Remarkably this produces a suitable model for the entire collection of natural numbers, from the barest of materials. Although each term in this sequence can be proved to exist using Separation, Pairing and Union, we cannot prove that there is a set containing all such terms without assuming the Axiom of Infinity.

The original set theoretical definition of the natural numbers defined each natural number n as the set of all sets with n elements (this can be managed without the apparent circularity of this brief summary). This definition (due to Frege and Russell) does not work in ZFC because the collections involved are too large to be sets. However, this approach does work in New Foundations and subsystems of NF known to be consistent. 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. ... Russell is a Scottish or French name derived from the colour red or from the fox animal: // Spanish - Quesada French - Roussel Italian - Rufino Latin - Rufus American - Rusty Notable people with the surname Russell include: William Russell (disambiguation page) Members of this family have held the title of Earl of Bedford... 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. ... 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. ...

Since relations, and most specifically functions, are defined to be sets of ordered pairs, and there are well-known constructions progressively building up the integers, rationals, reals, and complex numbers from sets of the natural numbers, we are able to model essentially all of the usual infrastructure of daily mathematical practice. 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... This article is about functions in mathematics. ... 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, there are a number of ways of defining the real number system as an ordered field. ... 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, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...

## Well-foundedness and hypersets

In 1917 Dimitri Mirimanoff (also spelled Dmitry Mirimanov) introduced the concept of well-foundedness: In mathematics, a binary relation, R, is well-founded (or wellfounded) on a class (set theory) X iff every non-empty subset of X has an R-minimal element; that is, for every non-empty subset S of X, there is an element m of S such that for every...

a set, x0, is well founded if and only if it has no infinite descending membership sequence: $cdots in x_2 in x_1 in x_0.$

In ZFC, there is no infinite descending ∈-sequence by the axiom of regularity (for a proof see Axiom of regularity). In fact, the axiom of regularity is often called the foundation axiom since it can be proved within ZFC- (that is, ZFC without the axiom of regularity) that well-foundedness implies regularity. â†” â‡” â‰¡ logical symbols representing iff. ... The axiom of regularity (also known as the axiom of foundation) is one of the axioms of Zermelo-Fraenkel set theory. ... The axiom of regularity (also known as the axiom of foundation) is one of the axioms of Zermelo-Fraenkel set theory. ...

In variants of ZFC without the axiom of regularity, the possibility of non-well-founded sets arises. When working in such a system, a set that is not necessarily well founded is called a hyperset. Clearly, if AA, then A is a non-well-founded hyperset. The axiom of regularity (also known as the axiom of foundation) is one of the axioms of Zermelo-Fraenkel set theory. ... Non-well founded set theories are variants of axiomatic set theory which allow sets to contain themselves and otherwise violate the rule of well-foundedness. ... Non-well founded set theories are variants of axiomatic set theory which allow sets to contain themselves and otherwise violate the rule of well-foundedness. ...

The theory of hypersets has been applied in computer science (process algebra and final semantics), linguistics (situation theory), and philosophy (work on the Liar Paradox). In the first half of the 20th century, various formalisms were proposed to capture the informal concept of computable function, μ-recursive functions, Turing Machines and the λ-calculus possibly being the most well-known examples today. ... // In philosophy and logic, the liar paradox encompasses paradoxical statements such as This sentence is false. ...

Three distinct anti-foundation axioms are well known:

1. AFA (‘Anti-Foundation Axiom’) — due to M. Forti and F. Honsell;
2. FAFA (‘Finsler’s AFA’) — due to P. Finsler;
3. SAFA (‘Scott’s AFA’) — due to Dana Scott.

The first of these, AFA, is based on accessible pointed graphs (apg) and states that two hypersets are equal if and only if they can be pictured by the same apg. Within this framework, it can be shown that the so-called Quine atom, formally defined by Q={Q}, exists and is unique. Dana Stewart Scott (born 1932) is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, California. ... 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. ...

It is worth emphasizing that hyperset theory is an extension of classical set theory rather than a replacement: the well-founded sets within a hyperset domain conform to classical set theory. The kinds of non-well-foundedness found in New Foundations or positive set theory (or more generally any set theory with a universal set which is an element of itself) are rather different. 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. ... In mathematical logic, positive set theory is an alternative set theory consisting of the following axioms: The axiom of extensionality: . The axiom of infinity: the von Neumann ordinal exists. ... Results from FactBites:

 Set theory (2182 words) Bolzano gave examples to show that, unlike for finite sets, the elements of an infinite set could be put in 1-1 correspondence with elements of one of its proper subsets. By this stage, however, set theory was beginning to have a major impact on other areas of mathematics. Analysis needed the set theory of Cantor, it could not afford to limit itself to intuitionist style mathematics in the spirit of Kronecker.
 Axiomatic set theory (2759 words) Initially controversial, set theory has come to play the role of a foundational theory in modern mathematics, in the sense of a theory invoked to justify assumptions made in mathematics concerning the existence of mathematical objects (such as numbers or functions) and their properties. Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. The important idea of Cantor's, which got set theory going as a new field of study, was to define two sets A and B to have the same number of members (the same cardinality) when there is a way of pairing off members of A exhaustively with members of B.
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