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Topic: Class set theory

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Naive set theory

Ordinal number

Absolute Infinite

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	Class (set theory) - Wikipedia, the free encyclopedia
		Some classes are sets (for instance, the class of all integers that are even), but others are not (for instance, the class of all ordinal numbers or the class of all sets).
		A proper class cannot be an element of a set or a class and is not subject to the Zermelo-Fraenkel axioms of set theory; thereby a number of paradoxes of naive set theory are avoided.
		The word "class" is sometimes used synonymously with "set," most notably in the term "equivalence class." This usage dates from a historical period where classes and sets were not distinguished as they are in modern terminology.
	en.wikipedia.org /wiki/Class_(set_theory) (345 words)

	Class article - Class English sociology economics social class computer science object-oriented - What-Means.com (Site not responding. Last check: 2007-10-21)
		In mathematical set theory, for classes as sets or collections of sets, see class (set theory).
		In education, a class is also short period of teaching in a particular subject.
		In sea warfare, where individual vessels of a class of naval watercraft share essentially the same design, see navy or, under an article for a specific country, the topic of its navy.
	www.what-means.com /encyclopedia/Class (262 words)

	All About Musical Set Theory (Site not responding. Last check: 2007-10-21)
		Set class analysis refers to the efforts of music theorists to reveal the systems that composers like Schoenberg and his followers used to organize the pitch content of their works.
		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.
		There is one occurence of interval class 1 (between the 2 and the 3), one occurence of interval class 6 (between the 3 and the 9) and one occurence of interval class 5 (between the 2 and the 9).
	www.jaytomlin.com /music/settheory/help.html (2147 words)

	Up to article - Up to mathematics equivalence class group theory group action eight - What-Means.com (Site not responding. Last check: 2007-10-21)
		In mathematics, the jargon term "up to xxxx" is used to describe a situation in which members of an equivalence class can be regarded as a single entity for some purpose.
		In group theory, for example, this may be a group action.
		This means that there are two equivalence classes of groups of order 4, if we consider groups to be equivalent if they are isomorphic.
	www.what-means.com /encyclopedia/Up_to (313 words)

	Class (set theory) - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-10-21)
		Classes that are not sets are called proper classes.
		The word "class" is sometimes used synonymously with "set," most notably in the term "equivalence class."
		This usage dates from a historical period where classes and sets were not distinguished as they are in modern terminology.
	encyclopedia.worldsearch.com /class_(set_theory).htm (403 words)

	Absolute Infinite - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-21)
		For if Ω* were consistent, then as a well-ordered set, a number D [delta] would belong to it which would be greater than all numbers of the system Ω; the number D, however, also belongs to the system Ω, because it comprises all numbers.
		There is a quick fix in Zermelo's system by his Axiom of Separation, which stipulates that sets cannot be independently defined by any arbitrary logically definable notion, but must be separated as subsets of sets already "given".
		Any such totality would itself have to be a set, thus lying somewhere within the hierarchy and thus failing to contain every set.
	www.bexley.us /project/wikipedia/index.php/Absolute_infinite (577 words)

	Brasil Nuts (Site not responding. Last check: 2007-10-21)
		Pitch-class set analysis was developed with a specific musical repertoire in mind, the atonal (non 12-tone) music of the first part of the 20th century.
		Forte’s answer: unordered pitch-class set theory was not developed within “a twelve-tone theoretical mold”, but was derived independently and inductively through the intensive study of a good deal of music.
		Set identification engages a number of complex analytical decisions, primarily in the domain of segmentation: the determination of those musical units that are to be regarded as structural.
	brasilnuts.blogspot.com /2005/09/forte-allen-pitch-class-set-analysis.html (671 words)

	Class - Wikipedia, the free encyclopedia
		Appliance classes, in the electric appliance manufacturing industry.
		Class (education), a group of students attending a specific course or lesson, or short for a classroom.
		In navy, a set of vessels that share essentially the same design
	en.wikipedia.org /wiki/Class (102 words)

	80.07.04: Logic and Set Theory
		A set is a well defined collection of “objects.” The term “well defined” means that the set is described in such a way that we can determine whether or not any given object belongs to that set.
		Equal sets are sets in which both elements of set A and elements of set B are the same written A> B. If A = {1,2,3} and B= {3,1,2} they are equal sets.
		Note: The definition for union says that in order for an element to be part of the solution set for AU B it must only be a member of either set A or of set B. The definition of the disjunction “or” also requires that only one statement p or q be true.
	www.cis.yale.edu /ynhti/curriculum/units/1980/7/80.07.04.x.html (3651 words)

