
	Where results make sense

Topic: Cardinality of the continuum

	PlanetMath: cardinality of the continuum
		The cardinality of the continuum, often denoted by
		"cardinality of the continuum" is owned by yark.
		This is version 12 of cardinality of the continuum, born on 2004-03-15, modified 2006-12-30.
	planetmath.org /encyclopedia/ContinuumMany.html (0 words)

	Infinite Ink: The Continuum Hypothesis FAQ
		The continuum hypothesis was proposed by Georg Cantor in 1877 after he showed that the real numbers cannot be put into one-to-one correspondence with the natural numbers.
		He used the Hebrew letter aleph to name the different levels of infinity: aleph_0 is the number of (or cardinality of) the natural numbers or any countably infinite set, and the next levels of infinity are aleph_1, aleph_2, aleph_3, et cetera.
		The pursuit of the continuum problem has motivated a lot of work in set theory and in mathematics in general.
	www.ii.com /math/ch/faq (0 words)

	Set Theory (Stanford Encyclopedia of Philosophy)
		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).
		As Cantor was unable to find any set of real numbers whose cardinal lies strictly between the countable and the continuum, he conjectured that the continuum is the next cardinal: the Continuum Hypothesis.
		Along with the theory of large cardinals it is used to gauge the consistency strength of mathematical statements.
	plato.stanford.edu /entries/set-theory (0 words)

	Ernst Friedrich Ferdinand Zermelo
		Cantor had put forward the continuum hypothesis in 1878, conjecturing that every infinite subset of the continuum is either countable or has the cardinality of the continuum.
		In 1902, Zermelo published his first work on set theory, which was on the addition of transfinite cardinals.
		Two years later, he succeeded in taking the first step suggested by Hilbert towards the continuum hypothesis when he proved that every set can be well ordered.
	www.stetson.edu /~efriedma/periodictable/html/Zr.html (773 words)

	Kids.Net.Au - Encyclopedia > Cardinality (Site not responding. Last check: )
		Cardinal numbers, or cardinals for short, are numbers used to denote the size of a mathematical set.
		It can also be proved that the cardinal $\aleph_0$ (aleph-0, where aleph is the first letter in the Hebrew alphabet, represented by the unicode character and#1488;) of the set of natural numbers is the smallest infinite cardinal, i.e., that any infinite set admits a subset of cardinality $\aleph_0$ .
		The latter cardinal number is also often denoted by c; it is the cardinality of the set of real numbers, or the continuum, whence the name.
	www.kids.net.au /encyclopedia-wiki/ca/Cardinality (1160 words)

	Continuum (mathematics) - Encyclopedia, History, Geography and Biography
		Somewhat more generally a continuum is a linearly ordered set that is "densely ordered", i.e., between any two members there is another, and lacks gaps, i.e., every non-empty subset with an upper bound has a least upper bound.
		The continuum hypothesis is sometimes stated by saying that no cardinality lies between that of the continuum and that of the natural numbers.
		One interesting subject in continuum theory is the existence of nontrivial indecomposable continua (continua which cannot be written as the union of two proper subcontinua).
	www.arikah.com /encyclopedia/Continuum_%28mathematics%29 (258 words)

	tScholars.com \| Uncountable set (Site not responding. Last check: )
		Cardinality refers to the size of a set; these are analyzed with the theory of cardinal numbers.
		The cardinality of R is often called the cardinality of the continuum and denoted by c or $\beth_1$ (beth-one).
		The cardinality of Î© is denoted $\aleph_1$ (aleph-one).
	www.tscholars.com /encyclopedia/Non-denumerable (487 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal (Site not responding. Last check: )
		In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set.
		Cardinality is also an area studied for its own sake as part of set theory, particularly in trying to describe the properties of large cardinals.
		The latter cardinal number is also often denoted by c; it is the cardinality of the continuum (the set of real numbers).
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=cardinal_number (2102 words)

	PlanetMath: cardinality of the continuum (Site not responding. Last check: )
		, so it could be either a successor cardinal or a limit cardinal, and either a regular cardinal or a singular cardinal.
		"cardinality of the continuum" is owned by yark.
		This is version 12 of cardinality of the continuum, born on 2004-03-15, modified 2006-12-30.
	www.planetmath.org /encyclopedia/ContinuumMany.html (158 words)

