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Topic: Cofinality

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	Cofinality
		The cofinality of A is the smallest cardinality of a cofinal subset.
		If A admits a totally ordered cofinal subset B, then we can find a subset of B which is well-ordered and cofinal in B (and hence in A).
		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.
	www.ebroadcast.com.au /lookup/encyclopedia/co/Cofinality.html (120 words)

	PlanetMath: cofinality
		The cofinality of any totally ordered set is necessarily a regular cardinal.
		This is version 22 of cofinality, born on 2002-02-19, modified 2006-10-12.
		Object id is 2205, canonical name is Cofinality.
	www.planetmath.org /encyclopedia/Cofinality.html (206 words)

	PlanetMath: another definition of cofinality
		The cofinality of a cardinal is always a regular cardinal and hence
		"another definition of cofinality" is owned by x_bas.
		This is version 5 of another definition of cofinality, born on 2003-08-20, modified 2003-08-20.
	planetmath.org /encyclopedia/AnotherDefinitionOfCofinality.html (83 words)

	PlanetMath: partitions less than cofinality
		This follows easily from the definition of cofinality.
		"partitions less than cofinality" is owned by Henry.
		This is version 2 of partitions less than cofinality, born on 2002-08-10, modified 2007-06-17.
	www.planetmath.org /encyclopedia/PartitionsLessThanCofinality.html (62 words)

	Ordinal
		A class of ordinals is said to be unbounded, or '''cofinal''', when given any ordinal, there is always some element of the class greater than it (then the class must be a proper class, i.e., it cannot be a set).
		The cofinality of a set of ordinals or any other well ordered set is the cofinality of the order type of that set.
		the cofinality of the cofinality of ''Î±'' is the same as the cofinality of ''Î±''.
	www.seattleluxury.com /encyclopedia/entry/ordinal (4001 words)

	Cofinality - Wikipedia, the free encyclopedia
		The cofinality of A is the least cardinality of a cofinal subset.
		Cofinality can also be similarly defined for a directed set and it is used to generalize the notion of a subsequence in a net.
		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.
	en.wikipedia.org /wiki/Cofinality (302 words)

	Cofinality -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-03)
		Cofinality is only an interesting concept if there is no (Click link for more info and facts about greatest element) greatest element in A since otherwise the cofinality is 1.
		Cofinality can also be similarly defined for a (Click link for more info and facts about directed set) directed set and it is used to generalize the notion of a (Something that follows something else) subsequence in a (An open fabric of string or rope or wire woven together at regular intervals) net.
		Moreover, any cofinal subset of B whose cardinality is equal to the cofinality of B is well-ordered and (Click link for more info and facts about order isomorphic) order isomorphic to its own cardinality.
	www.absoluteastronomy.com /encyclopedia/c/co/cofinality.htm (312 words)

	Piotr Koszmider's web page
		In particular this construction shows that it is consistent that the minimal character of a nonprincipal ultrafilter in a homomorphic image of an algebra A can be strictly less that the minimal size of a homomorphic image of A, answering a question of J. Monk.
		This proves, answering a question of Arhangel'skii, that it is consistent that there is a first countable compact space which has a continuous image without a point of countable character.
		of uncountable cofinality it is consistent that there is a countably tight Boolean algebra A with a distinguished ultrafilter
	www.ime.usp.br /~piotr/cv/ativpas199294.html (589 words)

	mmtheorems47 - Metamath Proof Explorer
		Cofinality is a function on the class of ordinal numbers.
		Cofinality is bounded by the cardinality of its argument.
		Value of the cofinality function at omega (the set of natural numbers).
	us.metamath.org /mpegif/mmtheorems47.html (915 words)

	Cofinality (Site not responding. Last check: 2007-11-03)
		The cofinality of A is theleast cardinality of a cofinal subset.
		If A admits a totally ordered cofinal subset B, then wecan find a subset of B which is well-ordered and cofinal in B (and hence in A).
		Cofinality can also be similarly defined for a directed set and it isused to generalize the notion of a subsequence in a net.
	www.therfcc.org /cofinality-329126.html (282 words)

	[No title]
		Only weak monotonicity is true, plus the fact that the cofinality of 2^x must be strictly more than x.
		For k singular of uncountable cofinality there are quite a few results.
		It was known from the 1910's or 1920's that ZFC proves the continuum cannot have countable cofinality (ie with AC the reals are not a countable union of sets each of cardinality strictly smaller than the continuum).
	www.math.niu.edu /~rusin/known-math/99/luzin_easton (1129 words)

