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This page is a list of articles related to set theory.

Articles on individual set theory topics[link]

Lists related to set theory[link]

• List of large cardinal properties
• List of properties of sets of reals

Set theorists[link]

Societies and organizations[link]

• Association for Symbolic Logic
• The Cabal

Look up list in Wiktionary, the free dictionary.

List or lists may refer to:

Enumerations[link]

• Any enumeration of a set of items
• A list of comma separated values, a common way of listing in everyday life and computing. (British usage: Comma-separated values)
• A mailing list
• An electronic mailing list
• An electoral list
• List (computing)
• Task lists, a list of tasks to be completed
• Listicle, an article in list format
• Watchlist
• Ranker

Other meanings[link]

• List (watercraft), the leaning to either port or starboard of a ship
• List, an alternative term for roll in flight dynamics
• Lists (jousting), a roped-off enclosure where medieval knights jousted
• SC Germania List, German rugby union club
• List College, an undergraduate division of the Jewish Theological Seminary of America

People[link]

• Friedrich List, German economist
• Guido von List, Austrian Armanist
• John List, murderer
• Liesbeth List, Dutch singer
• Maximilian List, SS officer and Commandant of Lager Sylt, a labour camp on Alderney
• Paul List, chess player
• Peyton List, the name of two American actresses:
  • Peyton List (actress born 1986), American actress in the TV series Mad Men
  • Peyton List (actress born 1998), American actress in the Disney Channel TV series Jessie
• Siegmund Wilhelm List, WWII German field marshal

Places[link]

• List, Schleswig-Holstein, the northernmost village in Germany, on the island Sylt

Work titles[link]

The List may refer to:

• The List (magazine), a Scottish magazine
• The List (South Park), an episode of the television show South Park
• The List (The X-Files), an episode of the television show The X-Files
• The List (2007 film), a film based on the novel by Robert Whitlow
• The List (album), an album by Rosanne Cash

See also[link]

• Liszt (surname)
• Listing (disambiguation)

This disambiguation page lists articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article.

A Venn diagram illustrating the intersection of two sets.

Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.

The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.

Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.

History[link]

Georg Cantor

Mathematical topics typically emerge and evolve through interactions among many researchers. Set theory, however, was founded by a single paper in 1874 by Georg Cantor: "On a Characteristic Property of All Real Algebraic Numbers".^[1][2]

Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, mathematicians had struggled with the concept of infinity. Especially notable is the work of Bernard Bolzano in the first half of the 19th century.^[3] Modern understanding of infinity began in 1867-71, with Cantor's work on number theory. An 1872 meeting between Cantor and Richard Dedekind influenced Cantor's thinking and culminated in Cantor's 1874 paper.

Cantor's work initially polarized the mathematicians of his day. While Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. Cantorian set theory eventually became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, his proof that there are more real numbers than integers, and the "infinity of infinities" ("Cantor's paradise") resulting from the power set operation. This utility of set theory led to the article "Mengenlehre" contributed in 1898 by Arthur Schoenflies to Klein's encyclopedia.

The next wave of excitement in set theory came around 1900, when it was discovered that Cantorian set theory gave rise to several contradictions, called antinomies or paradoxes. Bertrand Russell and Ernst Zermelo independently found the simplest and best known paradox, now called Russell's paradox: consider "the set of all sets that are not members of themselves", which leads to a contradiction since it must be a member of itself, and not a member of itself. In 1899 Cantor had himself posed the question "What is the cardinal number of the set of all sets?", and obtained a related paradox. Russell used his paradox as a theme in his 1903 review of continental mathematics in his Principles of Mathematics.

The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment. The work of Zermelo in 1908 and Abraham Fraenkel in 1922 resulted in the set of axioms ZFC, which became the canonical^{[dubious – discuss]} axioms for set theory. The work of analysts such as Henri Lebesgue demonstrated the great mathematical utility of set theory, which has since become woven into the fabric of modern mathematics. Set theory is commonly used as a foundational system, although in some areas category theory is thought to be a preferred foundation.

Basic concepts[link]

Set theory begins with a fundamental binary relation between an object o and a set A. If o is a member (or element) of A, write o ∈ A. Since sets are objects, the membership relation can relate sets as well.

A derived binary relation between two sets is the subset relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, denoted A ⊆ B. For example, {1,2} is a subset of {1,2,3} , but {1,4} is not. From this definition, it is clear that a set is a subset of itself; in cases where one wishes to avoid this, the term proper subset is defined to exclude this possibility.

