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Topic: Uncountable set

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	Uncountable set - Wikipedia, the free encyclopedia
		Not all uncountable sets have the same size; the sizes of infinite sets are analyzed with the theory of cardinal numbers.
		The diagonalization proof technique can also be used to show that several other sets are uncountable as well, for instance the set of all infinite sequences of natural numbers (and even the set of all infinite sequences consisting only of zeros and ones) and the set of all subsets of natural numbers.
		The Cantor set is an uncountable subset of R.
	en.wikipedia.org /wiki/Uncountable (370 words)

	uncountable set
		All uncountable sets are infinite, but not all infinite sets are uncountable.
		The best known uncountable set is the set of all real numbers.
		By contrast the set of all natural numbers, which represents the "smallest" type of infinity, is countable.
	www.daviddarling.info /encyclopedia/U/uncountable_set.html (111 words)

	Cantor set - Wikipedia, the free encyclopedia
		The Cantor set is the prototype of a fractal.
		A closed set in which every point is an accumulation point is also called a perfect set in topology, while a closed subset of the interval with no interior points is nowhere dense in the interval.
		The Cantor set is the set of all points on the Koch curve that intersect the original horizontal line segment.
	en.wikipedia.org /wiki/Cantor_set (1802 words)

	PlanetMath: Cantor set
		is the canonical example of an uncountable set of measure zero.
		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 27 of Cantor set, born on 2002-02-17, modified 2006-01-06.
	planetmath.org /encyclopedia/CantorSet.html (684 words)

	PlanetMath: uncountable
		Definition A set is uncountable if it is not countable.
		This is version 4 of uncountable, born on 2001-11-16, modified 2004-09-01.
		Object id is 884, canonical name is Uncountable.
	planetmath.org /encyclopedia/Uncountable.html (83 words)

	Peter Suber, "Infinite Sets"
		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.
		Our next theorem shows that some sets can indeed be larger than the natural numbers, and in Theorem 16 we will see that the additions to the naturals required to generate the real numbers do indeed create an uncountable set.
	www.earlham.edu /~peters/writing/infapp.htm (6879 words)

	All Elementary Mathematics - Study Guide - Sets - Basic notions. Examples of sets...
		A set and an element of a set concern with category of primary notions, for which it's impossible to formulate the strict definitions.
		For instance, a set of books in a library, a set of cars on a parking lot, a set of stars in the sky, a world of plants, a world of animals – these are examples of sets.
		An uncountable set is a set, elements of which can't be numbered.
	www.bymath.com /studyguide/sets/sec/sets1.htm (417 words)

	CmSc 365 Theory of Computation
		If it is difficult to represent the bijection explicitly as a function, we can prove that a set is countably infinite by showing a method to order the elements in the set so that if we follow the method of ordering we can visit each element in the set.
		In order to apply the principle the elements of the set have to be represented as infinite sequences of 0 and 1, and any infinite sequence of 0 and 1 has to be a representation of some element in the set.
		In many problems the set in consideration is obtained as a result of some set operation applied to countable/uncountable sets.
	www.simpson.edu /~sinapova/cmsc365-02/L02-Countability.htm (791 words)

	CmSc 365 Theory of Computation (Site not responding. Last check: 2007-10-13)
		Example 3: If A is uncountable set and B is countable, the set C = A - B is uncountable.
		A is a subset of C, and we have proved, that a subset of countable set must be countable.
		We have shown that the union of finite number of countable sets is countable.
	storm.simpson.edu /~sinapova/cmsc365-02/L02-direct.htm (250 words)

	Perfect Subset Property for co-analytic sets in ZF\P
		A perfect set is the set of all paths through a perfect tree, and can be coded by such perfect tree.
		The only uses of uncountable sets in the standard proof are routine manipulations of projective sets, so the proof goes through (with syntactic modifications) inside ZF\P. Consider a model M of ZF\P. The notions in this paragraph are assumed to be relativized to M.
		The constructible sets under the membership relation form a model L of ZFC\P. Suppose that every set is countable and perfect subset property holds for co-analytic sets.
	web.mit.edu /dmytro/www/other/PerfectSubsetsAndZFC.htm (917 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-13)
		Thus the set of primes and the set of composite numbers are both countable (since they are both subsets of the natural numbers).
		It is not possible to take the union of an uncountable set A and any set B (either finite, inifinite countable or infinite uncountable) and produce a countable set.
		Thus if we start with a set that has uncountably many elements (this is the set A), A U B will always contain everything in A with the possibility of adding more elements (specifically, the elements in B which are not in A).
	mathforum.org /library/drmath/view/60781.html (353 words)

	Infinite and uncountable data structures
		The set of all binary strings is known to be uncountable.
		The set of paths is isomorphic to the set of nodes (this can be taken as the definition of a tree as a prefix-complete set of paths).
		Because the set of all binary strings is uncountable, so is the set of the paths admitted by the tree.
	okmij.org /ftp/Computation/uncountable-sets.html (921 words)

	Prove the following statements
		A countable union of countable sets is countable.
		The set of polynomials with integer coefficients is countable.
		Suppose A is an unknown set and B = { 7 }.
	www.math.toronto.edu /jkorman/Math246Y/problemsincardinality.htm (351 words)

	Glossary (Site not responding. Last check: 2007-10-13)
		This is the (unprovable) conjecture that the cardinality of the continuum (i.e.
		Given a set, S, with a Linear Order (≤) defined on it, and a ∈ S a Section is defined as the set of all elements of S which are ≤ a.
		Transitive sets form the basis for the construction of the ordinals.
	www.jboden.demon.co.uk /SetTheory/glossary.html (381 words)

