Who developed set theory?
Mathematical logic
Set theory is the study of sets, which are abstract collections of objects. Many of the basic notions, such as ordinal and cardinal numbers, were developed informally by Cantor before formal axiomatizations of set theory were developed. The first such axiomatization, due to Zermelo (1908b), was extended slightly to become Zermelo–Fraenkel set theory (ZF), which is now the most widely used foundational theory for mathematics.
Paradoxes of set theory
Before set theory was introduced, the notion of the "size" of a set had been problematic. It had been discussed by Galileo Galilei and Bernard Bolzano, among others. Are there as many natural numbers as squares of natural numbers when measured by the method of enumeration?
19th century in science
In the later 19th century, Georg Cantor established the first foundations of set theory, which enabled the rigorous treatment of the notion of infinity and has become the common language of nearly all mathematics. Cantor's set theory, and the rise of mathematical logic in the hands of Peano, L. E. J. Brouwer, David Hilbert, Bertrand Russell, and A.N. Whitehead, initiated a long running debate on the foundations of mathematics.
Naive set theory
“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
Wacław Sierpiński
In 1907 Sierpiński first became interested in set theory when he came across a theorem which stated that points in the plane could be specified with a single coordinate. He wrote to Tadeusz Banachiewicz (then at Göttingen), asking how such a result was possible. He received the one-word reply 'Cantor'. Sierpiński began to study set theory and, in 1909, he gave the first ever lecture course devoted entirely to the subject.
Set theory
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.
Topology
Modern topology depends strongly on the ideas of set theory, developed by Georg Cantor in the later part of the 19th century. In addition to establishing the basic ideas of set theory, Cantor considered point sets in Euclidean space as part of his study of Fourier series. For further developments, see point-set topology and algebraic topology.
Paradoxes of set theory
For all its usefulness in resolving questions regarding infinite sets, naive set theory has some fatal flaws. In particular, it is prey to logical paradoxes such as those exposed by Russell's paradox. The discovery of these paradoxes revealed that not all sets which can be described in the language of naive set theory can actually be said to exist without creating a contradiction. The 20th century saw a resolution to these paradoxes in the development of the various axiomatizations of set theories such as ZFC and NBG in common use today. However, the gap between the very formalized and symbolic language of these theories and our typical informal use of mathematical language results in various paradoxical situations, as well as the philosophical question of exactly what it is that such formal systems actually propose to be talking about.
Zermelo–Fraenkel set theory
The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. However, the discovery of paradoxes in naive set theory, such as Russell's paradox, led to the desire for a more rigorous form of set theory that was free of these paradoxes.
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( ; ;  – January 6, 1918) was a German mathematician. He created set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are more numerous than the natural numbers. In fact, Cantor's method of proof of this theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.
Lévy hierarchy
In set theory and mathematical logic, the Lévy hierarchy, introduced by Azriel Lévy in 1965, is a hierarchy of formulas in the formal language of the Zermelo–Fraenkel set theory, which is typically called just the language of set theory. This is analogous to the arithmetical hierarchy which provides the classifications but for sentences of the language of arithmetic.
Set theory
An enrichment of ZFC called internal set theory was proposed by Edward Nelson in 1977.
Finite set
More generally, informal notions like set, and particularly finite set, may receive interpretations across a range of formal systems varying in their axiomatics and logical apparatus. The best known axiomatic set theories include Zermelo-Fraenkel set theory (ZF), Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), Von Neumann–Bernays–Gödel set theory (NBG), Non-well-founded set theory, Bertrand Russell's Type theory and all the theories of their various models. One may also choose among classical first-order logic, various higher-order logics and intuitionistic logic.
Cantor set
This set would have been considered abstract at the time when Cantor devised it. Cantor himself was led to it by practical concerns about the set of points where a trigonometric series might fail to converge. The discovery did much to set him on the course for developing an abstract, general theory of infinite sets.
Fuzzy set
In mathematics, fuzzy sets (aka uncertain sets) are somewhat like sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh and Dieter Klaua in 1965 as an extension of the classical notion of set.
At the same time, defined a more general kind of structure called an L-relation, which he studied in an abstract algebraic context. Fuzzy relations, which are used now in different areas, such as linguistics , decision-making , and clustering , are special cases of "L"-relations when "L" is the unit interval [0, 1].