	NationMaster - Encyclopedia: Cardinality of the continuum
		In mathematics, the cardinality of the continuum is the cardinal number of the set of real numbers R (sometimes called the continuum).
		The cardinality of a set is a property that describes the size of the set by describing it using a cardinal number.
		A variation on Cantor's diagonal argument can be used to prove Cantor's theorem which states that the cardinality of any set is strictly less than that of its power set, i.e.
	www.nationmaster.com /encyclopedia/Cardinality-of-the-continuum (1755 words)

	Cofinality (Site not responding. Last check: )
		The cofinality of A is the least cardinality of a cofinal subset.
		Moreover, any cofinal subset of B whose cardinality is equal to the cofinality of B is well-ordered and order-isomorphic to its own cardinality.
		For any infinite well-orderable cardinal number κ, an equivalent and useful definition is cf(κ) = the cardinality of the smallest collection of sets of strictly smaller cardinals such that their sum is κ; more precisely
	www.wapipedia.org /wikipedia/mobiletopic.aspx?cur_title=Cofinality (330 words)

	Cardinality - Article from FactBug.org - the fast Wikipedia mirror site (Site not responding. Last check: )
		Any set that has the same cardinality as the set of the natural numbers is said to be an infinite countable set, if the cardinality of such a set is less than that of the natural numbers then it is a finite set, otherwise the set is uncountable.
		The cardinality of the natural numbers is less than the cardinality of the real numbers (see: Cantor's diagonal argument).
		The continuum hypothesis states that there is no cardinal number between the cardinality of the reals and the cardinality of the natural numbers.
	www.factbug.org /cgi-bin/a.cgi?a=6174 (384 words)

	Cardinality - KnowledgeIsFun.com
		In mathematics, the cardinality of a set is a measure of the "number of elements of the set".
		Any set that has the same cardinality as the set of the natural numbers is said to be a countably infinite set.
		The relation of having the same cardinality is called equinumerosity, and this is an equivalence relation on the class of all sets.
	www.knowledgeisfun.com /C/Ca/Cardinality.php (699 words)

	Hamel dimension
		The dimension of a vector space V is the cardinality (i.e.
		However, the Hamel dimension depends on the base field, so while R has dimension 1 when considered as a vector space over itself, it has dimension c (the cardinality of the continuum) when considered as a vector space over Q (the rationals).
		Some simple formulae relate the Hamel dimension of a vector space with the cardinality of the base field and the cardinality of the space itself.
	www.ebroadcast.com.au /lookup/encyclopedia/di/Dimension_(linear_algebra).html (173 words)

	Real closed field
		Considered simply as a field with the cardinality of the continuum, there is up to isomorphism only one field which is not algebraically closed but which becomes so by adjoining the square root of minus one.
		The weight of F, which is the minimum size of a dense subset of F. These three cardinal numbers tell us much about the order properties of any real closed field, though it may be difficult to discover what they are, especially if we are not willing to invoke GHC.
		This is the most commonly used hyperreal number field in nonstandard analysis, and its uniqueness is equivalent to the continuum hypothesis.
	www.xasa.com /wiki/en/wikipedia/r/re/real_closed_field.html (913 words)

	24.97.7.148 (Site not responding. Last check: )
		You inserted a reference to aleph_c, but from the context it is clear that you meant simply c, the cardinality of the continuum.
		You also inserted an error: that aleph-one is by definition the next cardinal after aleph-null.
		Cantor did (in effect) accept the axiom of choice; nonetheless what he used as a definition of aleph-one is: the cardinality of the set of all countable ordinals.
	www.wapipedia.org /wikipedia/mobiletopic.aspx?cur_title=24.97.7.148 (242 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal (Site not responding. Last check: )
		Transfinite numbers are cardinal numbers or ordinal numbers that are larger than all finite numbers, yet not necessarily absolutely infinite.
		The continuum hypothesis states that there are no intermediate cardinal numbers between aleph-null and the cardinality of the continuum (the set of real numbers): that is to say, aleph-one is the cardinality of the set of real numbers.
		Some authors, for example Suppes, Rubin, use the term transfinite cardinal to refer to the cardinality of a Dedekind-infinite set, in contexts where this may not be equivalent to "infinite cardinal"; that is, in contexts where the axiom of countable choice is not assumed or is not known to hold.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=transfinite_numbers (359 words)