	11.8 Applications of Fuzzy Set Theory (Site not responding. Last check: 2007-10-21)
		Fuzzy set theory, to treat fuzziness in data, was proposed by Zadeh in 1965.
		In Fuzzy set theory the membership grade can be taken as a value intermediate between 0 and 1 although in the normal case of set theory membership the grade can be taken only as 0 or 1.
		Figure 11.8.1 shows a comparison between the normal case of set theory and fuzzy set theory.
	www.profc.udec.cl /~gabriel/tutoriales/rsnote/cp11/cp11-8.htm (242 words)

	Amazon.com: Books: Axiomatic Set Theory (Site not responding. Last check: 2007-10-21)
		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 /exec/obidos/tg/detail/-/0486616304?v=glance (2717 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.
		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 (2170 words)

	Set theory
		The history of set theory is rather different from the history of most other areas of mathematics.
		Rather than dismiss set theory because of the paradoxes, ways were sought to keep the main features of set theory yet eliminate the paradoxes.
		However the method of avoiding the paradoxes by introducing a 'theory of types' made it impossible to say that a class was or was not a member of itself.
	www-history.mcs.st-and.ac.uk /PrintHT/Beginnings_of_set_theory.html (2153 words)

	Property (philosophy) - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-21)
		It differs from the logical concept of class by not having any concept of extensionality, and from the philosophical concept of class in that a property is considered to be distinct from the objects which possess it.
		In mathematical terminology, given any element of a set X, a certain property p is either true or false.
		Any property gives rise in a natural way to the set {x: x has the property p} and the corresponding characteristic function.
	www.peekskill.us /project/wikipedia/index.php/Property_(philosophy) (227 words)

	Re: [Set Theory] Class of Ordinals well-ordered ?
		I define the class of ordinals Ord(X): Strictly speaking, ZF does not have proper classes.
		Ordinals are transitive sets with the property of being well-ordered by the set membership relation.
		There may be other sets (non-ordinals) that violate the regularity axiom, but ordinals are ordinals whether you have regularity or not.
	www.usenet.com /newsgroups/sci.math/msg23878.html (429 words)

	Class (Site not responding. Last check: 2007-10-21)
		In mathematicsmathematical set theory, for classes as sets or collections of sets, see class (set theory)/.
		In sea warfare, where individual vessels of a class of naval watercraft share essentially the same design, see navy or, under an article for a specific list of sovereign statescountry/, the topic of its navy.
		Class graduates may forward their current phone number and address to ocs1976@gmail.com or contact Charles Lognion at 942-5349.
	www.infothis.com /find/Class (455 words)

	A HYPERMEDIA ENVIRONMENT FOR EXPLORING PITCH-CLASS SETS IN BARTÓK'S MIKROKOSMOS
		In the theory of unordered pc sets, the generator is a Tn/TnI-type or prime form, and the rules are the operations of pitch class transposition (Tn), or inversion followed by transposition (TnI).
		Mikrokosmik Sets is a suite of hypermedia documents that facilitate exploration of Bartók's Mikrokosmos from the viewpoint of pitch-class set theory.
		Topics addressed include integer representation of pitch, pitch class, name class, and octave, and of the corresponding intervals between such elements, transposition and inversion of ordered sets, normalization of unordered sets, interval cycles, pc set-types (classes) interval class vectors, and the diatonic and pentatonic collections including their modes and subsets.
	music.utsa.edu /tdml/conf-V/V-Williams/V-Williams.html (2067 words)

	Don Gibson - Pitch-class Set Theory and Perception
		All of these experiments raise significant questions for the perceptibility of various concepts in set theory.
		Millar, J.K. The aural perception of pitch-class set relations: A computer-assisted investigation.
		Morris, R. A similarity index for pitch class sets.
	dactyl.som.ohio-state.edu /Gibson/research.summary.html (771 words)

	The Math Forum - Math Library - Set Theory (Site not responding. Last check: 2007-10-21)
		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 (2290 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		The term "class", when used by itself, has several meanings in English.
		In computer science, concerning classes in the object-oriented implementation of computer applications, see class (computer science).
		In mathematical set theory, for classes as sets or sets of sets, see class (set theory).
	www.online-encyclopedia.info /encyclopedia/c/cl/class.html (196 words)

	Forte Set Theory: Pitch Content Analysis (Site not responding. Last check: 2007-10-21)
		An interval vector is a 6-digit number which enumerates the appearances of each interval class (within a chord, melody, musical event).
		Sets of cardinality of less than 2 or more than 10 are not considered as all sets are subset-related to them.
		While these conditions make classification possible, they are also the primary weaknesses of the system in the eyes of its critics.
	music.acu.edu /www/reid/class/caim/forteset.html (186 words)

	Set Theory
		Handout for Lecture 1: A Short History of Set Theory.
		This course is an introduction to the fundamentals of set theory.
		Evaluation will be based upon the mid-term (20%), the final (30%) and the assignments to be set throughout the term (50%).
	www.nyu.edu /gsas/dept/philo/courses/settheory (419 words)

