Instruct: Given a question, retrieve relevant passages that can help answer the question.
Query: Who developed set theory?
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 Property of the Collection of All Real Algebraic Numbers".[1][2]
While René Thom is most known to the public for his development of catastrophe theory between 1968 and 1972, his earlier work was on differential topology. In the early 1950s it concerned what are now called Thom spaces, characteristic classes, cobordism theory, and the Thom transversality theorem. Another example of this line of work is the Thom conjecture, versions of which have been investigated using gauge theory. From the mid 1950s he moved into singularity theory, of which catastrophe theory is just one aspect, and in a series of deep (and at the time obscure) papers between 1960 and 1969 developed the theory of stratified sets and stratified maps, proving a basic stratified isotopy theorem describing the local conical structure of Whitney stratified sets, now known as the Thom–Mather isotopy theorem. Much of his work on stratified sets was developed so as to understand the notion of topologically stable maps, and to eventually prove the result that the set of topologically stable mappings between two smooth manifolds is a dense set. Thom's lectures on the stability of differentiable mappings, given at the University of Bonn in 1960, were written up by Harold Levine and published in the proceedings of a year long symposium on singularities at Liverpool University during 1969-70, edited by C. T. C. Wall. The proof of the density of topologically stable mappings was completed by John Mather in 1970, based on the ideas developed by Thom in the previous ten years. A coherent detailed account was published in 1976 by Cristopher Gibson, Klaus Wirthmüller, Andrew du Plessis, and Eduard Looijenga.
The Kripke–Platek axioms of set theory (KP), pronounced , are a system of axiomatic set theory developed by Saul Kripke and Richard Platek.
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".
Categorical set theory is any one of several versions of set theory developed from or treated in the context of mathematical category theory.
Claude Ponsard (1927–1990) was a French economist who worked in spatial economics and in the application of fuzzy set theory to economics. Bo Yuan and George J. Klir noted that Ponsard was a "pioneer who initiated the reformulation of economic theory by taking advantage of fuzzy set theory" in their book, "Fuzzy sets and fuzzy logic theory and applications" (1995).
Set Theory: An Introduction to Independence Proofs is a textbook and reference work in set theory by Kenneth Kunen. It starts from basic notions, including the ZFC axioms, and quickly develops combinatorial notions such as trees, Suslin's problem, ◊, and Martin's axiom. It develops some basic model theory (rather specifically aimed at models of set theory) and the theory of Gödel's constructible universe L.
An extension (named ARC) of Ackermann set theory was developed by F.A. Muller(2001), who stated that ARC "founds Cantorian set-theory as well as category-theory and therefore can pass as a founding theory of the whole of mathematics".
The two-factor theory (also known as Herzberg's motivation-hygiene theory and dual-factor theory) states that there are certain factors in the workplace that cause job satisfaction, while a separate set of factors cause dissatisfaction. It was developed by psychologist Frederick Herzberg, who theorized that job satisfaction and job dissatisfaction act independently of each other.
It is possible to give a precise meaning to the claim that SN is the universe of ordinary mathematics; it is a model of Zermelo set theory, the axiomatic set theory originally developed by Ernst Zermelo in 1908. Zermelo set theory was successful precisely because it was capable of axiomatising "ordinary" mathematics, fulfilling the programme begun by Cantor over 30 years earlier. But Zermelo set theory proved insufficient for the further development of axiomatic set theory and other work in the foundations of mathematics, especially model theory.
Von Neumann developed the axiom of limitation of size as a new method of identifying sets. ZFC identifies sets via its set building axioms. However, as Abraham Fraenkel pointed out: "The rather arbitrary character of the processes which are chosen in the axioms of Z [ZFC] as the basis of the theory, is justified by the historical development of set-theory rather than by logical arguments."
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:
In the mid-1980s, McLeod encountered the work of Dutch composer Peter Schat, who had developed a post-tonal compositional technique called the Tone Clock. This technique emphasised the role on forming the chromatic aggregate through the transposition and inversion of three-note chords (trichords). McLeod expanded this technique to encompass all 223 possible set-classes (to use the terminology of Allan Forte's pitch-class set theory), and also developed an exceptionally coherent labelling, categorisation and analytical approach to the chromatic universe. Her unpublished manuscript "Tone Clock Theory Expanded: Chromatic Maps I & II" explains the theoretical and philosophical basis behind her theory, and includes a rigorous listing of all set-classes (called Intervallic Prime Forms) with details notes and observations on their properties.
