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Topic: Countably infinite

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Countably many

Whole number

Cartesian product

Total order

Cardinal number

Number

Rational number

Bijection

Irrational number

Prime number

Real number

Empty function

Georg Cantor

Isomorphism

Numeral system

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	IFS Attractors: Rep-Tiles
		As the set of L-polyominoes is countably infinite, and this set is a subset of the rep-tiles with non-fractal boundaries, then the set of rep-tiles with non-fractal boundaries must be at least countably infinite.
		Proof: As the set of cis-homeolineomers is countably infinite, and this set is a subset of the rep-tiles with an uncountable number of sides, then the set of rep-tiles with an uncountable number of sides must be at least countably infinite.
		As the set of L-polyominoes is countably infinite, and this set is a subset of the rep-tiles with non-self-intersecting boundaries, then the set of rep-tiles with non-fractal boundaries must be at least countably infinite.
	www.meden.demon.co.uk /Fractals/reptiles.html (1506 words)

	Countable
		A set is called countably infinite if there exists a bijective mapping between it and the set N of all natural numbers.
		A countable set is a set which is either finite or countably infinite.
		The different sizes of infinite sets are investigated in the theory of cardinal numbers.
	www.ebroadcast.com.au /lookup/encyclopedia/co/Countable.html (873 words)

	PlanetMath: countably infinite
		As the name implies, any countably infinite set is both countable and infinite.
		Countably infinite sets are also sometimes called denumerable.
		This is version 3 of countably infinite, born on 2001-11-16, modified 2002-06-14.
	planetmath.org /encyclopedia/CountablyInfinite.html (62 words)

	Countable set - Wikipedia, the free encyclopedia
		In mathematics, a countable set is a set with the same cardinality (i.e., number of elements) as some subset of the set of natural numbers.
		Note that countable set is sometimes given a more specific definition: sometimes, it is defined as a set with the same cardinality as the set of natural numbers.
		Technically, a countably infinite set is any set which, in spite of its boundlessness, can be shown equinumerous to the natural numbers — nothing more, nothing less.
	en.wikipedia.org /wiki/Countable (1566 words)

	Infinity (Site not responding. Last check: 2007-10-09)
		Infinite quantities can be discomfitting, and most applications of mathematics shy away from dealing with them.
		The simplest infinity is countable infinity or enumerable infinity.
		Since this definition of equivalent size applies to infinite sets, it can be used to show that the cardinality of all the real numbers is the same as the cardinality of the reals in the interval (0, 1).
	www.mattababy.org /~belmonte/Publications/Books/CSaW/5_infinity.html (1568 words)

	Number Definitions
		The complex numbers are uncountably infinite, are closed under the four basic operations (other than dividing by 0), and have additive and multiplicative identity and inverses (other than 0).
		A countably infinite set can be put into one-to-one correspondence with the counting numbers {1, 2, 3, 4, 5,...}.
		For example, the positive even numbers are countably infinite, since we can find a one-to-one mapping of {2, 4, 6, 8, 10,...} onto the counting numbers.
	www.learner.org /channel/courses/learningmath/number/keyterms.html (2526 words)

	Constructive Mathematics
		The concept of infinitely small entities (the notion of limits was not made explicit until much later) was troubling in an age when mathematical objects were considered to have the same reality as, say, the electron is thought to have today.
		Cantor pointed out that certain infinite sets, such as the even numbers (top row) and the rational numbers (bottom row), are countable in the sense that they can be put into a one-to-one correspondence with the natural numbers 1, 2, 3...
		Infinite sets forced them to confront the quite different concept of an actual, or completed, infinity: a collection of infinitely many objects that can be considered simultaneously.
	digitalphysics.org /Publications/Cal79/html/cmath.htm (8036 words)

	Countable set Summary (Site not responding. Last check: 2007-10-09)
		Thus infinite sets that are countable have the same cardinality as the integers.
		In this situation the set theoretic notion of cardinality is an important way to think about the size of infinite sets and the notion of countability captures the idea that an infinite set has the same size as the set of integers.
		Every infinite subset S of the natural numbers is countable since the function that takes the least element in S to 1 and the next to least element to 2 and the next to 3 and so on, is a correspondence.
	www.bookrags.com /Countable_set (2497 words)