Just as arithmetic features binary operations on numbers, set theory features binary operations on sets. The:

• Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of A, or B, or both. The union of {1, 2, 3} and {2, 3, 4} is the set {1, 2, 3, 4} .
• Intersection of the sets A and B, denoted A ∩ B, is the set of all objects that are members of both A and B. The intersection of {1, 2, 3} and {2, 3, 4} is the set {2, 3} .
• Set difference of U and A, denoted U \ A is the set of all members of U that are not members of A. The set difference {1,2,3} \ {2,3,4} is {1} , while, conversely, the set difference {2,3,4} \ {1,2,3} is {4} . When A is a subset of U, the set difference U \ A is also called the complement of A in U. In this case, if the choice of U is clear from the context, the notation A^c is sometimes used instead of U \ A, particularly if U is a universal set as in the study of Venn diagrams.
• Symmetric difference of sets A and B is the set of all objects that are a member of exactly one of A and B (elements which are in one of the sets, but not in both). For instance, for the sets {1,2,3} and {2,3,4} , the symmetric difference set is {1,4} . It is the set difference of the union and the intersection, (A ∪ B) \ (A ∩ B).
• Cartesian product of A and B, denoted A × B, is the set whose members are all possible ordered pairs (a,b) where a is a member of A and b is a member of B. The cartesian product of {1, 2} and {red, white} is {(1, red), (1, white), (2, red), (2, white)}.
• Power set of a set A is the set whose members are all possible subsets of A. For example, the power set of {1, 2} is { {}, {1}, {2}, {1,2} } .

Some basic sets of central importance are the empty set (the unique set containing no elements), the set of natural numbers, and the set of real numbers.

Some ontology[link]

An initial segment of the von Neumann hierarchy.

A set is pure if all of its members are sets, all members of its members are sets, and so on. For example, the set {{}} containing only the empty set is a nonempty pure set. In modern set theory, it is common to restrict attention to the von Neumann universe of pure sets, and many systems of axiomatic set theory are designed to axiomatize the pure sets only. There are many technical advantages to this restriction, and little generality is lost, since essentially all mathematical concepts can be modeled by pure sets. Sets in the von Neumann universe are organized into a cumulative hierarchy, based on how deeply their members, members of members, etc. are nested. Each set in this hierarchy is assigned (by transfinite recursion) an ordinal number α, known as its rank. The rank of a pure set X is defined to be one more than the least upper bound of the ranks of all members of X. For example, the empty set is assigned rank 0, while the set {{}} containing only the empty set is assigned rank 1. For each ordinal α, the set V_α is defined to consist of all pure sets with rank less than α. The entire von Neumann universe is denoted V.

Axiomatic set theory[link]

Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using Venn diagrams. The intuitive approach tacitly assumes that a set may be formed from the class of all objects satisfying any particular defining condition. This assumption gives rise to paradoxes, the simplest and best known of which are Russell's paradox and the Burali-Forti paradox. Axiomatic set theory was originally devised to rid set theory of such paradoxes.^[4]

The most widely studied systems of axiomatic set theory imply that all sets form a cumulative hierarchy. Such systems come in two flavors, those whose ontology consists of:

• Sets alone. This includes the most common axiomatic set theory, Zermelo–Fraenkel set theory (ZFC), which includes the axiom of choice. Fragments of ZFC include:
  • Zermelo set theory, which replaces the axiom schema of replacement with that of separation;
  • General set theory, a small fragment of Zermelo set theory sufficient for the Peano axioms and finite sets;
  • Kripke-Platek set theory, which omits the axioms of infinity, powerset, and choice, and weakens the axiom schemata of separation and replacement.
• Sets and proper classes. This includes Von Neumann-Bernays-Gödel set theory, which has the same strength as ZFC for theorems about sets alone, and Morse-Kelley set theory, which is stronger than ZFC.

The above systems can be modified to allow urelements, objects that can be members of sets but that are not themselves sets and do not have any members.

The systems of New Foundations NFU (allowing urelements) and NF (lacking them) are not based on a cumulative hierarchy. NF and NFU include a "set of everything," relative to which every set has a complement. In these systems urelements matter, because NF, but not NFU, produces sets for which the axiom of choice does not hold.

Systems of constructive set theory, such as CST, CZF, and IZF, embed their set axioms in intuitionistic logic instead of first order logic. Yet other systems accept standard first order logic but feature a nonstandard membership relation. These include rough set theory and fuzzy set theory, in which the value of an atomic formula embodying the membership relation is not simply True or False. The Boolean-valued models of ZFC are a related subject.

An enrichment of ZFC called Internal Set Theory was proposed by Edward Nelson in 1977.

Applications[link]

Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse as graphs, manifolds, rings, and vector spaces can all be defined as sets satisfying various (axiomatic) properties. Equivalence and order relations are ubiquitous in mathematics, and the theory of mathematical relations can be described in set theory.

Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of Principia Mathematica, it has been claimed that most or even all mathematical theorems can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using first or second order logic. For example, properties of the natural and real numbers can be derived within set theory, as each number system can be identified with a set of equivalence classes under a suitable equivalence relation whose field is some infinite set.

Set theory as a foundation for mathematical analysis, topology, abstract algebra, and discrete mathematics is likewise uncontroversial; mathematicians accept that (in principle) theorems in these areas can be derived from the relevant definitions and the axioms of set theory. Few full derivations of complex mathematical theorems from set theory have been formally verified, however, because such formal derivations are often much longer than the natural language proofs mathematicians commonly present. One verification project, Metamath, includes derivations of more than 10,000 theorems starting from the ZFC axioms and using first order logic.