	Cofinality (Site not responding. Last check: 2007-11-03)
		En matemáticas, especialmente en teoría de la orden, un subconjunto B de un grupo parcialmente pedido A es cofinal si para cada a en A hay un b en B tales que un ≤ b.
		Si A admite un subconjunto totalmente pedido B del cofinal, entonces podemos encontrar un subconjunto de B bien-se pida que y cofinal en B (y por lo tanto en A).
		Cofinality se puede también definir semejantemente para un grupo dirigido y se utiliza para generalizar la noción de un subsequence en una red.
	www.yotor.net /wiki/es/co/Cofinality.htm (317 words)

	Antimeta: Not Countably Many
		However, it is clear that nothing with cofinality A is a possible size of the universe, which rules out A itself (so the universe must be uncountable, answering Brian Weatherson's first question negatively), and aleph_A, and aleph_(aleph_A), and aleph_(A+A), and epsilon_0 (the first fixed point of the aleph function, ie aleph_(aleph_(...))).
		In addition, the only 2^K that can have cofinality aleph_1 is 2^A, so either the atomic part is required to have cofinality at least aleph_2, or 2^A is at least aleph_(aleph_1), in which case the universe is required to have at least that cardinality, which rules out uncountably many cardinalities.
		But at any rate, the universe is not countable, and does not have countable cofinality, and is also at least the size of the continuum.
	www.ocf.berkeley.edu /~easwaran/blog/2005/04/not_countably_many.html (983 words)

	sci.math: Well-orderings of infinite cardinals
		In the case when k is not regular, let m be its cofinality.
		The cofinality of an ordinal a, denoted cf(a), is the least ordinal
		The cofinality of the limit ordinal alpha is the smallest
	sci.tech-archive.net /Archive/sci.math/2004-12/5442.html (423 words)

	Transactions of the American Mathematical Society
		Depending on the large cardinals we start with, this forcing can blow the power of a cardinal together with changing its cofinality to a prescribed value.
		M. Magidor, Changing Cofinality Of Cardinals, Fundamenta Mathematicae 99 (1978), 61-71 MR 57:5754
		M. Gitik, Changing Cofinalities and the Non-Stationary Ideal, Israel Journal of Mathematics 56 (1986), 280-314 MR 89b:03086
	www.ams.org /tran/2003-355-05/S0002-9947-03-03202-1/home.html (391 words)

	[ShTh:524] (Site not responding. Last check: 2007-11-03)
		A group G that is not finitely generated can be written as the union of a chain of proper subgroups.
		The cofinality spectrum of G, written CF(S), is the set of regular cardinals lambda such that G can be expressed as the union of a chain of lambda proper subgroups.
		The cofinality of G, written c(G), is the least element of CF(G).
	www.math.rutgers.edu /pub/shelarch/all/524_abs.html (158 words)

	Cofinality - Definition up Erdmond.Com (Site not responding. Last check: 2007-11-03)
		The fact that a countable union of countable sets is countable implies that the cofinality of the cardinality of the continuum must be uncountable, and hence we have :2^{\aleph_0}\neq\aleph_\omega, the ordinal number ω being the first infinite ordinal; this is because : \aleph_\omega = \bigcup_{n.
		Many more interesting results relating cardinal numbers and cofinality follow from a useful theorem of König (e.g., κ cf(κ) and κ κ) for any infinite cardinal κ).
		Cofinality can also be similarly defined for a directed_set and it is used to generalize the notion of a subsequence in a net.
	www.erdmond.com /Cofinality.html (281 words)

	Ordinal Calculator
		But then {epsilon(b): bcofinal in omega(c+1), which being an isolated cardinal cannot have a cofinal subset of a lesser cardinality (AC used here).
		As for (c2) and (d1), a cofinal subset of the required type can easily be constructed and it can be shown that any greater cofinality of the result would imply a greater cofinality of the operand (the contradiction).
		If calculating in a theory that lacks AC, however, the cofinality of each element of E must be known and added to E.
	www.volny.cz /behounek/logic/papers/ordcalc/index.html (1378 words)

	Weakly inaccessible cardinal
		A cardinal number κ > ‭א‬0 is called weakly inaccessible iff cf(κ) = κ, where cf denotes the cofinality.
		Assuming that ZFC is consistent, the existence of weakly inaccessible cardinals provably cannot be proved in ZFC.
		I say, though, falling to the ground, for that would be a fear unworthy.
	www.termsdefined.net /we/weakly-inaccessible-cardinal.html (188 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		The least \htmladdnormallink{cardinality}{http://planetmath.org/encyclopedia/Cardinality.html} of a cofinal set of $P$ is called the \emph{cofinality} of $P$.
		Equivalently, the cofinality of $P$ is the least \htmladdnormallink{ordinal}{http://planetmath.org/encyclopedia/2787.html} $\alpha$ such that there is a cofinal function $f\colon\alpha\to P$.
		The cofinality of $P$ is written $\cf{P}$, or $\cof{P}$.
	www.ma.utexas.edu /~jcorneli/e/work%20folder/FEM-2004-08-16/TeX/03E04--Cofinality.tex (210 words)

