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Topic: Axiomatic set theory


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Set

  
  Set theory
Bolzano gave examples to show that, unlike for finite sets, the elements of an infinite set could be put in 1-1 correspondence with elements of one of its proper subsets.
By this stage, however, set theory was beginning to have a major impact on other areas of mathematics.
Analysis needed the set theory of Cantor, it could not afford to limit itself to intuitionist style mathematics in the spirit of Kronecker.
www-groups.dcs.st-and.ac.uk /~history/HistTopics/Beginnings_of_set_theory.html   (2182 words)

  
  Axiomatic set theory
Initially controversial, set theory has come to play the role of a foundational theory in modern mathematics, in the sense of a theory invoked to justify assumptions made in mathematics concerning the existence of mathematical objects (such as numbers or functions) and their properties.
Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century.
The important idea of Cantor's, which got set theory going as a new field of study, was to define two sets A and B to have the same number of members (the same cardinality) when there is a way of pairing off members of A exhaustively with members of B.
www.brainyencyclopedia.com /encyclopedia/a/ax/axiomatic_set_theory.html   (2759 words)

  
  Set theory - Wikipedia, the free encyclopedia
In axiomatic set theory, the concepts of sets and set membership are defined indirectly by first postulating certain axioms which specify their properties.
Naive set theory is the original set theory developed by mathematicians at the end of the 19th century.
Axiomatic set theory is a rigorous axiomatic theory developed in response to the discovery of serious flaws (such as Russell's paradox) in naïve set theory.
en.wikipedia.org /wiki/Set_theory   (391 words)

  
 Facts about topic: (Axiomatic set theory)   (Site not responding. Last check: )
Initially controversial, set theory has come to play the role of a foundational theory (additional info and facts about foundational theory) in modern mathematics, in the sense of a theory invoked to justify assumptions made in mathematics concerning the existence of mathematical objects (such as numbers or functions) and their properties.
To address these problems, set theory had to be re-constructed, this time using an axiom ((logic) a proposition that is not susceptible of proof or disproof; its truth is assumed to be self-evident) atic approach.
The most frequent objection to set theory is the constructivist (An artist of the school of constructivism) view that mathematics is loosely related to computation and that naive set theory (additional info and facts about naive set theory) is being formalised with the addition of noncomputational elements.
www.absoluteastronomy.com /encyclopedia/a/ax/axiomatic_set_theory.htm   (2977 words)

  
 Amazon.com: Axiomatic Set Theory: Books: Patrick Suppes   (Site not responding. Last check: )
Set theory, the theory of types, and mathematical logic are still very important though in computer science and in artificial intelligence, due to the needs in these fields for knowledge representation, computational models of intelligence, and automated reasoning.
The notion of a set is defined formally, and then the axiom of extensionality, which gives a criterion for two sets being equal, and the axiom schema schema of separation.
The theory of denumerable sets is then discussed, followed by one of the most fascinating concepts in all of mathematics: the theory of transfinite and infinite cardinals.
www.amazon.com /Axiomatic-Set-Theory-Patrick-Suppes/dp/0486616304   (2366 words)

  
 ScienceDaily: Mathematics   (Site not responding. Last check: )
A "Unified Theory" For Calculus (January 29, 2003) -- A University of Missouri-Rolla mathematician's research into a "unified theory" of continuous and discrete calculus is gaining the attention of mathematicians worldwide for numerous applications,...
Probability theory -- Probability theory is the mathematical study of phenomena characterized by randomness or uncertainty.
Experiment -- In the scientific method, an experiment is a set of actions and observations, performed in the context of solving a particular problem or question, to support or falsify a hypothesis or research...
www.sciencedaily.com /encyclopedia/mathematics   (1370 words)

  
 Axiomatic set theory - Wikipedia, the free encyclopedia
Cantor's development of set theory was still "naïve" in the sense that he did not have a precise axiomatization in mind.
The most frequent objection to set theory is the constructivist view that mathematics is loosely related to computation and that naive set theory is being formalised with the addition of noncomputational elements.
Topos theory can be used to interpret various alternatives to set theory such as constructivism, fuzzy set theory, finite set theory, and computable set theory.
en.wikipedia.org /wiki/Axiomatic_set_theory   (2664 words)