Areas of study[link]

Set theory is a major area of research in mathematics, with many interrelated subfields.

Combinatorial set theory[link]

Combinatorial set theory concerns extensions of finite combinatorics to infinite sets. This includes the study of cardinal arithmetic and the study of extensions of Ramsey's theorem such as the Erdős–Rado theorem.

Descriptive set theory[link]

Descriptive set theory is the study of subsets of the real line and, more generally, subsets of Polish spaces. It begins with the study of pointclasses in the Borel hierarchy and extends to the study of more complex hierarchies such as the projective hierarchy and the Wadge hierarchy. Many properties of Borel sets can be established in ZFC, but proving these properties hold for more complicated sets requires additional axioms related to determinacy and large cardinals.

The field of effective descriptive set theory is between set theory and recursion theory. It includes the study of lightface pointclasses, and is closely related to hyperarithmetical theory. In many cases, results of classical descriptive set theory have effective versions; in some cases, new results are obtained by proving the effective version first and then extending ("relativizing") it to make it more broadly applicable.

A recent area of research concerns Borel equivalence relations and more complicated definable equivalence relations. This has important applications to the study of invariants in many fields of mathematics.

Fuzzy set theory[link]

In set theory as Cantor defined and Zermelo and Fraenkel axiomatized, an object is either a member of a set or not. In fuzzy set theory this condition was relaxed by Lotfi A. Zadeh so an object has a degree of membership in a set, as number between 0 and 1. For example, the degree of membership of a person in the set of "tall people" is more flexible than a simple yes or no answer and can be a real number such as 0.75.

Inner model theory[link]

An inner model of Zermelo–Fraenkel set theory (ZF) is a transitive class that includes all the ordinals and satisfies all the axioms of ZF. The canonical example is the constructible universe L developed by Gödel. One reason that the study of inner models is of interest is that it can be used to prove consistency results. For example, it can be shown that regardless whether a model V of ZF satisfies the continuum hypothesis or the axiom of choice, the inner model L constructed inside the original model will satisfy both the generalized continuum hypothesis and the axiom of choice. Thus the assumption that ZF is consistent (has any model whatsoever) implies that ZF together with these two principles is consistent.

The study of inner models is common in the study of determinacy and large cardinals, especially when considering axioms such as the axiom of determinacy that contradict the axiom of choice. Even if a fixed model of set theory satisfies the axiom of choice, it is possible for an inner model to fail to satisfy the axiom of choice. For example, the existence of sufficiently large cardinals implies that there is an inner model satisfying the axiom of determinacy (and thus not satisfying the axiom of choice).^[5]

Large cardinals[link]

A large cardinal is a cardinal number with an extra property. Many such properties are studied, including inaccessible cardinals, measurable cardinals, and many more. These properties typically imply the cardinal number must be very large, with the existence of a cardinal with the specified property unprovable in Zermelo-Fraenkel set theory.

Determinacy[link]

Determinacy refers to the fact that, under appropriate assumptions, certain two-player games of perfect information are determined from the start in the sense that one player must have a winning strategy. The existence of these strategies has important consequences in descriptive set theory, as the assumption that a broader class of games is determined often implies that a broader class of sets will have a topological property. The axiom of determinacy (AD) is an important object of study; although incompatible with the axiom of choice, AD implies that all subsets of the real line are well behaved (in particular, measurable and with the perfect set property). AD can be used to prove that the Wadge degrees have an elegant structure.

Forcing[link]

Paul Cohen invented the method of forcing while searching for a model of ZFC in which the axiom of choice or the continuum hypothesis fails. Forcing adjoins to some given model of set theory additional sets in order to create a larger model with properties determined (i.e. "forced") by the construction and the original model. For example, Cohen's construction adjoins additional subsets of the natural numbers without changing any of the cardinal numbers of the original model. Forcing is also one of two methods for proving relative consistency by finitistic methods, the other method being Boolean-valued models.

Cardinal invariants[link]

A cardinal invariant is a property of the real line measured by a cardinal number. For example, a well-studied invariant is the smallest cardinality of a collection of meagre sets of reals whose union is the entire real line. These are invariants in the sense that any two isomorphic models of set theory must give the same cardinal for each invariant. Many cardinal invariants have been studied, and the relationships between them are often complex and related to axioms of set theory.

Set-theoretic topology[link]

Set-theoretic topology studies questions of general topology that are set-theoretic in nature or that require advanced methods of set theory for their solution. Many of these theorems are independent of ZFC, requiring stronger axioms for their proof. A famous problem is the normal Moore space question, a question in general topology that was the subject of intense research. The answer to the normal Moore space question was eventually proved to be independent of ZFC.