	Ernest Schimmerling (Site not responding. Last check: 2007-11-03)
		The possible behaviors of the continuum function $\kappa \mapsto 2^\kappa$ restricted to regular cardinals was mostly understood in 1970.
		By the late 1980's, Shelah had developed techniques, known collectively as {\em PCF theory}, which could also be used to prove theorems about cardinals of countable cofinality.
		Shelah uses these methods to obtain new results of the form: if $\aleph_\alpha$ is a singular strong limit cardinal of uncountable cofinality and $\aleph_\alpha$ is small, then $(2^(2^{\aleph_\alpha})^+$ is also small.
	www.math.cmu.edu /users/eschimme/seminar/raff.html (238 words)

	Ordinal Calculator
		But then {epsilon(b): bcofinal in omega(c+1), which being an isolated cardinal cannot have a cofinal subset of a lesser cardinality (AC used here).
		As for (c2) and (d1), a cofinal subset of the required type can easily be constructed and it can be shown that any greater cofinality of the result would imply a greater cofinality of the operand (the contradiction).
		If calculating in a theory that lacks AC, however, the cofinality of each element of E must be known and added to E.
	ikaros.ff.cuni.cz /~behounek/ordinalc.htm (1403 words)

	Joerg's Recent Work
		This paper evolved from a series of lectures given at the seminar in Kobe in spring 2001 and, in more compressed form, at the Shelah conference in Beer Sheva in May 2001.
		We show it is consistent the almost disjointness number a has countable cofinality.
		We show that g leq cof(Sym(omega)) where g is the groupwise density number and cof(Sym(omega)) is the cofinality of the group of all permutations of the natural numbers.
	kurt.scitec.kobe-u.ac.jp /~brendle/current.html (1206 words)

	ABSTRACTS DROSTE
		This is a strong form of uncountable cofinality for $G$, where each $H_i$ is a subgroup of $G$.
		Our favored example is the group \bsym of all bounded permutations of the rationals $\Q$ which has uncountable cofinality but countable strong cofinality.
		The cofinality of a group G is the cardinality of the length of a shortest chain of proper subgroups terminating at G.
	www.math.tu-dresden.de /alg/droabal.html (3181 words)

	ABSTRACTS DROSTE
		This is a strong form of uncountable cofinality for $G$, where each $H_i$ is a subgroup of $G$.
		Our favored example is the group \bsym of all bounded permutations of the rationals $\Q$ which has uncountable cofinality but countable strong cofinality.
		The cofinality of a group G is the cardinality of the length of a shortest chain of proper subgroups terminating at G.
	www.informatik.uni-leipzig.de /theo/pers/droste/droabal.html (3181 words)

	Infinite Ink: Cardinal Numbers
		Another way to say this is to say that it is possible to represent k as the supremum of fewer than k smaller ordinals.
		any finite cardinal greater than 1, i.e., in {2, 3, 4,....}, has cofinality 1.
		,...}, where a is any ordinal, has cofinality w.
	www.ii.com /math/cardinals (1276 words)

	Ernest Schimmerling (Site not responding. Last check: 2007-11-03)
		It is know that GCH is not strong enough to guarantee the existence of $\omega_1$-Suslin trees, but for $\omega_2$, this problem remains open.
		It is fairly easy to show (under CH) that $\Diamond_{\omega_2}(S)$ implies the existence of $\omega_2$-Suslin trees if $S$ is a stationary subset of $\omega_2$ of cofinality $\omega_1$ points.
		GCH also implies $\Diamond_{\omega_2}(S)$ when $S$ is a set of cofinality $\omega$ points, but it does not guarantee that a nonreflecting stationary set exists.
	www.math.cmu.edu /users/eschimme/seminar/dore2.html (171 words)

	Talk:Cofinality - Wikipedia, the free encyclopedia
		There is this statement: "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." Is this correct?
		The last part "order isomorphic to its own cardinality." would imply that B is a cardinal?!
		I don't know much about cofinality, but I deleted the statement that the cofinality of an infinite cardinal is itself, as I'm pretty sure this is wrong.
	www.wikipedia.org /wiki/Talk:Cofinality (140 words)

	JSTOR: Journal of Symbolic Logic: Vol. 47, No. 2, pp. 275-288 (Site not responding. Last check: 2007-11-03)
		Let $\kappa_B$ be the least cardinal for which the Baire category theorem fails for the real line $\mathbf{R}$.
		Thus $\kappa_B$ is the least $\kappa$ such that the real line can be covered by $\kappa$ many nowhere dense sets.
		It is shown that $\kappa_B$ cannot have countable cofinality.
	www.math.wisc.edu /~miller/res/ctble.htm (94 words)

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