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Structural Set Theory
In mathematics, a structural set theory is an approach to set theory that emphasizes the aspect of sets as abstract structures. It is in contrast to a more traditional ZFC set-theory, which emphasizes membership. A prime example is Lawvere's Elementary Theory of the Category of Sets, which identifies sets in terms of relations to each other through functions. Another example is SEAR (Sets, Elements, And Relations). The adjective "structural" comes from the structuralism Structuralism is an intellectual current and methodological approach, primarily in the social sciences, that interprets elements of human culture by way of their relationship to a broader system. It works to uncover the structural patterns t ... in the philosophy of mathematics. References * * François G. DoraisBack to Cantor? a blog post Further reading structural set theory in nLab Set theory {{improve categories, date=December 2024 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of ''naive set theory''. After the discovery of Paradoxes of set theory, paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox), various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied. Set the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Theory Of The Category Of Sets
Elementary may refer to: Arts, entertainment, and media Music * ''Elementary'' (Cindy Morgan album), 2001 * ''Elementary'' (The End album), 2007 * ''Elementary'', a Melvin "Wah-Wah Watson" Ragin album, 1977 Other uses in arts, entertainment, and media * ''Elementary'' (TV series), a 2012 American drama television series * "Elementary, my dear Watson", a catchphrase of Sherlock Holmes Education * Elementary and Secondary Education Act, US * Elementary education, or primary education, the first years of formal, structured education * Elementary Education Act 1870, England and Wales * Elementary school, a school providing elementary or primary education Science and technology * ELEMENTARY, a class of objects in computational complexity theory * Elementary, a widget set based on the Enlightenment Foundation Libraries * Elementary abelian group, an abelian group in which every nontrivial element is of prime order * Elementary algebra * Elementary arithmetic * Elementary c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Structuralism (philosophy Of Mathematics)
Structuralism is a theory in the philosophy of mathematics that holds that mathematical theories describe structures of mathematical objects. Mathematical objects are exhaustively defined by their place in such structures. Consequently, structuralism maintains that mathematical objects do not possess any intrinsic properties but are defined by their external relations in a system. For instance, structuralism holds that the number 1 is exhaustively defined by being the successor of 0 in the structure of the theory of natural numbers. By generalization of this example, any natural number is defined by its respective place in that theory. Other examples of mathematical objects might include lines and planes in geometry, or elements and operations in abstract algebra. Structuralism is an epistemologically realistic view in that it holds that mathematical statements have an objective truth value. However, its central claim only relates to what ''kind'' of entity a mathematical o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