Objections to set theory as a foundation for mathematics[link]

From set theory's inception, some mathematicians objected to it as a foundation for mathematics. The most common objection to set theory, one Kronecker voiced in set theory's earliest years, starts from the constructivist view that mathematics is loosely related to computation. If this view is granted, then the treatment of infinite sets, both in naive and in axiomatic set theory, introduces into mathematics methods and objects that are not computable even in principle. Ludwig Wittgenstein questioned the way Zermelo–Fraenkel set theory handled infinities.^{[citation needed]} Wittgenstein's views about the foundations of mathematics were later criticised by Georg Kreisel and Paul Bernays, and closely investigated by Crispin Wright, among others.

Category theorists have proposed topos theory as an alternative to traditional axiomatic set theory. Topos theory can interpret various alternatives to that theory, such as constructivism, finite set theory, and computable set theory.^[6]

See also[link]

	Set theory portal
	Mathematics portal

• Category theory
• List of set theory topics
• Musical set theory concerns the application of combinatorics and group theory to music; beyond the fact that it uses finite sets it has nothing to do with mathematical set theory of any kind. In the last two decades, transformational theory in music has taken the concepts of mathematical set theory more rigorously (see Lewin 1987).
• Relational model - Borrows from Set Theory.

Notes[link]

Further reading[link]

• Devlin, Keith, (2nd ed.) 1993. The Joy of Sets. Springer Verlag, ISBN 0-387-94094-4
• Ferreirós, Jose, 2007 (1999). Labyrinth of Thought: A history of set theory and its role in modern mathematics. Basel, Birkhäuser. ISBN 978-3-7643-8349-7
• Johnson, Philip, 1972. A History of Set Theory. Prindle, Weber & Schmidt ISBN 0-87150-154-6
• Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. North-Holland, ISBN 0-444-85401-0.
• Tiles, Mary, 2004 (1989). The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise. Dover Publications.

External links[link]

Wikibooks has a book on the topic of Set Theory

Wikibooks has a book on the topic of Discrete mathematics/Set theory

• Foreman, M., Akihiro Kanamori, eds. Handbook of Set Theory. 3 vols., 2010. Each chapter surveys some aspect of contemporary research in set theory. Does not cover established elementary set theory, on which see Devlin (1993).
• Arthur Schoenflies (1898) Mengenlehre in Klein's encyclopedia.

Look up set in Wiktionary, the free dictionary.

Set may refer to:

Mathematics and programming[link]

• Set (mathematics), A collection of well defined and distinct objects
• Set theory
• Category of sets
• Set (computer science), a data type that is a collection of unique values
  • Set (C++), an implementation of Set (computer science) in the C++ language Standard Template Library (STL)

Chemistry[link]

• SET, single electron transfer

Psychology[link]

• Set (psychology), a set of expectations which shapes perception or thought

Technology[link]

• Set, to make/become solid: see Solidification
• A television set
• SET, Secure Electronic Transaction, a standard protocol for securing credit card transactions over insecure networks
• SET, Single-Electron Transistor, a device to amplify currents in nanoelectronics
• SET, single-ended triode, a type of electronic amplifier
• SET Awards, The Science, Engineering & Technology Student of the Year Awards
• Set (command), a command for manipulating environment variables in Unix and Microsoft operating-systems
• Software Engineer in Test, a Quality Assurance job title in some software companies
• Saw set, the process of setting the teeth of a saw so each tooth protrudes to the side of the blade

Arts[link]

Theatre[link]

• Theatrical scenery
• Set construction, construction of scenery for theatrical, movie and video game production.

Dancing[link]

• Set, the basic square formation in square dancing

Music[link]

• DJ set or DJ mix, a musical performance by a DJ
• Set theory (music), dealing with concepts for categorizing musical objects and describing their relationships
• Set (music), a collection of discrete entities, for example pitch sets, duration sets, and timbre sets
• Set (Thompson Twins album)
• Set (Alex Chilton album)
• Set list, a list of songs to be performed at a concert
• In the professional jargon of music production and critique, a "set" refers to an album of songs.

Fiction[link]

• Set or Father Set, a fictional deity in the Conan adventure novels by Robert E. Howard and their Marvel Comics interpretation

Sport and games[link]

• A chess set
• Set, a signal used in American football
• Set (darts), when one player wins three legs
• Set (game), a card game
• Set, a unit of play in tennis
• Set, a term for three of a kind in poker
• Set, a unit of play in volleyball (formerly "game")
• Set, a group of repetitions in weight training
• The set, a pitching position in baseball (also called the stretch)

Crime[link]

• Set, a sub-group of a street gang

Religion[link]

• Set or Seth, a Biblical character, a son of Adam and Eve
• Set (mythology), an ancient Egyptian deity

Miscellaneous[link]

• Set and setting, coined by Timothy Leary to describe the mindset and location of hallucinogenic experiences
• Seṭ and aniṭ roots, in Sanskrit grammar
• Set (video game), a group of items that adds specific bonuses
• Set (river), a river in northeastern Spain
• Set or sett, a badger's den
• Set, a small tuber or bulb used instead of seed, especially:
  • Potato set
  • Onion set