Paradoxes of set theory
After all this, the version of the "set of all sets" paradox conceived by Bertrand Russell in 1903 led to a serious crisis in set theory. Russell recognized that the statement "x" = "x" is true for every set, and thus the set of all sets is defined by {"x" | "x" = "x"}. In 1906 he constructed several paradox sets, the most famous of which is the set of all sets which do not contain themselves. Russell himself explained this abstract idea by means of some very concrete pictures. One example, known as the Barber paradox, states: The male barber who shaves all and only men who don't shave themselves has to shave himself only if he does not shave himself.
Algebra
Sets: Rather than just considering the different types of numbers, abstract algebra deals with the more general concept of "sets": a collection of all objects (called elements) selected by property specific for the set. All collections of the familiar types of numbers are sets. Other examples of sets include the set of all two-by-two matrices, the set of all second-degree polynomials ("ax" + "bx" + "c"), the set of all two dimensional vectors in the plane, and the various finite groups such as the cyclic groups, which are the groups of integers modulo "n". Set theory is a branch of logic and not technically a branch of algebra.
Set (mathematics)
A set is a well-defined collection of distinct objects. The objects that make up a set (also known as the set's "elements" or "members") can be anything: numbers, people, letters of the alphabet, other sets, and so on. Georg Cantor, one of the founders of set theory, gave the following definition of a set at the beginning of his "Beiträge zur Begründung der transfiniten Mengenlehre":
History of logic
The logicist project received a near-fatal setback with the discovery of a paradox in 1901 by Bertrand Russell. This proved Frege's naive set theory led to a contradiction. Frege's theory contained the axiom that for any formal criterion, there is a set of all objects that meet the criterion. Russell showed that a set containing exactly the sets that are not members of themselves would contradict its own definition (if it is not a member of itself, it is a member of itself, and if it is a member of itself, it is not). This contradiction is now known as Russell's paradox. One important method of resolving this paradox was proposed by Ernst Zermelo. Zermelo set theory was the first axiomatic set theory. It was developed into the now-canonical Zermelo–Fraenkel set theory (ZF). Russell's paradox symbolically is as follows:
Axiom schema of replacement
Its publication by Adolf Fraenkel in 1922 is what makes modern set theory Zermelo-"Fraenkel" set theory (ZFC). The axiom was independently discovered and announced by Thoralf Skolem later in the same year (and published in 1923). Zermelo himself incorporated Fraenkel's axiom in his revised system he published in 1930, which also included as a new axiom von Neumann's axiom of foundation. Although it is Skolem's first order version of the axiom list that we use today, he usually gets no credit since each individual axiom was developed earlier by either Zermelo or Fraenkel. The phrase “Zermelo-Fraenkel set theory” was first used in print by von Neumann in 1928.
Mathematical logic
Two famous statements in set theory are the axiom of choice and the continuum hypothesis. The axiom of choice, first stated by Zermelo (1904), was proved independent of ZF by Fraenkel (1922), but has come to be widely accepted by mathematicians. It states that given a collection of nonempty sets there is a single set "C" that contains exactly one element from each set in the collection. The set "C" is said to "choose" one element from each set in the collection. While the ability to make such a choice is considered obvious by some, since each set in the collection is nonempty, the lack of a general, concrete rule by which the choice can be made renders the axiom nonconstructive. Stefan Banach and Alfred Tarski (1924) showed that the axiom of choice can be used to decompose a solid ball into a finite number of pieces which can then be rearranged, with no scaling, to make two solid balls of the original size. This theorem, known as the Banach–Tarski paradox, is one of many counterintuitive results of the axiom of choice.
History of logic
In set theory, the method of forcing revolutionized the field by providing a robust method for constructing models and obtaining independence results. Paul Cohen introduced this method in 1963 to prove the independence of the continuum hypothesis and the axiom of choice from Zermelo–Fraenkel set theory. His technique, which was simplified and extended soon after its introduction, has since been applied to many other problems in all areas of mathematical logic.
Tricotyledon theory of system design
In systems engineering, the tricotyledon theory of system design (T3SD) is a mathematical theory of system design developed by A. Wayne Wymore. T3SD consists of a language for describing systems and requirements, which is based on set theory, a mathematical systems model based on port automata, and a precise definition of the different types of system requirements and relationships between requirements.
Biordered set
A biordered set ("boset") is a mathematical object that occurs in the description of the structure of the set of idempotents in a semigroup. The concept and the terminology were developed by K S S Nambooripad in the early 1970s.
