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Topic: Set theory


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In the News (Fri 14 Dec 18)

  
  Set Theory (Stanford Encyclopedia of Philosophy)
The language of set theory, in its simplicity, is sufficiently universal to formalize all mathematical concepts and thus set theory, along with Predicate Calculus, constitutes the true Foundations of Mathematics.
There are four main directions of current research in set theory, all intertwined and all aiming at the ultimate goal of the theory: to describe the structure of the mathematical universe.
Rather, sets are introduced either informally, and are understood as something self-evident, or, as is now standard in modern mathematics, axiomatically, and their properties are postulated by the appropriate formal axioms.
plato.stanford.edu /entries/set-theory   (0 words)

  
  PlanetMath: set theory
Set theory is special among mathematical theories, in two ways: It plays a central role in putting mathematics on a reliable axiomatic foundation, and it provides the basic language and apparatus in which most of mathematics is expressed.
A category is not a set, and a functor is not a mapping, despite similarities in both cases.
This is version 8 of set theory, born on 2003-01-01, modified 2003-02-07.
planetmath.org /encyclopedia/SetTheory.html   (937 words)

  
  Axiomatic set theory - Wikipedia, the free encyclopedia
Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century.
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.
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.
en.wikipedia.org /wiki/Axiomatic_set_theory   (2693 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   (406 words)

  
 Set theory Summary
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 branch of mathematics developed in response to the discovery of serious flaws (such as Russell's paradox) in naïve set theory.
www.bookrags.com /Set_theory   (2598 words)

  
 Set theory
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.
www.brainyencyclopedia.com /encyclopedia/s/se/set_theory.html   (466 words)

  
 Set Theory   (Site not responding. Last check: )
The Universal set in a Venn diagram is drawn as a rectangle.
The intersection of two sets is the set of all elements in common to the two sets.
The intersection of the set of hollow spheres and the set of white balls are those elements that are in both circles of yarn (as demonstrated in class on the table).
shark.comfsm.fm /~dleeling/math/cogapp/setheory.html   (1375 words)

  
 All About Musical Set Theory
Keep in mind that sets and set classes determined pitch content only; the composers remained free to fashion all other aspects of the music according to their artistic desires (at least until super-serialism, a philosophy of subjecting every aspect of the music to serial techniques, came into fashion in the 1950s).
The set (2,9,10), for example, is not in normal form because the interval between 2 and 9 (7) is larger than the intervals between 9 and 10 (1) or between 10 and 2 (4).
Sets with the same prime form contain the same number of pitches and the same collection of intervals between its pitches, hence they are in some sense aurally "equivalent," in much the same way that all major chords are aurally equivalent in tonal music.
www.jaytomlin.com /music/settheory/help.html   (2147 words)

  
 Set Theory Primer
For instance, the statement that the major chord is a subset of the 12-note set, although true, is insignificant, because all sets are subsets of the 12-note set; i.e., the statement is not discriminating.
By this criteria it is assumed that the perception of the similarity of small sets is easier than is the perception of the similarity of large ones; i.e., the similarity of, say, 9-note sets would be more difficult to perceive than their complements, 3-note sets.
Two pc sets of the same cardinality can be mapped to one another, with the exception of one pc in one of the sets, which must be within a semitone of a match with the unmatched pc of the other set.
solomonsmusic.net /setheory.htm   (4701 words)

  
 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.
Cantor's early work was in number theory and he published a number of articles on this topic between 1867 and 1871.
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   (0 words)

  
 Quantifying Complexity Theory
the calculus of Newton and Liebnitz, the topology of Poincaré, non-Euclidean geometry of Riemann, statistics of Boltzmann, set theory of Cantor and renormalization of Wilson.
This theory takes the view that systems are best regarded as wholes, and studied as such, rejecting the traditional emphasis on simplification and reduction as inadequate techniques on which to base this sort of scientific work.
The approaches used in complexity theory are based on a number of new mathematical techniques, originating from fields as diverse as physics, biology, artificial intelligence, politics and telecommunications, and this interdisciplinary viewpoint is the crucial aspect, reflecting the general applicability of the theory to systems in all areas.
www.calresco.org /lucas/quantify.htm   (0 words)