Other acronyms[link]

• Sanlih Entertainment Television, a television channel in Taiwan
• Scalar expectancy theory
• Serial endosymbiosis theory, a theory concerning the origins of mitochondria and plastids
• Similarity Enhanced Transfer, a technology used to speed up download rate of files on file sharing networks
• Simulated Emergency Test, an amateur radio training exercise
• Single Embryo Transfer
• Societatea Pentru Exploatări Tehnice, a Romanian aeronautics company of the 1920s to 1940s
• Society of Engineering Technologists, a student organization at Texas Tech University
• Sony Entertainment Television, a television channel owned by Sony
• South Eastern Trains, a rail company operating in South East England
• Stock Exchange of Thailand, the national stock exchange of Thailand
  • SET Index, an index for the Stock Exchange of Thailand
• Strategic Energy Technologies Plan (EU SET Plan)
• Study of Exceptional Talent, a program for gifted students
• Single-Ended Triode, a type of electronic amplifier
• Social entropy Theory, application of entropy concept to maintenance or disorganisation in society
• Northeastern University, School of Engineering Technology
• Southeast Toyota Distributors LLC, a distributor of Toyota vehicles
• Suzuki SET, Suzuki Exhaust Tuning, a device for altering pressure waves within motorcycle exhausts

See also[link]

• Sett (disambiguation)
• Setting (disambiguation)
• Setup (disambiguation)
• Seti (disambiguation)

This disambiguation page lists articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article.

The English word theory was derived from a technical term in philosophy in Ancient Greek. The word theoria, θεωρία, meant "a looking at, viewing, beholding", and referring to contemplation or speculation, as opposed to action.^[1] Theory is especially often contrasted to "practice" (from Greek praxis, πρᾶξις) a Greek term for "doing", which is opposed to theory because theory involved no doing apart from itself.

A classical example of the distinction between theoretical and practical uses the discipline of medicine: Medical theory and theorizing involves trying to understand the causes and nature of health and sickness, while the practical side of medicine is trying to make people healthy. These two things are related but can be independent, because it is possible to research health and sickness without curing specific patients, and it is possible to cure a patient without knowing how the cure worked.^[2]

By extension of the philosophical meaning, "theoria" is also a word still used in theological contexts.

In modern contexts, while theories in the arts and philosophy may address ideas and empirical phenomena which are not easily measurable, in modern science the term "theory", or "scientific theory" is generally understood to refer to a proposed explanation of empirical phenomena, made in a way consistent with scientific method. Such theories are preferably described in such a way that any scientist in the field is in a position to understand and either provide empirical support ("verify") or empirically contradict ("falsify") it. In this modern scientific context the distinction between theory and practice corresponds roughly to the distinction between theoretical science and technology or applied science.

A common distinction made in science is between theories and hypotheses. Hypotheses are individual empirically testable conjectures; while theories are collections of hypotheses that are logically linked together into a coherent explanation of some aspect of reality and which have individually or jointly received some empirical support.

Ancient uses[link]

Although it has more mundane meanings in Greek, the word θεωρία apparently developed special uses early in the recorded history of the Greek language. In the book From Religion to Philosophy, Francis Cornford suggests that the Orphics used the word "theory" to mean 'passionate sympathetic contemplation'.^[3] Pythagoras changed the word to mean a passionate sympathetic contemplation of mathematical and scientific knowledge, because he considered such intellectual pursuits the way to reach the highest plane of existence. Pythagoras emphasized subduing emotions and bodily desires in order to enable the intellect to function at the higher plane of theory. Thus it was Pythagoras who gave the word "theory" the specific meaning which leads to the classical and modern concept of a distinction between theory as uninvolved, neutral thinking, and practice.^[4]

In Aristotle's terminology, as has already been mentioned above, theory is contrasted with praxis or practice, which remains the case today. For Aristotle, both practice and theory involve thinking, but the aims are different. Theoretical contemplation considers things which humans do not move or change, such as nature, so it has no human aim apart from itself and the knowledge it helps create. On the other hand, praxis involves thinking, but always with an aim to desired actions, whereby humans cause change or movement themselves for their own ends. Any human movement which involves no conscious choice and thinking could not be an example of praxis or doing.^[5]

Theories formally and scientifically[link]

Theories are analytical tools for understanding, explaining, and making predictions about a given subject matter. There are theories in many and varied fields of study, including the arts and sciences. A formal theory is syntactic in nature and is only meaningful when given a semantic component by applying it to some content (i.e. facts and relationships of the actual historical world as it is unfolding). Theories in various fields of study are expressed in natural language, but are always constructed in such a way that their general form is identical to a theory as it is expressed in the formal language of mathematical logic. Theories may be expressed mathematically, symbolically, or in common language, but are generally expected to follow principles of rational thought or logic.