The defining properties of a biordered set are expressed in terms of two quasiorders defined on the set and hence the name biordered set. Patrick Jordan, while a master's student at University of Sydney, introduced in 2002 the term boset as an abbreviation of biordered set.
Set theory
In 1906 English readers gained the book "Theory of Sets of Points" by husband and wife William Henry Young and Grace Chisholm Young, published by Cambridge University Press.
Set (card game)
Set is a real-time card game designed by Marsha Falco in 1974 and published by Set Enterprises in 1991. The deck consists of 81 cards varying in four features: number (one, two, or three); symbol (diamond, squiggle, oval); shading (solid, striped, or open); and color (red, green, or purple). Each possible combination of features (e.g., a card with three striped green diamonds) appears precisely once in the deck.
Ω-logic
In set theory, Ω-logic is an infinitary logic and deductive system proposed by as part of an attempt to generalize the theory of determinacy of pointclasses to cover the structure formula_1. Just as the axiom of projective determinacy yields a canonical theory of formula_2, he sought to find axioms that would give a canonical theory for the larger structure. The theory he developed involves a controversial argument that the continuum hypothesis is false.
Sauer–Shelah lemma
In combinatorial mathematics and extremal set theory, the Sauer–Shelah lemma states that every family of sets with small VC dimension consists of a small number of sets. It is named after Norbert Sauer and Saharon Shelah, who published it independently of each other in 1972. The same result was also published slightly earlier and again independently, by Vladimir Vapnik and Alexey Chervonenkis, after whom the VC dimension is named. In his paper containing the lemma, Shelah gives credit also to Micha Perles, and for this reason the lemma has also been called the Perles–Sauer–Shelah lemma.
John von Neumann
The axiomatization of mathematics, on the model of Euclid's "Elements", had reached new levels of rigour and breadth at the end of the 19th century, particularly in arithmetic, thanks to the axiom schema of Richard Dedekind and Charles Sanders Peirce, and in geometry, thanks to Hilbert's axioms. But at the beginning of the 20th century, efforts to base mathematics on naive set theory suffered a setback due to Russell's paradox (on the set of all sets that do not belong to themselves). The problem of an adequate axiomatization of set theory was resolved implicitly about twenty years later by Ernst Zermelo and Abraham Fraenkel. Zermelo–Fraenkel set theory provided a series of principles that allowed for the construction of the sets used in the everyday practice of mathematics, but they did not explicitly exclude the possibility of the existence of a set that belongs to itself. In his doctoral thesis of 1925, von Neumann demonstrated two techniques to exclude such sets—the "axiom of foundation" and the notion of "class."
Discrete mathematics
Set theory is the branch of mathematics that studies sets, which are collections of objects, such as {blue, white, red} or the (infinite) set of all prime numbers. Partially ordered sets and sets with other relations have applications in several areas.
Rough set
In computer science, a rough set, first described by Polish computer scientist Zdzisław I. Pawlak, is a formal approximation of a crisp set (i.e., conventional set) in terms of a pair of sets which give the "lower" and the "upper" approximation of the original set. In the standard version of rough set theory (Pawlak 1991), the lower- and upper-approximation sets are crisp sets, but in other variations, the approximating sets may be fuzzy sets.
Systematics – study of multi-term systems
Systematics is the name given by John Godolphin Bennett (1897–1974) to a branch of systems science that he developed in the mid-twentieth century. Also referred to as the theory of Multi-Term Systems or Bennettian Systematics, it focuses on types, levels, and degrees of complexity in systems, the qualities emergent at these levels, and the ability to represent and practically deal with ("understand") complexity using abstract models. Thus to understand the notions of sameness and difference requires a system or universe of discourse with a minimum of two terms or elements. To understand the concept of relatedness requires three, and so on.
Naive set theory
In naïve 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 1915 in his article "Beiträge zur Begründung der transfiniten Mengenlehre":
Mathematical and theoretical biology
Modelling physiological systemsMolecular set theory (MST) is a mathematical formulation of the wide-sense chemical kinetics of biomolecular reactions in terms of sets of molecules and their chemical transformations represented by set-theoretical mappings between molecular sets. It was introduced by Anthony Bartholomay, and its applications were developed in mathematical biology and especially in mathematical medicine.
In a more general sense, MST is the theory of molecular categories defined as categories of molecular sets and their chemical transformations represented as set-theoretical mappings of molecular sets. The theory has also contributed to biostatistics and the formulation of clinical biochemistry problems in mathematical formulations of pathological, biochemical changes of interest to Physiology, Clinical Biochemistry and Medicine.
