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Topic: Cantor set

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Cantor space

Uncountable

Set of uniqueness

Cantor dust

Countable

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Cantor function

Accumulation point

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Closed set

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Hausdorff dimension

Georg Cantor

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	Cantor, Georg - Famous mathematicians pictures, posters, gifts items, note cards, greeting cards, and prints
		He distinguished between countable and uncountable sets, and was able to prove that the set of all rational numbers Q is countable, while the set off all real numbers R is uncountable, and therefore, though both were infinite, R was strictly larger.
		The graphic set which backs Cantor's image began with an algorithm to generate the Cantor set, to which color was applied, and then universal operators related to color transition and magnification, ultimately resulting in a unique image whose essence was the Cantor set.
		Cantor came came to the conclusion that the Absolute was beyond man's reach, and identified this concept with God.
	mathematicianspictures.com /Mathematicians/Cantor.htm (509 words)

	PlanetMath: closure and interior of Cantor set (Site not responding. Last check: 2007-11-07)
		Since the Cantor set is the intersection of all these sets and intersections of closed sets are closed, it follows that the Cantor set is closed.
		To accomplish this, the ternary characterization of the Cantor set is useful.
		This is version 1 of closure and interior of Cantor set, born on 2006-03-03.
	www.planetmath.org /encyclopedia/ClosureAndInteriorOfCantorSet.html (246 words)

	PlanetMath: Cantor set (Site not responding. Last check: 2007-11-07)
		The Cantor set therefore has the same cardinality as the set of sequences of 0's and 1's, which is
		However, it is possible to construct Cantor sets with positive measure as well; the key is to remove less and less as we proceed.
		This is version 31 of Cantor set, born on 2002-02-17, modified 2006-07-28.
	www.planetmath.org /encyclopedia/CantorSet.html (604 words)

	Cantor set Summary
		The Cantor set is the prototype of a fractal.
		Since the Cantor set is the complement of a union of open sets, it itself is a closed subset of the reals, and therefore a complete metric space.
		Cantor himself was led to it by practical concerns about the set of points where a trigonometric series might fail to converge.
	www.bookrags.com /Cantor_set (2571 words)

	Cantor set
		The Cantor set can be generalized to a motif where the middle section 1 - 2a is omitted.
		The Sierpinski curve is a two-dimensional variant on the Cantor set.
		The Menger sponge is a three-dimensional variant on the Cantor set.
	www.2dcurves.com /fractal/fractalc.html (114 words)

	PlanetMath: Cantor set (Site not responding. Last check: 2007-11-07)
		The Cantor set therefore has the same cardinality as the set of sequences of 0's and 1's, which is
		However, it is possible to construct Cantor sets with positive measure as well; the key is to remove less and less as we proceed.
		This is version 31 of Cantor set, born on 2002-02-17, modified 2006-07-28.
	planetmath.org /encyclopedia/CantorSet.html (517 words)

	Famous Fractals - Cantor Set (Site not responding. Last check: 2007-11-07)
		Cantor Set is one of the most famous fractal of all, yet it is the most simple one.
		The Cantor Set is composed of 2 identical shapes, each of which is 1/3 the size of the entire figure.
		The pattern of the Cantor Set was found in the rings of Saturn and in the spectra of some molecules.
	library.thinkquest.org /26242/full/fm/fm3.html (242 words)

	Set theory
		Cantor's early work was in number theory and he published a number of articles on this topic between 1867 and 1871.
		In the same paper Cantor shows that the real numbers cannot be put into one-one correspondence with the natural numbers using an argument with nested intervals which is more complex than that used today (which is in fact due to Cantor in a later paper of 1891).
		Cantor claimed that mathematics is quite free and any concepts may be introduced subject only to the condition that they are free of contradiction and defined in terms of previously accepted concepts.
	www-gap.dcs.st-and.ac.uk /~history/HistTopics/Beginnings_of_set_theory.html (2182 words)

	The Cantor Set - Examples of Chaos - IMO (Site not responding. Last check: 2007-11-07)
		The Cantor Middle-Thirds Set is an example of a fractal on the real number line.
		This set is seen on the interval between 0 and 1 on the number line.
		The construction of the Cantor Set is pretty simple: (In this example, a line represents a set of numbers, and removing a section is analogous to taking out that part of the set):
	library.thinkquest.org /2647/chaos/cantor.htm (325 words)