  
 The Math Forum - Math Library - Set Theory
A tutorial on sets, convering the definition of sets and their elements, union, intersection, subsets, and sets of numbers.
Research in the group is concentrated on axiomatic set theory, in particular: Inner models and large cardinals; Descriptive set theory and Determinacy; Consistency strengths; and Forcing.
Set theory was introduced by W. Quine in 1937.
mathforum.org /library/topics/set_theory   (2258 words)

  
 Set Theory and Logic - Numericana
The collection of sets that are not members of themselves thus includes all sets and it is not a set itself.
is the set of all subsets of A.
A function f from set A to set B is said to be surjective when every element of B is the image of some element of A.
home.att.net /~numericana/answer/sets.htm   (3874 words)

  
 Set Theory :: 3DSoftware.com
For a set to be finite (and countable), none of the elements of the set are duplicated.
The complement of C is the set of all elements in the superset B that are not in C.
A mapping (to map a set of points) is a transformation of elements of one set into the elements of another set.
www.3dsoftware.com /Math/Programming/SetTheory   (2035 words)

  
 Peter Suber, "Infinite Sets"
Set A is a subset of set B iff all the members of A are also members of B. Notation.
Set A is a proper subset of set B iff all the members of A are also members of B, but not all the members of B are members of A. Notation.
Two sets can be put into one-to-one correspondence iff their members can be paired off such that each member of the first set has exactly one counterpart in the second set, and each member of the second set has exactly one counterpart in the first set.
www.earlham.edu /~peters/writing/infapp.htm   (6879 words)

  
 Set Theory
Set Theory and Venn Diagrams are very useful for clarifying and understanding classifications and definitions around the form of 'This is an X, it is not a Y' or 'X and Y have some things in common'.
In set theory, this means identifying to which sets it belongs.
Sets can also be nested, so one set is completely within another set (and is hence a subset).
changingminds.org /disciplines/argument/syllogisms/set_theory.htm   (490 words)

  
 RDA, JSC, DCAM, RDF, FRBR « Bibliographic Wilderness
In some ways I’m more attracted to Svenonius’ ’set theory’ approach to the abstract entities such as works (eg a work is a set of documents) rather than the top down approach of FRBR.
I like Svenonius’ set theory approach too, and don’t actually think it’s at all incompatible with the FRBR entity-relational model.
The thing I like about the set theory approach is that it gets (for me) the order of things right: starting from concrete things (documents, web pages, etc.) we abstract a work from them.
bibwild.wordpress.com /2007/05/04/rda-jsc-dcam-rdf-frbr   (609 words)

  
 Referativni Zhurnal Classification
Matrix theory 271.17.29.19.17 Determinants and their generalizations 271.17.29.19.21 Matrix equations 271.17.29.19.25 Eigenvalues of matrices 271.17.29.19.33 Special classes of matrices 271.17.29.21 Systems of linear equations and inequalities 271.17.29.31 Polylinear algebra.
Axiomatics 271.19.17.17.17.17 Investigation of topological spaces and continuous mappings by homological methods 271.19.17.17.17.17.15 Homology theory of dimension 271.19.17.17.17.17.21 Spectral sequence of a continuous mapping 271.19.17.17.17.17.27 Homology theory of fixed points and coincidence points 271.19.17.17.17.17.33 Homology manifolds 271.19.17.17.17.19 Homology and cohomology with nonabelian coefficients 271.19.17.17.17.25 Homotopy and cohomotopy groups: definitions and basic properties.
Theory of finite differences 271.23.19.15.17 Finite-difference equations 271.23.19.15.17.21 Recurrent relations and series 271.23.19.19 Functional equations and inequalities 271.23.21 Integral transformations, operational calculus 271.23.21.17 Laplace transform 271.23.21.19 Fourier integral and Fourier transform 271.23.21.21 Other integral transformations and their inversions.
www.ams.org /mathweb/Classif/RZhClassification.html   (0 words)

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