Universal set
In set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, the conception of a universal set leads to a paradox (Russell's paradox) and is consequently not allowed. However, some non-standard variants of set theory include a universal set.
Felix Hausdorff
The first six chapters deal with the basic concepts of the general set theory. At the beginning Hausdorff sets forth a detailed set algebra with some pioneering new concepts (differences chains, set rings and set fields, formula_20- and formula_21-systems). These introductory paragraphs on sets and their connections included, for example, the modern set-theoretic notion of functions. Next followed in Chapters 3 to 5 the classical theory of cardinal numbers, order types and ordinals. In the sixth chapter "Relations between ordered and well-ordered sets" Hausdorff presents, among other things, the most important results of his own research on ordered sets.
Set theory
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.
Infinity (philosophy)
Modern discussion of the infinite is now regarded as part of set theory and mathematics. Contemporary philosophers of mathematics engage with the topic of infinity and generally acknowledge its role in mathematical practice. But, although set theory is now widely accepted, this was not always so. Influenced by L.E.J Brouwer and verificationism in part, Wittgenstein (April 1889 Vienna - April 1951 Cambridge, England ), made an impassioned attack upon axiomatic set theory, and upon the idea of the actual infinite, during his "middle period".
Georg Cantor
Cantor introduced fundamental constructions in set theory, such as the power set of a set "A", which is the set of all possible subsets of "A". He later proved that the size of the power set of "A" is strictly larger than the size of "A", even when "A" is an infinite set; this result soon became known as Cantor's theorem. Cantor developed an entire theory and arithmetic of infinite sets, called cardinals and ordinals, which extended the arithmetic of the natural numbers. His notation for the cardinal numbers was the Hebrew letter formula_1 (aleph) with a natural number subscript; for the ordinals he employed the Greek letter ω (omega). This notation is still in use today.
Set theory
The next wave of excitement in set theory came around 1900, when it was discovered that some interpretations of 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 "The Principles of Mathematics".
Zermelo–Fraenkel set theory
In 1908, Ernst Zermelo proposed the first axiomatic set theory, Zermelo set theory. However, as first pointed out by Abraham Fraenkel in a 1921 letter to Zermelo, this theory was incapable of proving the existence of certain sets and cardinal numbers whose existence was taken for granted by most set theorists of the time, notably the cardinal number ℵ and the set {Z, ℘(Z), ℘(℘(Z))...}, where Z is any infinite set and ℘ is the power set operation. Moreover, one of Zermelo's axioms invoked a concept, that of a "definite" property, whose operational meaning was not clear. In 1922, Fraenkel and Thoralf Skolem independently proposed operationalizing a "definite" property as one that could be formulated as a well-formed formula in a first-order logic whose atomic formulas were limited to set membership and identity. They also independently proposed replacing the axiom schema of specification with the axiom schema of replacement. Appending this schema, as well as the axiom of regularity (first proposed by Dimitry Mirimanoff in 1917), to Zermelo set theory yields the theory denoted by ZF. Adding to ZF either the axiom of choice (AC) or a statement that is equivalent to it yields ZFC.
Kuratowski's free set theorem
Kuratowski's free set theorem, named after Kazimierz Kuratowski, is a result of set theory, an area of mathematics. It is a result which has been largely forgotten for almost 50 years, but has been applied recently in solving several lattice theory problems, such as the congruence lattice problem.
Gyula Kőnig
Nevertheless, Kőnig continued his efforts to disprove parts of set theory. In 1905 he published a paper proving that not all sets could be well-ordered.
Set (music)
In the theory of serial music, however, some authors (notably Milton Babbitt) use the term "set" where others would use "row" or "series", namely to denote an ordered collection (such as a twelve-tone row) used to structure a work. These authors speak of "twelve tone sets", "time-point sets", "derived sets", etc. (See below.) This is a different usage of the term "set" from that described above (and referred to in the term "set theory").
For these authors, a "set form" (or "row form") is a particular arrangement of such an ordered set: the prime form (original order), inverse (upside down), retrograde (backwards), and retrograde inverse (backwards and upside down).
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.