	Infinite Ink: The Continuum Hypothesis by Nancy McGough
		The lowest level is called "countable infinity" and higher levels are called "uncountable infinities." The natural numbers are an example of a countably infinite set and the real numbers are an example of an uncountably infinite set.
		Any infinite set of real numbers is either countably infinite or has the same cardinality as the entire set of real numbers.
		But, within this system every set of reals is either countable or has the cardinality of all the reals so the first three of the six versions of CH listed in section 1.1 hold.
	www.ii.com /math/ch (4563 words)

	Cantor's diagonal argument
		Cantor's diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite.
		(1) Assume that the interval (0,1) is countably infinite.
		The diagonal argument is an example of reductio ad absurdum because it proves a certain proposition (the interval (0,1) isn't countably infinite) by showing that the assumption of its negation leads to a contradiction.
	www.wordlookup.net /ca/cantor's-diagonal-argument.html (896 words)

	Finite and Infinite Sets and Alphabets
		Though this is a promising way to characterize infinite sets, it is nonetheless somewhat controversial, and we will not affirm it.
		A set that is equinumerous with N is said to be countably infinite; a set that is either countably infinite or finite is said to be countable.
		Notice that all countably infinite sets can be listed in a sequence that starts with the first element of the set, then passes to the second, and then the third, and so on; we can thus say that such a set is enumerable.
	www.rpi.edu /~faheyj2/SB/LCU/lcu.driver/node13.html (665 words)

	Cantor's Diagonal Proof
		A set of objects is said to be countably infinite if the elements can be placed in a 1-to-1 correspondence with the integers 0,1,2,3,..
		The new number is certainly in the set of real numbers, and it's certainly not on the countably infinite list from which it was generated.
		I suggest that it reflects a misunderstanding of infinity to believe that such numbers can be exactly represented by an infinite decimal expansion, since the existence of numbers with such expansions is purely hypothetical, and that it is better to say that such numbers can only be closely approximated by a finite decimal expansion.
	www.mathpages.com /home/kmath371.htm (1582 words)

	[No title]
		A set is "countably infinite" if it is equinumerous with N, and "countable" if it is finite or countably infinite.
		A is a subset of N which is countably infinite.
		Because programs are countable and the functions they generate are uncountable, there are more functions than program and there are some problems you might want to solve for which no program can be written in a given programming language.
	ranger.uta.edu /~cook/tcs/l3.html (821 words)

	Countably Infinite (Site not responding. Last check: 2007-10-09)
		We also say that it is countably infinite, since there is a way to list (count) them and eventually come to every predefined number: 0, 1, 2, 3, 4, 5,...
		We know by now that there are countably infinite sets; N is an example.
		It is not clear whether there are infinite sets which are not countable, but this is indeed the case, see UncountablyInfinite.
	c2.com /cgi/wiki?CountablyInfinite (275 words)

	Cardinality issues
		the "arithmetic of cardinalities" is especially non-intuitive when infinite sets are involved; in general, we may recognize several "orders" of infinity
		we say that a set is countably infinite if its members can be put in one-to-one correspondence with the natural numbers we say that a set is uncountable if its members can not be put in one-to-one correspondence with the natural numbers
		for infinite sets, the cardinality of the powerset is strictly larger than that of the base set
	www.willamette.edu /~fruehr/446/lectures/review2.html (193 words)

	Grand-Admiral Petry at NEMO: uniform convergence
		And its correlative trouble is that the All declaration of digits is specious: Infinite All is not the same-as nor extension-of, finite all.
		The third trouble is that the suggestion that a number can be specified for all its digits to be sufficiently different, is intrinsically a Zenoic paradox: At every stage in the development its 'discovered' number may be further in the list...
		Merely to assume the infinite all can compass itself to exclusion to break-through or-not in the infinite-all case, is not a proof but a self-paradoxic assumption: For example of the paradox itself consider the infinite sequence,.0,.10,.110,.1110,.11110,...
	members.tripod.com /~GrandAdmiralPetry/uniform.html (880 words)