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:
The development of category theory in the middle of the 20th century showed the usefulness of set theories guaranteeing the existence of larger classes than does ZFC, such as Von Neumann–Bernays–Gödel set theory or Tarski–Grothendieck set theory, albeit that in very many cases the use of large cardinal axioms or Grothendieck Universes is formally eliminable.
In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic includes the mathematical study of logic and the applications of formal logic to other areas of mathematics; set theory is the branch of mathematics that studies sets or collections of objects. Category theory, which deals in an abstract way with mathematical structures and relationships between them, is still in development. The phrase "crisis of foundations" describes the search for a rigorous foundation for mathematics that took place from approximately 1900 to 1930. Some disagreement about the foundations of mathematics continues to the present day. The crisis of foundations was stimulated by a number of controversies at the time, including the controversy over Cantor's set theory and the Brouwer–Hilbert controversy.
Appropriation in education is often understood and analyzed through activity theory. This theory was developed by Aleksei N. Leontiev and focuses on understanding the socio-cultural context (specifically the setting) learning occurs in. Activity theory is predicated on the assumption that a person's frameworks for thinking are developed and carried out in specific settings, and that these settings mediate cognitive development. Since appropriation also places a strong emphasis on setting, these theories complement each other when being used to analyze learning environments.
The use of the term "quasi-set" follows a suggestion in da Costa's 1980 monograph "Ensaio sobre os Fundamentos da Lógica" (see da Costa and Krause 1994), in which he explored possible semantics for what he called "Schrödinger Logics". In these logics, the concept of identity is restricted to some objects of the domain, and has motivation in Schrödinger's claim that the concept of identity does not make sense for elementary particles (Schrödinger 1952). Thus in order to provide a semantics that fits the logic, da Costa submitted that "a theory of quasi-sets should be developed", encompassing "standard sets" as particular cases, yet da Costa did not develop this theory in any concrete way. To the same end and independently of da Costa, Dalla Chiara and di Francia (1993) proposed a theory of "quasets" to enable a semantic treatment of the language of microphysics. The first quasi-set theory was proposed by D. Krause in his PhD thesis, in 1990 (see Krause 1992). A related physics theory, based on the logic of adding fundamental indistinguishability to equality and inequality, was developed and elaborated independently in the book "The Theory of Indistinguishables" by A. F. Parker-Rhodes.
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.
Presheaf models for constructive set theory were introduced by Nicola Gambino in 2004. They are analogous to the Presheaf models for intuitionistic set theory developed by Dana Scott in the 1980s (which remained unpublished).
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.
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.
General Systems Theory (GST) laid the foundation to systemic thinking. Ludwig Von Bertalanffy was known as the founder of the original principles of GST. Prior to 1968, when GST was introduced in Bertalanffy’s book, "General System Theory: Foundations, Development, Applications", the traditional approach to development used linear thinking or cause-and-effect thinking. GST was influenced by many different types of theories such as “Chaos theory, complexity theory, catastrophe theory, cybernetics, fuzzy set theory, and learning matrices” In 1990, Peter Senge, author of "The Fifth Discipline", wove systemic thinking approach into development.
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".
In the chapters on "point sets"—the topological chapters—Hausdorff developed for the first time, based on the known neighborhood axioms, a systematic theory of topological spaces, where in addition he added the separation axiom later named after him. This theory emerges from a comprehensive synthesis of earlier approaches of other mathematicians and Hausdorff's own reflections on the problem of space. The concepts and theorems of classical point set theory formula_22 are—as far as possible—transferred to the general case, and thus become part of the newly created general or set-theoretic topology. But Hausdorff not only performed this "translation work", but he developed also basic construction method of topology as nucleation (interior, dense-in-itself core) and shell formation (closure), and he works the fundamental importance of the concept of open set (called "area" by him) and of the compactness introduced by Fréchet. He also founded and developed the theory of the connected set, particularly through the introduction of the terms "component" and "quasi-component".