Absoluteness
Issues of absoluteness are particularly important in set theory and model theory, fields where multiple structures are considered simultaneously. In model theory, several basic results and definitions are motivated by absoluteness. In set theory, the issue of which properties of sets are absolute is well studied. The Shoenfield absoluteness theorem, due to Joseph Shoenfield (1961), establishes the absoluteness of a large class of formulas between a model of set theory and its constructible universe, with important methodological consequences. The absoluteness of large cardinal axioms is also studied, with positive and negative results known.
Arithmetization of analysis
An important spinoff of the arithmetization of analysis is set theory. Naive set theory was created by Cantor and others after arithmetization was completed as a way to study the singularities of functions appearing in calculus.
Discrete mathematics
In discrete mathematics, countable sets (including finite sets) are the main focus. The beginning of set theory as a branch of mathematics is usually marked by Georg Cantor's work distinguishing between different kinds of infinite set, motivated by the study of trigonometric series, and further development of the theory of infinite sets is outside the scope of discrete mathematics. Indeed, contemporary work in descriptive set theory makes extensive use of traditional continuous mathematics.
Paradoxes of set theory
Based upon work of the German mathematician Leopold Löwenheim (1915) the Norwegian logician Thoralf Skolem showed in 1922 that every consistent theory of first-order predicate calculus, such as set theory, has an at most countable model. However, Cantor's theorem proves that there are uncountable sets. The root of this seeming paradox is that the countability or noncountability of a set is not always absolute, but can depend on the model in which the cardinality is measured. It is possible for a set to be uncountable in one model of set theory but countable in a larger model (because the bijections that establish countability are in the larger model but not the smaller one).
Systems theory in anthropology
British anthropologist, Gregory Bateson, is the most influential and one of the earliest founders of System Theory in anthropology. He developed an interdisciplinary approach that included communication theory, cybernetics, and mathematical logic. In his collection of essays, "The Sacred Unity", Bateson argues that there are "ecological systems, social systems, and the individual organism plus the environment with which it interacts is itself a system in this technical sense." By adding environment with systems, Bateson closes the gap between the dualities such as subject and object. "Playing upon the differences between formalization and process, or crystallization and randomness, Bateson sought to transcend other dualisms–mind versus nature, organism versus environment, concept versus context, and subject versus object." Bateson set out the general rule of systems theory. He says:
Alternative set theory
Some of the alternative set theories are:Specifically, Alternative Set Theory (or AST) refers to a particular set theory developed in the 1970s and 1980s by Petr Vopěnka and his students. It builds on some ideas of the theory of semisets, but also introduces more radical changes: for example, all sets are "formally" finite, which means that sets in AST satisfy the law of mathematical induction for set-formulas (more precisely: the part of AST that consists of axioms related to sets only is equivalent to the Zermelo–Fraenkel (or ZF) set theory, in which the axiom of infinity is replaced by its negation). However, some of these sets contain subclasses that are not sets, which makes them different from Cantor (ZF) finite sets and they are called infinite in AST.
Named set theory
A name is given to a person, place, or thing to identify it. For example, parents can give their child a name or scientist can give an element a name. Examples of named sets include,
Henri Poincaré (1908) wrote that without a name no object exists in science or mathematics. Examples of such mathematical objects and their names as applications of named sets include,Therefore, any set is actually a named set of the form (X, ∈, "X" ) where X is a set (without a name), "X" is the name of this set and connects elements from X to the name "X". That is why any description of set theory begins with a fundamental binary relation ∈. The next is basic binary relation ⊆ between two sets called the subset relation, or set inclusion. Some basic named sets of central importance are the empty named set (the unique set containing no objects, no names and the empty connection), the named set of natural numbers, in which numbers have names in natural languages and numerical systems (for example, the number ten has names: "10", "ten", "1010", "diez", "dix", "zehn", "X", and many others) and the set of real numbers, in which numbers are assigned point of a straight line as their names.
Von Neumann–Bernays–Gödel set theory
In 1931, Bernays sent a letter containing his set theory to Kurt Gödel. Gödel simplified Bernays' theory by making every set a class, which allowed him to use just one sort and one membership primitive. He also weakened some of Bernays' axioms and replaced von Neumann's choice axiom with the equivalent axiom of global choice. Gödel used his axioms in his 1940 monograph on the relative consistency of global choice and the generalized continuum hypothesis.
Set theory
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, such as Russell's paradox, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with or without the axiom of choice, are the best-known.
Cantor's theorem
Ernst Zermelo has a theorem (which he calls "Cantor's Theorem") that is identical to the form above in the paper that became the foundation of modern set theory ("Untersuchungen über die Grundlagen der Mengenlehre I"), published in 1908. See Zermelo set theory.