	NationMaster - Encyclopedia: Hamel dimension
		In mathematics, the dimension of a vector space V is the cardinality (i.e.
		All bases of a vector space have equal cardinality (see dimension theorem for vector spaces) and so the Hamel dimension of a vector space is uniquely defined.
		Hamel dimension Hamel dimension The dimension of a vector space V is the...sometimes called Hamel dimension when it is necessary to distinguish it from other types of...other types of dimension.
	www.nationmaster.com /encyclopedia/Hamel-dimension (641 words)

	Cardinality - Definition, explanation
		We say that a set A has cardinality greater than or equal to the cardinality of B (and B has cardinality less than or equal to the cardinality of A) if there exists a 1-1 function from B into A.
		For example, the set R of all real numbers has cardinality strictly greater than the cardinality of the set N of all natural numbers, because the inclusion map i : N → R is 1-1, but it can be shown that there does not exist a 1-1 and onto function from N to R.
		Note that, up until this point, we have only defined the term "cardinality" in a strictly functional role: we have not actually defined the "cardinality" of a set as a specified object itself.
	www.calsky.com /lexikon/en/txt/c/ca/cardinality.php (701 words)

	The Dispatch - Serving the Lexington, NC - News (Site not responding. Last check: )
		The uncountability of a set is closely related to its cardinal number; a set is uncountable if its cardinal number is larger than that of the natural numbers.
		The diagonalization proof technique can also be used to show that several other sets are uncountable, such as 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 the set of natural numbers.
		is now called the continuum hypothesis and is known to be independent of the Zermelo-Frankel axioms for set theory (including the axiom of choice).
	www.the-dispatch.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=uncountable_set (601 words)

	Peter Suber, "Glossary of First-Order Logic"
		The numerical continuum is the series of real numbers; the linear continuum is the series of points on a geometrical line.
		Set theory in which either the generalized continuum hypothesis or the axiom of choice is an axiom.
		Set theory in which either the negation of the generalized continuum hypothesis (GCH) or the negation of the axiom of choice (AC) is an axiom.
	www.earlham.edu /~peters/courses/logsys/glossary.htm (0 words)

	Science Fair Projects - Uncountable set
		Explicitly, a set X is uncountable iff there does not exist a surjective function from the natural numbers N to X.
		Not all uncountable sets have the same size; the sizes of infinite sets are analyzed with the theory of cardinal numbers.
		The cardinality of R is often called the cardinality of the continuum and denoted by c or
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Uncountable_set (495 words)

	How big can a manifold be? (Site not responding. Last check: )
		So cardinality of the union of X is greater than or equal to b, and = therefore not equal to c.
		So cardinality of the union of X is greater than or equal to b, and therefore not equal to c.
		For 1-manifolds, the answer to your question appears to be that the cardinality of the continuum is the limit.
	www.forum-one.org /new-6080310-4346.html (2896 words)

	Math Help Forum - View Single Post - The minimal uncountable well-ordered set
		If you find such a set then it cannot be stricly greater than the cardinality of the integers and strictly less than the cardinality of the continuum, because as you said this would violate the Countinuum hypothesis thus we have two possiblities.
		1)The continuum hypothesis is false and the cardinality of the continuum is NOT the minimal uncountable set.
		Because (and I might be wrong on this) the continuum hypothesis is independent from ZFC (even with the Axiom of Choice).
	www.mathhelpforum.com /math-help/5391-post2.html (144 words)

	The Continuum Hypothesis
		Recall that sets A and B are said to be of the same cardinality iff there is a bijection from A to B.
		A set is countable iff it's either finite or of the same cardinality of N =
		Every infinite subset of R is either countable or of the same cardinality as R.
	www.rpi.edu /~faheyj2/SB/LCU/lcu.driver/node67.html (132 words)

	Blogster.com - Science
		The cardinality of the real line for all ordinal numbers is alef-null.
		For all real numbers in the cuts between the ordinals the cardinality is alef-null for each cut.
		This leads to a cardinality of alef-null times alef-null for all ordinals and reals on the real line.
	blogster.com /Computing_and_Technology/Science/articles.rss (0 words)

	S.O.S. Mathematics CyberBoard :: View topic - bijection between two uncountable sets
		I was under the assumption that if sets were uncountable they had the same cardinality.
		continuum is one of 2 words in the english language with a double uu I've suggested this contraction
		The most common are vacuum and continuum; the less common ones are menstruum, residuum, triduum, and the distinctly rare duumvir and duumvirate."
	www.sosmath.com /CBB/viewtopic.php?t=24831&start=0&postdays=0&postorder=asc&highlight= (427 words)

Try your search on: Qwika (all wikis)