Theory is constructed of a set of sentences which consist entirely of true statements about the subject matter under consideration. However, the truth of any one of these statements is always relative to the whole theory. Therefore the same statement may be true with respect to one theory, and not true with respect to another. This is, in ordinary language, where statements such as "He is a terrible person" cannot be judged to be true or false without reference to some interpretation of who "He" is and for that matter what a "terrible person" is under the theory.^[6]

Sometimes two theories have exactly the same explanatory power because they make the same predictions. A pair of such theories is called indistinguishable, and the choice between them reduces to convenience or philosophical preference.

The form of theories is studied formally in mathematical logic, especially in model theory. When theories are studied in mathematics, they are usually expressed in some formal language and their statements are closed under application of certain procedures called rules of inference. A special case of this, an axiomatic theory, consists of axioms (or axiom schemata) and rules of inference. A theorem is a statement that can be derived from those axioms by application of these rules of inference. Theories used in applications are abstractions of observed phenomena and the resulting theorems provide solutions to real-world problems. Obvious examples include arithmetic (abstracting concepts of number), geometry (concepts of space), and probability (concepts of randomness and likelihood).

Gödel's incompleteness theorem shows that no consistent, recursively enumerable theory (that is, one whose theorems form a recursively enumerable set) in which the concept of natural numbers can be expressed, can include all true statements about them. As a result, some domains of knowledge cannot be formalized, accurately and completely, as mathematical theories. (Here, formalizing accurately and completely means that all true propositions—and only true propositions—are derivable within the mathematical system.) This limitation, however, in no way precludes the construction of mathematical theories that formalize large bodies of scientific knowledge.

Underdetermination[link]

A theory is underdetermined (also called indeterminacy of data to theory) if, given the available evidence cited to support the theory, there is a rival theory which is inconsistent with it that is at least as consistent with the evidence. Underdetermination is an epistemological issue about the relation of evidence to conclusions.

Intertheoretic reduction and elimination[link]

If there is a new theory which is better at explaining and predicting phenomena than an older theory (i.e. it has more explanatory power), we are justified in believing that the newer theory describes reality more correctly. This is called an intertheoretic reduction because the terms of the old theory can be reduced to the terms of the new one. For instance, our historical understanding about "sound," "light" and "heat" have today been reduced to "wave compressions and rarefactions," "electromagnetic waves," and "molecular kinetic energy," respectively. These terms which are identified with each other are called intertheoretic identities. When an old theory and a new one are parallel in this way, we can conclude that we are describing the same reality, only more completely.

In cases where a new theory uses new terms which do not reduce to terms of an older one, but rather replace them entirely because they are actually a misrepresentation it is called an intertheoretic elimination. For instance, the obsolete scientific theory that put forward an understanding of heat transfer in terms of the movement of caloric fluid was eliminated when a theory of heat as energy replaced it. Also, the theory that phlogiston is a substance released from burning and rusting material was eliminated with the new understanding of the reactivity of oxygen.

Theories vs. theorems[link]

Theories are distinct from theorems. Theorems are derived deductively from objections according to a formal system of rules, sometimes as an end in itself and sometimes as a first step in testing or applying a theory in a concrete situation; theorems are said to be true in the sense that the conclusions of a theorem are logical consequences of the objections. Theories are abstract and conceptual, and to this end they are always considered true. They are supported or challenged by observations in the world. They are 'rigorously tentative', meaning that they are proposed as true and expected to satisfy careful examination to account for the possibility of faulty inference or incorrect observation. Sometimes theories are incorrect, meaning that an explicit set of observations contradicts some fundamental objection or application of the theory, but more often theories are corrected to conform to new observations, by restricting the class of phenomena the theory applies to or changing the assertions made.

"Sometimes a hypothesis never reaches the point of being considered a theory because the answer is not found to derive its assertions analytically or not applied empirically."

Philosophical theories[link]

Theories whose subject matter consists not in empirical data, but rather in ideas are in the realm of philosophical theories as contrasted with scientific theories. At least some of the elementary theorems of a philosophical theory are statements whose truth cannot necessarily be scientifically tested through empirical observation.

Fields of study are sometimes named "theory" because their basis is some initial set of objections describing the field's approach to a subject matter. These assumptions are the elementary theorems of the particular theory, and can be thought of as the axioms of that field. Some commonly known examples include set theory and number theory; however literary theory, critical theory, and music theory are also of the same form.

Metatheory[link]

One form of philosophical theory is a metatheory or meta-theory. A metatheory is a theory whose subject matter is some other theory. In other words it is a theory about a theory. Statements made in the metatheory about the theory are called metatheorems.

Political theories[link]

A political theory is an ethical theory about the law and government. Often the term "political theory" refers to a general view, or specific ethic, political belief or attitude, about politics.