Ordinal definable set
In mathematical set theory, a set "S" is said to be ordinal definable if, informally, it can be defined in terms of a finite number of ordinals by a first order formula. Ordinal definable sets were introduced by .
Georg Cantor's first set theory article
Georg Cantor published his first set theory article in 1874, and it contains the first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is "Cantor's revolutionary discovery" that the set of all real numbers is uncountably, rather than countably, infinite. This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument. The title of the article, "On a Property of the Collection of All Real Algebraic Numbers" (""Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen""), refers to its first theorem: the set of real algebraic numbers is countable. In 1879, Cantor modified his uncountability proof by using the topological notion of a set being dense in an interval.
History of the function concept
Set theory began with the work of the logicians with the notion of "class" (modern "set") for example , Jevons (1880), , and . It was given a push by Georg Cantor's attempt to define the infinite in set-theoretic treatment (1870–1890) and a subsequent discovery of an antinomy (contradiction, paradox) in this treatment (Cantor's paradox), by Russell's discovery (1902) of an antinomy in Frege's 1879 (Russell's paradox), by the discovery of more antinomies in the early 20th century (e.g., the 1897 Burali-Forti paradox and the 1905 Richard paradox), and by resistance to Russell's complex treatment of logic and dislike of his axiom of reducibility (1908, 1910–1913) that he proposed as a means to evade the antinomies.
S (set theory)
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.
Skolem's paradox
In 1925, von Neumann presented a novel axiomatisation of set theory, which developed into NBG set theory. Very much aware of Skolem's 1922 paper, von Neumann investigated countable models of his axioms in detail. In his concluding remarks, Von Neumann comments that there is no categorical axiomatisation of set theory, or any other theory with an infinite model. Speaking of the impact of Skolem's paradox, he wrote,
Sperner's theorem
Sperner's theorem, in discrete mathematics, describes the largest possible families of finite sets none of which contain any other sets in the family. It is one of the central results in extremal set theory, and is named after Emanuel Sperner, who published it in 1928.
Diatonic set theory
Music theorists working in diatonic set theory include Eytan Agmon, Gerald J. Balzano, Norman Carey, David Clampitt, John Clough, Jay Rahn, and mathematician Jack Douthett. A number of key concepts were first formulated by David Rothenberg, who published in the journal "Mathematical Systems Theory", and Erv Wilson, working entirely outside of the academic world.
Set theory
From set theory's inception, some mathematicians have 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. The feasibility of constructivism as a substitute foundation for mathematics was greatly increased by Errett Bishop's influential book "Foundations of Constructive Analysis".
Von Neumann–Bernays–Gödel set theory
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel set theory (ZFC). NBG introduces the notion of class, which is a collection of sets defined by a formula whose quantifiers range only over sets. NBG can define classes that are larger than sets, such as the class of all sets and the class of all ordinals. Morse–Kelley set theory (MK) allows classes to be defined by formulas whose quantifiers range over classes. NBG is finitely axiomatizable, while ZFC and MK are not.
Naive Set Theory (book)
Naive Set Theory is a mathematics textbook by Paul Halmos providing an undergraduate introduction to set theory. Originally published by "Van Nostrand" in 1960, it was reprinted in the Springer-Verlag Undergraduate Texts in Mathematics series in 1974.
Rough set
The following section contains an overview of the basic framework of rough set theory, as originally proposed by Zdzisław I. Pawlak, along with some of the key definitions. More formal properties and boundaries of rough sets can be found in Pawlak (1991) and cited references. The initial and basic theory of rough sets is sometimes referred to as ""Pawlak Rough Sets"" or ""classical rough sets"", as a means to distinguish from more recent extensions and generalizations.
Naive set theory
Unlike axiomatic set theories, which are defined using formal logic, naïve 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.
Set theory
Set theory begins with a fundamental binary relation between an object and a set . If is a member (or element) of , the notation is used. Since sets are objects, the membership relation can relate sets as well.
Mathematical logic
Zermelo (1908b) provided the first set of axioms for set theory. These axioms, together with the additional axiom of replacement proposed by Abraham Fraenkel, are now called Zermelo–Fraenkel set theory (ZF). Zermelo's axioms incorporated the principle of limitation of size to avoid Russell's paradox.