	Set Theory: Cantor
		We include Cantor in our historical overview, not because of his direct contribution to logic and the formalization of mathematics, but rather because he initiated the study of infinite sets and numbers which have provided such fascinating material, and difficulties, for logicians.
		Cantor's last two papers on set theory, Contributions to the foundations of infinite set theory, 1895/1897, give his most polished study of cardinal and ordinal numbers and their arithmetic.
		By the end of the nineteenth century Cantor was aware of the paradoxes one could encounter in his set theory, e.g., the set of everything thinkable leads to contradictions, as well as the set of all cardinals and the set of all ordinals.
	www.math.uwaterloo.ca /~snburris/htdocs/scav/cantor/cantor.html (1069 words)

	Cantor, Georg - Famous mathematicians pictures, posters, gifts items, note cards, greeting cards, and prints
		Cantor's image is flanked by the "Aleph", the first letter of the Hebrew alphabet, which Cantor used (accompanied by subscripts) in his descriptions of transfinite numbers -- quite simply numbers which were not finite.
		He distinguished between countable and uncountable sets, and was able to prove that the set of all rational numbers Q is countable, while the set off all real numbers R is uncountable, and therefore, though both were infinite, R was strictly larger.
		The graphic set which backs Cantor's image began with an algorithm to generate the Cantor set, to which color was applied, and then universal operators related to color transition and magnification, ultimately resulting in a unique image whose essence was the Cantor set.
	www.mathematicianspictures.com /Mathematicians/Cantor.htm (509 words)

	Cantor set (Site not responding. Last check: 2007-11-07)
		The Cantor set is the prototype of a fractal.
		Since the Cantor set is the complement of a union of open set s, it itself is a closed subset of the reals, and therefore a complete metric space.
		Thus the Cantor set is nowhere dense in the unit interval and totally disconnected.
	www.serebella.com /encyclopedia/article-Cantor_set.html (1205 words)

	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.
		Cantor observed that many infinite sets of numbers are countable: the set of all integers, the set of all rational numbers, and also the set of all algebraic numbers.
		Cantor's discovery of uncountable sets led him to the subsequent development of ordinal and cardinal numbers, with their underlying order and arithmetic, as well as to a plethora of fundamental questions that begged to be answered (such as the Continuum Hypothesis).
	plato.stanford.edu /entries/set-theory (3292 words)

	Cantor set and function
		When I was a freshman, a graduate student showed me the Cantor set, and remarked that although there were supposed to be points in the set other than the endpoints, he had never been able to find any.
		The derivative is, of course, discontinuous; it's undefined at points of the Cantor set C
		Cantor function is as famous as it is useful for other exceptional constructs.
	www.cut-the-knot.org /do_you_know/cantor.shtml (644 words)

	Cantor Set (PRIME)
		We can extend this to a function, called the Cantor function, from the entire unit interval onto itself, by simply agreeing to let its value on the missing intervals be the constant values which equal the values of the original function on the endpoints of those intervals.
		Thus, the Cantor set (which is the complement of this union) is closed.
		The Cantor set is an instructively simple example of a fractal, demonstrating that our geometrical intuitions about space (even such simple spaces as the unit interval) can fail to capture much of the deep structure inherent in those very intuitions.
	www.mathacademy.com /pr/prime/articles/cantset (1244 words)

	Example 5.2.13(b): Properties of the Cantor Set (Site not responding. Last check: 2007-11-07)
		The Cantor set is perfect and hence uncountable.
		Next, we need to show that every point in the Cantor set is a limit point.
		} converges to x, and since all endpoints of the subintervals are contained in the Cantor set, we have found a sequence of numbers contained in C that converges to x.
	www.shu.edu /projects/reals/topo/answers/cantor2.html (248 words)

	Cantor Set, Mu-Ency at MROB (Site not responding. Last check: 2007-11-07)
		A set of points, no two of which are touching, but each of which is a limit point of other points in the set.
		the Cantor middle-thirds set: A set of points on a line segment that is the result of an infinite number of steps, in which each step consists of removing the middle-third of the segment(s) remaining from the previous step.
		A Cantor set has the same number of points as a line segment or any other continuum, but zero measure.
	www.mrob.com /pub/muency/cantorset.html (153 words)

	Andrzej Gutek: Publications
		Extensions of homeomorphisms on the Cantor set, Proceedings of the International Conference on Geometric Topology, PWN, Warszawa 1980, pp.
		Solenoids and homeomorphisms on the Cantor set, Commentationes Mathematicae 21 (1979), pp.
		On extending homeomorphisms on the Cantor set, Topological Structures II, Mathematical Centre Tracts 115 (1979), pp.
	www.agutek.com /math/publications.htm (357 words)