	Cantor's square w x w table with diagonal
		I_w as countable set of rows > (with a single 1), then so is the set of their complements > (with a single 0), and we need not interprete them at all > as group elements.
		Cantor's proof of "uncountable" then boils down simply to checking the number of ones, say in the complemented diagonal which is NOT in N obviously because it does not contain just _one_ 1, but none.
		Of course, to prevent confusion the term "(un)countable" should not be used, and I suggest "sequential dimension" (seq_dim) as the minimal number of generators to generate a covering closure (that is: of which the set/system is a sub_closure).
	home.iae.nl /users/benschop/cantor.htm (9039 words)

	Uncountable Sets (Site not responding. Last check: 2007-10-13)
		Of course any infinite subsets of the natural numbers (such as the set of prime numbers) is a countable set.
		The set of all rational numbers is a countable set.
		One of the best known examples of an uncountable set is the set of all real numbers.
	www.math.uic.edu /~lewis/las100/uncount.html (352 words)

	[No title]
		until the set is exhausted, and the last number used is the number of objects in the set.
		The set of all touring machines is countable, but the subset of them which halt is uncountable.
		For example, if you have sets A and B, one presumes that you can create the sets A U B and A n B. Using the information available, you want to know what can be created.
	orion.math.iastate.edu /hentzel/class.166.05/Apr.27 (1397 words)

	Uncountable Infinity
		the set of all infinite sequences of 0's and 1's is uncountable
		The power set of a given set S is the set of all subsets of S, denoted by P(S).
		When dealing with sets of sets, one has to be careful that one does not by accident construct a logical impossibility.
	pirate.shu.edu /projects/reals/infinity/uncntble.html (911 words)

	Set Cardinality - GameDev.Net Discussion Forums
		This can be demonstrated by considering the uncountable set, the real numbers and the subset, the set of decimal numbers between 0 and 1.
		The set of all words is countable (easy to prove, since you can just enumerate all words of length 0,1,2,...), so we can assign a natural number to each word.
		Some people were confused by the wording and thought it was uncountable, but "uncountable" was refering to the set of all languages.
	www.gamedev.net /community/forums/topic.asp?topic_id=316799 (1043 words)

	Commentary: MAT335
		The cardinality of a set is the number of elements in the set.
		We showed in class that the cardinality of the set of integers Z, which we'll denote by card(Z), and the cardinality of the rational numbers Q, card(Q), are the same.
		These sets are both uncountable (in fact, they have the same cardinality, which is also the cardinality of R, and R has infinite length).
	www.sfu.ca /~rpyke/335/W00/7jan.html (932 words)

	Set Theory. Zermelo-Fraenkel Axioms. Russell's Paradox. Infinity. By K.Podnieks (Site not responding. Last check: 2007-10-13)
		Thus, the set theory C1+C1'+C21+C22+C23 seems to be equivalent (in the sense of Section 3.2) to PA (defined in Section 3.1).
		To prove the existence of uncountable sets, the power-set axiom C24 must be applied additionally: by Cantor's Theorem, the set of all sets of natural numbers P(w) is uncountable.
		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.
	www.all4resumes.com /~podnieks/gt2.html (8336 words)

	[No title]
		If A is an uncountable set, and B is a finite set, then the most we can say about (A union B) is that it is Answer.
		If A is an uncountable set, and B is a countable set, then the most we can say about the set A-B (the elements of A which are not in B) is that it is Answer.
		If A is a countable set, and B is an uncountable set, then the most we can say about the set A-B (the elements of A which are not in B) is that it is Comment.
	www.math.ucla.edu /~tao/java/MultipleChoice/countable.txt (804 words)

	How Big is Infinity?
		That is, a set is countably infinite if it is possible to devise a systematic way of pointing to each of its elements in turn, and counting them: one, two, three,...
		is an uncountable set, representing a new and larger kind of infinite size.
		Although Georg Cantor's discovery of the differences among the cardinalities of various infinite sets came as quite a shock to the mathematical community in the 1870's, it is now well understood that there is a whole new type of arithmetic based on the transfinite numbers, the cardinalities of infinite sets.
	www.jcu.edu /math/vignettes/infinity.htm (859 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		The whole set of integers, Z, is also countable: Z = { 0, 1, -1, 2, -2, 3, -3, } That is, Z = { a1, a2, a3, } where an = n/2 if n is even, and -(n-1)/2 if n is odd.
		Without the distinction between countable and uncountable sets, we couldn’t have uniform distributions.
		All sets are of three kinds: finite; countably infinite; uncountable.
	www.swarthmore.edu /NatSci/wstromq1/stat53/CountableSets.doc (936 words)

	The Continuum Hypothesis
		There is no onto function from R, the set of real numbers, to P(R), the set of all subsets of the real number.
		Brouwer: (1881-1966) (Rejection of the law of excluded middle for infinite sets) He rejected in mathematical proofs the Principle of the Excluded Middle, which states that any mathematical statement is either true or false.
		In 1918 he published a set theory, in 1919 a measure theory and in 1923 a theory of functions all developed without using the Principle of the Excluded Middle.
	www.humboldt.edu /~mef2/Courses/m446s02n2.html (515 words)

	More on Example 2 (Site not responding. Last check: 2007-10-13)
		Since we've managed to identify this set with the interval [0,1], this means that [0,1] can't be counted either.
		A set is uncountable if and only if every countably infinite subset is proper.
		Another, even more curious, example of an uncountable set is provided by the Cantor set, the next installment in our saga.
	personal.bgsu.edu /~carother/infinite/Infinite4.html (280 words)

	No Title
		This is often phrased as saying the countable union of countable sets is countable.
		Theorem 3 The set of all infinite sequences of elements of the set
		Theorem 4 Let Y be the set of all subsets of a countably infinite set X.
	www.rpi.edu /~piperb/analysis1/countable (235 words)

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