Skolem's paradox
One of the earliest results in set theory, published by Georg Cantor in 1874, was the existence of uncountable sets, such as the powerset of the natural numbers, the set of real numbers, and the Cantor set. An infinite set "X" is countable if there is a function that gives a one-to-one correspondence between "X" and the natural numbers, and is uncountable if there is no such correspondence function. When Zermelo proposed his axioms for set theory in 1908, he proved Cantor's theorem from them to demonstrate their strength.
Von Neumann–Bernays–Gödel set theory
John von Neumann introduced classes into set theory in 1925. The primitive notions of his theory were function and argument. Using these notions, he defined class and set. Paul Bernays reformulated von Neumann's theory by taking class and set as primitive notions. Kurt Gödel simplified Bernays' theory for his relative consistency proof of the axiom of choice and the generalized continuum hypothesis.
Ordered pair
Another way to rigorously deal with ordered pairs is to define them formally in the context of set theory. This can be done in several ways and has the advantage that existence and the characteristic property can be proven from the axioms that define the set theory. One of the most cited versions of this definition is due to Kuratowski (see below) and his definition was used in the second edition of Bourbaki's "Theory of Sets", published in 1970. Even those mathematical textbooks that give an informal definition of ordered pairs will often mention the formal definition of Kuratowski in an exercise.
Derived set (mathematics)
The concept was first introduced by Georg Cantor in 1872 and he developed set theory in large part to study derived sets on the real line.
Von Neumann universe
In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory (ZFC), is often used to provide an interpretation or motivation of the axioms of ZFC.
Nested set model
The nested set model is a particular technique for representing nested sets (also known as trees or hierarchies) in relational databases.
The term was apparently introduced by Joe Celko; others describe the same technique using different terms.
Scott–Potter set theory
Potter (1990, 2004) clarified and simplified the approach of Scott (1974), and showed how the resulting axiomatic set theory can do what is expected of such theory, namely grounding the cardinal and ordinal numbers, Peano arithmetic and the other usual number systems, and the theory of relations.
Anthony Bartholomay
Anthony Francis Bartholomay (1919–1975) was a mathematician who introduced molecular set theory, a topic on which he wrote books.
Felix Hausdorff
Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, function theory, and functional analysis.
Constantin Carathéodory
He is credited with the Carathéodory extension theorem which is fundamental to modern set theory. Later Carathéodory extended the theory from sets to Boolean algebras.
Von Neumann–Bernays–Gödel set theory
Von Neumann's work in set theory was influenced by Georg Cantor's articles, Ernst Zermelo's 1908 axioms for set theory, and the 1922 critiques of Zermelo's set theory that were given independently by Abraham Fraenkel and Thoralf Skolem. Both Fraenkel and Skolem pointed out that Zermelo's axioms cannot prove the existence of the set {"Z", "Z", "Z", ...} where "Z" is the set of natural numbers and "Z" is the power set of "Z". They then introduced the axiom of replacement, which would guarantee the existence of such sets. However, they were reluctant to adopt this axiom: Fraenkel stated "that Replacement was too strong an axiom for 'general set theory'", while "Skolem only wrote that 'we could introduce' Replacement".
Wacław Sierpiński
Wacław Franciszek Sierpiński () (14 March 1882 – 21 October 1969) was a Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions and topology. He published over 700 papers and 50 books.
Set theory
Set theory is a branch of mathematical logic that studies sets, which informally 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 to define nearly all mathematical objects.
Minimal model (set theory)
In set theory, the minimal model is the minimal standard model of ZFC.
The minimal model was introduced by and rediscovered by .
Naive set theory
The first development of set theory was a naïve set theory. It was created at the end of the 19th century by Georg Cantor as part of his study of infinite sets and developed by Gottlob Frege in his "Begriffsschrift".
Von Neumann universe
Existence and uniqueness of the general transfinite recursive definition of sets was demonstrated in 1928 by von Neumann for both Zermelo-Fraenkel set theory and Neumann's own set theory (which later developed into NBG set theory). In neither of these papers did he apply his transfinite recursive method to construct the universe of all sets. The presentations of the von Neumann universe by Bernays and Mendelson both give credit to von Neumann for the transfinite induction construction method, although not for its application to the construction of the universe of ordinary sets.
Mostowski model
In mathematical set theory, the Mostowski model is a model of set theory with atoms where the full axiom of choice fails, but every set can be linearly ordered. It was introduced by . The Mostowski model can be constructed as the permutation model corresponding to the group of all automorphisms of the ordered set of rational numbers and the ideal of finite subsets of the rational numbers.