Scientific theories[link]

In scientific usage, the term "theory" is reserved for explanations of phenomena which meet basic requirements about the kinds of empirical observations made, the methods of classification used, and the consistency of the theory in its application among members of the class to which it pertains. These requirements vary across different scientific fields of knowledge, but in general theories are expected to be functional and parsimonious: i.e. a theory should be the simplest possible tool that can be used to effectively address the given class of phenomena. Such theories are constructed from elementary assumptions that are motivated by empirical data about observable phenomena. A scientific theory is used as a plausible general principle or body of principles offered to explain a phenomenon.^[7]

A scientific theory is a deductive theory, in that its content is based on some formal system of logic and on basic axioms. In a deductive theory, any sentence which is a logical consequence of one or more of the axioms is also a sentence of that theory.^[6]

A major concern in construction of scientific theories is the problem of demarcation, i.e., distinguishing those ideas that are properly studied by the sciences and those that are not.

Theories are intended to be an accurate, predictive description of the natural world.

Theories as models[link]

Scientific theories are constructed to explain and predict phenomena (e.g., inanimate things, events, or behavior of animals). A scientific theory can be thought of as a model of reality. According to Stephen Hawking in A Brief History of Time, "a theory is a good theory if it satisfies two requirements: It must accurately describe a large class of observations on the basis of a model which contains only a few arbitrary elements, and it must make definite predictions about the results of future observations". He goes on to state, "any physical theory is always provisional, in the sense that it is only a hypothesis; you can never prove it. No matter how many times the results of experiments agree with some theory, you can never be sure that the next time the result will not contradict the theory. On the other hand, you can disprove a theory by finding even a single observation which disagrees with the predictions of the theory". The "unprovable but falsifiable" nature of theories is a consequence of the necessity of using inductive logic.^[8]

Theories in physics[link]

In physics the term theory is generally used for a mathematical framework—derived from a small set of basic postulates (usually symmetries, like equality of locations in space or in time, or identity of electrons, etc.)—which is capable of producing experimental predictions for a given category of physical systems. One good example is classical electromagnetism, which encompasses results derived from gauge symmetry (sometimes called gauge invariance) in a form of a few equations called Maxwell's equations. Note that the specific theoretical aspects of classical electromagnetic theory, which have been consistently and successfully replicated for well over a century, are termed "laws of electromagnetism", reflecting that they are today taken for granted. Within electromagnetic theory generally, there are numerous hypotheses about how electromagnetism applies to specific situations. Many of these hypotheses are already considered to be adequately tested, with new ones always in the making and perhaps untested.

Pedagogical definition[link]

In pedagogical contexts or in official pronouncements by official organizations of scientists a definition such as the following may be promulgated.

According to the United States National Academy of Sciences,

Some scientific explanations are so well established that no new evidence is likely to alter them. The explanation becomes a scientific theory. In everyday language a theory means a hunch or speculation. Not so in science. In science, the word theory refers to a comprehensive explanation of an important feature of nature supported by facts gathered over time. Theories also allow scientists to make predictions about as yet unobserved phenomena,^[9]

Look up theory in Wiktionary, the free dictionary.

According to the American Association for the Advancement of Science,

A scientific theory is a well-substantiated explanation of some aspect of the natural world, based on a body of facts that have been repeatedly confirmed through observation and experiment. Such fact-supported theories are not "guesses" but reliable accounts of the real world. The theory of biological evolution is more than "just a theory." It is as factual an explanation of the universe as the atomic theory of matter or the germ theory of disease. Our understanding of gravity is still a work in progress. But the phenomenon of gravity, like evolution, is an accepted fact.^[10]

These definitions firmly mark things termed "theories" as being well supported by evidence, although scientists sometimes also use the word "theory" to describe untested but intricate hypotheses.^{[citation needed]}

[edit] The term theoretical

The term theoretical is sometimes informally used in place of hypothetical to describe a result that is predicted, but has not yet been adequately tested by observation or experiment. A hypothesis is a prediction which has yet to be tested, while a theory is a prediction-making conceptual framework that is consistent with data.

List of notable theories[link]