	Cantor & Nissel - Products Introduction
		The Cantor and Nissel initiative to supply single-use trial sets in PMMA to a practitioner's preferred design has ensured consistency - and even an upsurge - in the RGP market.
		Our free trial set policy means that the busy practitioner can continue to perform trials as before, but with single-use lenses.
		Each set is accompanied by a supply of re-order forms for replacement of the single-use lenses as they are disposed of.
	www.cantor-nissel.co.uk /practitioners.html (297 words)

	Georg Cantor Biography \| World of Mathematics
		The set of all algebraic numbers, containing, for example, the square root of 2, is properly larger than the set of all rational numbers (that is, quotients of integers).
		Where he had previously shown that the set of algebraic numbers is equipollent with the set of positive integers, Cantor then proved that the set of points on any surface such as a plane is equipollent to the set of all real numbers.
		The novelty and courage of Cantor's contributions are in his refusal to consider this a contradiction and in using it to define infinite sets of equal power.
	www.bookrags.com /biography/georg-cantor-wom (1479 words)

	Cantor's Concept of Infinity:
		Cantor was keenly aware of the paradoxes inherent in such constructions as the set of everything, the set of all sets, and even Taw.
		Cantor produced a classic example of contingent rationality when he drew the distinction between transfinite numbers, which exist in the human mind, and Absolute Infinity, which is beyond all human determination, and exists only in the mind of God.
		Cantor was explicitly opposed to the prevailing materialism of his scientific community, which regarded the physical universe as eternal and unbounded.
	www.asa3.org /ASA/PSCF/1993/PSCF3-93Hedman.html (5863 words)

	GEORG CANTOR
		Two sets M and N are equivalent if they can be put into a one-to-one correspondence; that is, if it is possible by some rule to associate each element of M with exactly one element of N, and vice versa.
		Cantor also defined a set which is equivalent to the set of all natural numbers as a denumberable set.
		Cantor determined that although not all infinite sets are denumberable, the set of all positive and negative rational numbers is denumberable and the set of all algebraic numbers (roots of polynomials with integer coefficients) is denumberable.
	www.engr.iupui.edu /~orr/webpages/cpt120/mathbios/gcant.htm (1376 words)

	Georg Cantor, Set Theory and Transfinite numbers
		Together they set the basis for set theory, and their somewhat obvious proof schemes are now called Zermelo-Fraenkel Theory (ZF) and are the starting point for all set theory study.
		Set theory differentiates between the number of elements in a set and the value of the number of elements in a set.
		This is the question Cantor asked himself and found the answer is NO. That is to say, the number of members of the infinite natural set would not be equal to those in the real set.
	www.geocities.com /roble_wais/cantor_set_theory.htm (1619 words)

	Body
		Cantor sets play an important role in real analysis, particularly in furnishing counter examples and exotic or pathological functions.
		Most frequently, we meet Cantor set of zero measure but the construction is canonical enough to apply to give Cantor set of positive measure.
		The first Cantor set we shall construct is the Cantor set of measure zero.
	www.math.nus.edu.sg /~matngtb/Calculus/Cantor/cantor.htm (2458 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 (3862 words)

	Georg Cantor
		Set theory has been widely adopted in mathematics and philosophy, but the controversy surrounding it at the turn of the century remains of great interest.
		This fairly detailed Georg Cantor biography is from the School of Mathematics and Statistics at the University of St Andrews, Scotland.
		Extract from Cantor's Uber einen die trigonometrischen Reihen betraffenden Lerhrsatz which is one of his first publications on the theory of functions.
	www.erraticimpact.com /~19thcentury/html/cantor.htm (527 words)

	Cantor's Paradox
		Cantor's paradox, sometimes called the paradox of the greatest cardinal, expresses what its second name would imply--that there is no cardinal larger than every other cardinal.
		These conclusions are sound, given that Cantor stated in his letter that the ``totality of everything thinkable'' ([11], p.
		He also stated that the set of all cardinals was an inconsistent multiplicity, which follows from conclusion B of his letter ([11], p.
	www.u.arizona.edu /~miller/finalreport/node3.html (444 words)

	Georg Cantor's set theory proof of the existence of number larger than infinity still fascinates me to this day
		There are some aspect of set theory that I omit from this essay in the interest of space and clarity.
		The point of this essay is to interest the reader in this area of set theory mathematics, not give him or her a lesson in it.
		This is the question Cantor asked himself and found the answer is NO. That is to say, the number of members of the infinite natural set would not be equal to those in the real set.
	members.tripod.com /~Robleh/cantor_set_theory.htm (1751 words)

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