Naive set theory
Naïve set theory is any of several theories of sets used in the discussion of the foundations of mathematics.
Named set theory
Named set theory is a branch of theoretical mathematics that studies the structures of names. The named set is a theoretical concept that generalizes the structure of a name described by Frege. Its generalization bridges the descriptivists theory of a name, and its triad structure (name, sensation and reference), with mathematical structures that define mathematical names using triplets. It deploys the former to view the latter at a higher abstract level that unifies a name and its relationship to a mathematical structure as a constructed reference. This enables all names in science and technology to be treated as named sets or as systems of named sets.
Morass (set theory)
In axiomatic set theory, a mathematical discipline, a morass is an infinite combinatorial structure, used to create "large" structures from a "small" number of "small" approximations. They were invented by Ronald Jensen for his proof that cardinal transfer theorems hold under the axiom of constructibility. A far less complex but equivalent variant known as a simplified morass was introduced by Velleman, and the term morass is now often used to mean these simpler structures.
Anthony Morse
Anthony Perry Morse (1911–1984) was an American mathematician who worked in both analysis, especially measure theory, and in the foundations of mathematics. He is best known as the co-creator, together with John L. Kelley, of Morse–Kelley set theory. This theory first appeared in print in Kelley's "General Topology". Morse's own version appeared later in "A Theory of Sets".
Set theory
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:
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.
Zermelo set theory
Mac Lane set theory, introduced by , is Zermelo set theory with the axiom of separation restricted to first-order formulas in which every quantifier is bounded,
Mac Lane set theory is similar in strength to topos theory with a natural number object, or to the system in Principia mathematica. It is strong enough to carry out almost all ordinary mathematics not directly connected with set theory or logic.
Set theory
Ludwig Wittgenstein condemned set theory. He wrote that "set theory is wrong", since it builds on the "nonsense" of fictitious symbolism, has "pernicious idioms", and that it is nonsensical to talk about "all numbers". Wittgenstein's views about the foundations of mathematics were later criticised by Georg Kreisel and Paul Bernays, and investigated by Crispin Wright, among others.
Indian logic
This later school began around eastern India and Bengal, and developed theories resembling modern logic, such as Gottlob Frege's "distinction between sense and reference of proper names" and his "definition of number," as well as the Navya-Nyaya theory of "restrictive conditions for universals" anticipating some of the developments in modern set theory. Udayana in particular developed theories on "restrictive conditions for universals" and "infinite regress" that anticipated aspects of modern set theory. According to Kisor Kumar Chakrabarti:
Bass–Serre theory
Bass–Serre theory was developed by Jean-Pierre Serre in the 1970s and formalized in "Trees", Serre's 1977 monograph (developed in collaboration with Hyman Bass) on the subject. Serre's original motivation was to understand the structure of certain algebraic groups whose Bruhat–Tits buildings are trees. However, the theory quickly became a standard tool of geometric group theory and geometric topology, particularly the study of 3-manifolds. Subsequent work of Bass contributed substantially to the formalization and development of basic tools of the theory and currently the term "Bass–Serre theory" is widely used to describe the subject.
Set theory
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.
Set theory
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.
Part–whole theory
The Cantor–Peano concept of set did not become canonical until about 1910, when the first volume of "Principia Mathematica" appeared, and right after Ernst Zermelo proposed the first axiomatization of set theory in 1908.
General set theory
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.
Felix Hausdorff
According to former notions, set theory included not only the general set theory and the theory of sets of points, but also dimension and measure theory. Hausdorff's work was the first textbook which presented all of set theory in this broad sense, systematically and with full proofs. Hausdorff was aware of how easily the human mind can err while also seeking for rigor and truth. So he proposed in the preface of the work:
Grundzüge der Mengenlehre
First published in April 1914, "Grundzüge der Mengenlehre" was the first comprehensive introduction to set theory. Besides the systematic treatment of known results in set theory, the book also contains chapters on measure theory and topology, which were then still considered parts of set theory. Hausdorff presented and developed original material which was later to become the basis for those areas. In 1927 Hausdorff published an extensively revised second edition under the title "Mengenlehre" (German for "Set Theory"), with many of the topics of the first edition omitted. In 1935 there was a third German edition, which in 1957 was translated by John R. Aumann "et al." into English under the title "Set Theory".