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• Astronomy: Big Bang Theory
• Biology: Cell theory — Evolution — Germ theory
• Chemistry: Molecular theory — Kinetic theory of gases — Molecular orbital theory — Valence bond theory — Transition state theory — RRKM theory — Chemical graph theory — Flory-Huggins solution theory — Marcus theory — Lewis theory (successor to Brønsted–Lowry acid–base theory) — HSAB theory — Debye–Hückel theory — Thermodynamic theory of polymer elasticity — Reptation theory — Polymer field theory — Møller–Plesset perturbation theory — Density Functional Theory — Frontier molecular orbital theory — Polyhedral skeletal electron pair theory — Baeyer strain theory — Quantum theory of atoms in molecules — Collision theory — Ligand field theory (successor to Crystal field theory) — Variational Transition State Theory — Benson group increment theory — Specific ion interaction theory
• Climatology: Climate change theory (due to anthropogenic activity)
• Economics: Macroeconomic theory — Microeconomic theory
• Education: Constructivist theory — Critical pedagogy theory — Education theory — Multiple intelligence theory — Progressive education theory
• Engineering: Circuit theory — Control theory — Signal theory — Systems theory — Information theory
• Film: Film Theory
• Geology: Plate tectonics
• Humanities: Critical theory
• Linguistics: X-bar theory — Government and Binding — Principles and Parameters
• Literature: Literary theory
• Mathematics: Approximation theory — Arakelov theory — Asymptotic theory — Bifurcation theory — Catastrophe theory — Category theory — Chaos theory — Choquet theory — Coding theory — Combinatorial game theory — Deformation theory — Dimension theory — Ergodic theory — Field theory — Galois theory — Game theory — Graph theory — Group theory — Hodge theory — Homology theory — Homotopy theory — Ideal theory — Intersection theory — Invariant theory — Iwasawa theory — K-theory — KK-theory — Knot theory — L-theory — Lie theory — Littlewood–Paley theory — Matrix theory — Measure theory — Model theory — Morse theory — Nevanlinna theory — Number theory — Obstruction theory — Operator theory — PCF theory — Perturbation theory — Potential theory — Probability theory — Ramsey theory — Rational choice theory — Representation theory — Ring theory — Set theory — Shape theory — Small cancellation theory — Spectral theory — Stability theory — Stable theory — Sturm–Liouville theory — Twistor theory
• Music: Music theory
• Philosophy: Proof theory — Speculative reason — Theory of truth — Type theory — Value theory — Virtue theory
• Physics: Acoustic theory — Antenna theory — Atomic theory — BCS theory — Landau theory — M-theory — Perturbation theory — Theory of relativity (successor to classical mechanics) — Quantum field theory — Scattering theory — String theory
• Planetary science: Giant impact theory
• Psychology: Cognitive dissonance theory - Attachment Theory - Object permanance - Poverty of stimulus - Attribution Theory - Self fulfilling prophecy - Stockholm syndrome
• Sociology: Sociological theory — Social theory — Critical theory
• Sports: Chess theory
• Statistics: Extreme value theory
• Theatre: Performance theory
• Visual Art: Aesthetics — Art Educational theory — Architecture — Composition — Anatomy — Color theory — Perspective — Visual perception — Geometry — Manifolds
• Other: Obsolete scientific theories — Phlogiston theory

See also[link]

Notes[link]

1. ^ The word "theory" was used in Greek philosophy, for example, that of Plato. It is a statement of how and why particular facts are related. It is related to words for θεωρός "spectator", θέα thea "a view" + ὁρᾶν horan "to see", literally "looking at a show". See for example dictionary entries at Perseus website. The word has been in use in English since at least the late 16th century.Harper, Douglas. "theory". Online Etymology Dictionary. http://www.etymonline.com/index.php?term=theory. Retrieved 2008-07-18. 
2. ^ See for example Hippocrates Praeceptiones, Part 1.
3. ^ Cornford, Francis Macdonald (November 8, 1991). From religion to philosophy: a study in the origins of western speculation. Princeton University Press. ISBN 978-0‐691‐02076‐1. 
4. ^ Russell, Bertrand, History of Western Philosophy
5. ^ The LSJ cites two passages of Aristotle as examples, both from the Metaphysics and involving the definition of natural science: 11.1064a17, "it is clear that natural science (φυσικὴν ἐπιστήμην) must be neither practical (πρακτικὴν) nor productive (ποιητικὴν), but speculative (θεωρητικὴν)" and 6.1025b25, "Thus if every intellectual activity [διάνοια] is either practical or productive or speculative (θεωρητική), physics (φυσικὴ) will be a speculative [θεωρητική] science". So Aristotle actually made a three way distinction between practical, theoretical and productive or technical - or between doing, contemplating or making. All three types involve thinking, but are distinguished by what causes the objects of thought to move or change.
6. ^ ^{a b} Curry, Haskell, Foundations of Mathematical Logic
7. ^ Merriam-Webster.com Merriam-Webster Dictionary: Theory in Science
8. ^ Stephen Hawking, A Brief History of Time New York: Bantam, p. 7
9. ^ National Academy of Sciences (2005), Science, Evolution, and Creationism, a brochure on the book of the same title.
10. ^ AAAS Evolution Resources

References[link]

• Matson, Ronald, "Comparing scientific laws and theories", Biology, Kennesaw State University, http://science.kennesaw.edu/~rmatson/3380theory.html .
• Davidson Reynolds, Paul (1971). A primer in theory construction. Boston: Allyn and Bacon.
• Hawking, Stephen (1996). "The Illustrated A Brief History of Time" (Updated and expanded ed.). New York: Bantam Books, p. 15.
• Popper, Karl (1963), Conjectures and Refutations, Routledge and Kegan Paul, London, UK, pp. 33–39. Reprinted in Theodore Schick (ed., 2000), Readings in the Philosophy of Science, Mayfield Publishing Company, Mountain View, CA, USA, pp. 9–13.
• Zima, Peter V. (2007). What is theory? Cultural theory as discourse and dialogue. London: Continuum (translated from: Was ist Theorie? Theoriebegriff und Dialogische Theorie in der Kultur- und Sozialwissenschaften. Tübingen: A. Franke Verlag, 2004).