	MTO 3.3: Ethan T. Haimo, Linear Analysis--A Cure for Pitch-Class Set Analysis?
		In Latham's view, pitch-class set analysis of Schoenberg's music remains a viable approach because Forte has developed a more sophisticated method for its application--linear analysis--a method that (presumably) is free from the problems raised in my article.
		However, there is nothing in any of these more recent articles to indicate that pitch-class set analysis, as preached and practiced in his earlier essays, has been replaced by this new method.
		Instead, an objective pitch-class set analysis (which is a type two analysis) of Schoenberg's music simply yields an unwieldy, random, almost unlimited, mass of sets.
	www.societymusictheory.org /mto/issues/mto.97.3.3/mto.97.3.3.haimo_essay.html (1954 words)

	Amazon.com: Books: Elements of Set Theory (Site not responding. Last check: 2007-10-21)
		Some knowledge of set theory is necessary part of the background everyone needs for further study of mathematics.
		It is an insightful development of set theory, both as a foundation for mathematics and a distinctive mathematical discipline in its own right.
		Set theory can be developed from a "naive" or an "axiomatic" perspective.
	www.amazon.com /exec/obidos/tg/detail/-/0122384407?v=glance (1348 words)

	2 Styles of VSEPR Theory Molecular Models (Site not responding. Last check: 2007-10-21)
		The economical student set in the lower right corner includes 50 pieces, all of which are shown.
		The Class and Large Class Set versions have 10 and 25 times as many pieces respectively.
		The group to the far right is the advanced VSEPR theory set, shown in its entirety.
	www.indigo.com /models/gphmodel/vsepr-theory-molecular-model-sets.html?from=sciencebase (126 words)

	Diatonic connotations of pitch-class sets (Site not responding. Last check: 2007-10-21)
		This is a music-theoretical study of the relationship of two-, three-, four-, five-, and six-member subsets to the major (pure minor), harmonic minor, and melodic (ascending) minor reference collections, using pitch-class set analytic techniques.
		Several new terms are introduced to facilitate the application of pitch-class set theory to descriptions of tonal pitch relations, and to retain characteristic intervallic relationships in tonal music typically not found in discussions of atonal pitch-class relations.
		Further, it is suggested that the probability of a specific key interpretation by a listener may depend on the scale-degree functions of the tones.
	www.nici.kun.nl /Publications/1997/11214.html (167 words)

	MTO 11.3: Dissertation Listings
		As described in chapter 2, pitch-class set theory and transformation theory form the theoretical basis for the present study.
		Specifically, it is Martino's preoccupation with symmetrical set classes that functions as a link in his compositional development from the octatonic to the chromatic.
		Indeed, as the analytical chapters attest, it seems that Martino was eventually drawn to the symmetrical pitch-class relations that result from the resources of the total chromatic?resources that are significantly greater in number than those of the octatonic collection.
	www.music-theory.org /mto/issues/mto.05.11.3/diss.11.3.html (1234 words)

	Introduction to the Music Theory Cipher_Part 2 page 2
		In Set Theory, after starting with that simple line of numbers, it’s practitioners almost immediately head for left field, adding layers of transformations and reductions, looking for patterns.
		At any rate, the use of chromatic numbers (including zero) in Set Theory is further precedence, another place in music theory (albeit an obscure, advanced, and atonal form of Western music theory) where such (initial) numbers are recognized as being “natural” and are used every day.
		The point is to use both sets of numbers, to integrate them, as needed and for as long as they’re needed, to help you understand the elements of Classical Western music theory and the necks of string instruments (from guitar to violin).
	www.thecipher.com /introduction-part2-pg2.html (2753 words)

	Set Theory at FAU (Site not responding. Last check: 2007-10-21)
		The main purpose of this course is to acquaint students with some of the very basic concepts of abstract mathematics, such as sets, classes, cardinal and ordinal numbers.
		We shall cover the first two chapters of the textbook (which seem to contain most of the results of set theory and axiomatics needed by working mathematicians whose field is not set theory, mathematical logic or the foundations of mathematics), plus a selection of topics from later chapters.
		Homework is due at the beginning of the Wednesday class of the next week.
	www.math.fau.edu /schonbek/htmdocs/sthsu8.html (151 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		In this paper, I seek to resolve these inconsistencies by defining the harmonic sequence as the pairing of a repeating series of a root motions and a repeating series of voice-leading motions.
		I discuss the reduction graph, a tool for relating sequences whose patterns are of different cardinalities, and employ it to elucidate a three-chord sequence by Bach and a four-chord sequence by Wagner.
		I then outline a classification of sequences by their degree of voice-leading smoothness and discuss a set of sequences whose voice leading is exclusively parsimonious.
	theory.esm.rochester.edu /th591/abs/ricc0402.html (168 words)

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