  
 Encyclopedia: Axiomatic set theory   (Site not responding. Last check: )
In abstract mathematics, naive set theory1 was the first development of set theory, which was later to be reconstructed as axiomatic set theory.
Set theory is a branch of mathematics and computer science created principally by the German mathematician Georg Cantor at the end of the 19th century.
Henri Poincaré said "set theory is a disease from which mathematics will one day recover", [note that this quote is part of the folklore of mathematics, but it's hard to find the original quote] and Errett Bishop dismissed set theory as God's mathematics, which we should leave for God to do.
www.nationmaster.com /encyclopedia/Axiomatic-set-theory   (976 words)

  
 Set - Encyclopedia.WorldSearch   (Site not responding. Last check: )
Sets are one of the most important and fundamental concepts in modern mathematics.
Basic set theory, having only been invented at the end of the 19th century, is now a ubiquitous part of mathematics education, being introduced as early as elementary school.
Set identity does not depend on the order in which the elements are listed, nor on whether there are repetitions in the list.
encyclopedia.worldsearch.com /set.htm   (1490 words)

  
 Present status of axiomatic set theory (from set theory) --  Encyclopædia Britannica
The foundations of axiomatic set theory are in a state of significant change as a result of new discoveries.
The theory is less valuable in direct application to ordinary experience than as a basis for precise and adaptable terminology for the definition of complex and sophisticated mathematical concepts.
in mathematics and mechanics, theory that studies systems behaving unpredictably and randomly despite their seeming simplicity and fact that forces involved are supposedly governed by well-understood physical laws; applications of theory are diverse, including study of turbulent flow of fluids, irregularities in heartbeat, traffic jams, population dynamics, chemical...
www.britannica.com /eb/article-24045   (858 words)

  
 Set theory article - Set theory sets Naive theory Axiomatic theory Russell's Paradox logic fuzzy - What-Means.com   (Site not responding. Last check: )
It has a central role in modern mathematical theory, providing the basic language in which most of mathematics is expressed.
Axiomatic set theory is a rigorous axiomatic theory developed in response to the discovery of serious flaws (such as Russell's Paradox) in naive set theory.
Musical set theory may be considered the application of mathematical set theory to music.
www.what-means.com /encyclopedia/Set_theory   (183 words)

  
 Set Theory. Zermelo-Fraenkel Axioms. Russell's Paradox. Infinity. By K.Podnieks
set theory, axioms, Zermelo, Fraenkel, Frankel, infinity, Cantor, Frege, Russell, paradox, formal, axiomatic, Russell paradox, axiom, axiomatic set theory, comprehension, axiom of infinity, ZF, ZFC
The set theory adopting the axiom of extensionality (C1), the axiom C1', the separation axiom schema (C21), the pairing axiom (C22), the union axiom (C23), the power-set axiom (C24), the replacement axiom schema (C25), the axiom of infinity (C26) and the axiom of regularity (C3), is called Zermelo-Fraenkel set theory, and is denoted by ZF.
The set theory ZF+AC is denoted traditionally by ZFC.
www.ltn.lv /~podnieks/gt2.html   (8336 words)

  
 Axiomatic Set Theory by Paul Bernays, ISBN 0486666379 - Resin patio set - Set Plays: Organizing and Coaching Dead Ball ...
Axiomatic Set Theory by Paul Bernays, ISBN 0486666379
An in-depth look at set plays and their importance in the game of soccer, this book examines the fact that a high percentage of goals are scored as the result of dead ball situations.
An innovative problem-oriented introduction to set theory, this volume is intended for undergraduate courses in which students work in groups on projects and present their solutions to the class.
www.resinpatiosetstore.com /Axiomatic-Set-Theory-by-Paul-Bernays,-ISBN-0486666379/Page/423695   (590 words)

  
 Axiomatic Set Theory   (Site not responding. Last check: )
I will use “collection” for the ordinary idea of set but bear in mind that some particular collections that you may know, may not have set counterparts that are forced to exist by what ever collection of axioms are on the table at some moment.
The set of subsets of integers is, in ordinary logic, uncountable.
The proof of uncountability of the integer subsets is by Cantor and uses arguments that are perfectly valid in elementary set theory.
www.cap-lore.com /MathPhys/SetAxioms.html   (694 words)

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