	PlanetMath: countably categorical structures
		A countably infinite structure is called countably categorical (also called
		Cross-references: oligomorphic automorphism group, isomorphic, theory, countable, categorical, structure, countably infinite
		This is version 2 of countably categorical structures, born on 2005-05-12, modified 2005-05-12.
	planetmath.org /encyclopedia/CountablyCategoricalStructures.html (94 words)

	Notes on Chapter 4
		"Countably infinite" means the values can be put into a sequence so that there is a first, second, third,...
		For example, the set of all integers {...-2, -1, 0, 1, 2,...} is countably infinite because it can be put into the order 0, -1, 1, -2, 2, -3,...
		Figure 4.8 shows the 50% probability mass at zero using a bar (as in a bar chart): the pdf at zero, were it defined, would have to be infinitely large (because the width of the bin at zero is infinitely small).
	www.quantdec.com /envstats/notes/text/chap4.htm (2620 words)

	A countably infinite collection of countably infinite sets (Site not responding. Last check: 2007-10-09)
		Yes, the union of a countably infinite collection of countably infinite sets is countably infinite.
		Arrange the elements of each of your countably infinite sets in a row.
		Now create your large set by stacking each of the rows to form an array consisting of a countable infinite collection of countably infinite rows.
	mathcentral.uregina.ca /QQ/database/QQ.09.04/feroz1.html (115 words)

	transfinite number. The Columbia Encyclopedia, Sixth Edition. 2001-05
		cardinal or ordinal number designating the magnitude (power) or order of an infinite set; the theory of transfinite numbers was introduced by Georg Cantor in 1874.
		(aleph-null) is assigned to the countably infinite set of all positive integers {1, 2, 3, … n, … }.
		It can be proved that all countably infinite sets, among which are the set of all rational numbers and the set of all algebraic numbers, have the cardinal number
	www.bartleby.com /65/tr/transfin.html (286 words)

	Math Is Fun Forum / Countably Infinite
		Are you allowed to use that the rationals are countably infinite?
		Are you allowed to use that the union between two countably infinite sets is countably infinite?
		Are you allowed to use that a subset of a countably infinite set is countably infinite (or countably finite)?
	www.mathsisfun.com /forum/viewtopic.php?id=3445 (168 words)

	Infinity
		At most countably infinite many descriptions (each description is finite) may be constructed from a given finite alphabet.
		There are uncountably infinite many relations in a countably infinite universe of constants.
		Therefore, in a countably infinite universe of constants, there are relations that cannot be described.
	cs.wwc.edu /~aabyan/CII/Indescribable.html (791 words)

	Irrational digits countably infinite?
		The fact that you CAN do that means that the set is countably infinite.
		A set is countably infinite if it can be put in a 1 to 1 relation with the set of all natural numbers- "listing" a set, so that there is a "first", a "second", etc. is obviously doing that.
		In fact, considering terminating decimals as ending with an infinite string of 0s (0.5 is 0.500000...) then the decimal expansions of ALL numbers are countably infinite.
	www.physicsforums.com /showthread.php?t=67737 (401 words)

	The cartesian product of a countably infinite collection of countably infinite sets
		The cartesian product of a countably infinite collection of countably infinite sets is uncountable.
		Let N to be the set of positive integers and consider the cartesian product of countably many copies of N. This is the set S of sequences of positive integers.
		I am going to assume that S is countable and hence can be put into a one-to-one correspondence with N and then use this fact to produce an member of S that is outside this correspondence.
	mathcentral.uregina.ca /QQ/database/QQ.02.06/geetha1.html (243 words)

	ReuvenLax : Back to Math
		By countable, I mean a set that is in one-to-one correspondence with the natural numbers.
		The complaint was that countable often is used to mean either countably infinite _or_ finite.
		V is bounded by [0,2] and is can be shown to be countable by the same diagonalization argument Cantor used to show NxN is countable.
	blogs.msdn.com /reuvenlax/archive/2005/05/23/420971.aspx (435 words)

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