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Topic: Limit ordinals

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	Limit ordinal - Wikipedia, the free encyclopedia
		A limit ordinal is an ordinal number which is neither zero nor a successor ordinal.
		The classes of successor ordinals and limit ordinals (of various cofinalities) as well as zero exhaust the entire class of ordinals, so these cases are often used in proofs by transfinite induction or definitions by transfinite recursion.
		Limit ordinals represent a sort of "turning point" in such procedures, in which one must use limiting operations such as taking the union over all preceding ordinals.
	en.wikipedia.org /wiki/Limit_ordinal (486 words)

	Ordinal number - Wikipedia, the free encyclopedia
		Ordinals may be categorized as: zero, successor ordinals, and limit ordinals (of various cofinalities).
		Ordinals may be used to label the elements of any given well-ordered set (the smallest element being labeled 0, the one after that 1, the next one 2, "and so on") and to measure the "length" of the whole set by the least ordinal which is not a label for an element of the set.
		Any ordinal can be made into a topological space by endowing it with the order topology (since, being well-ordered, an ordinal is in particular totally ordered): in the absence of indication to the contrary, it is always that order topology which is meant when an ordinal is thought of as a topological space.
	www.wikipedia.org /wiki/Ordinal_number (4254 words)

	Learn more about Ordinal number in the online encyclopedia. (Site not responding. Last check: 2007-10-31)
		Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. The mathematician Georg Cantor showed in 1897 how to extend this concept beyond the natural numbers to the infinite and how to do arithmetic with these transfinite ordinals.
		The smallest uncountable ordinal may be identified with the set of all countable ordinals, and is usually denoted by ω
		Ordinals which don't have an immediate predecessor can always be written as a limit like this and are called limit ordinals.
	www.onlineencyclopedia.org /o/or/ordinal_number.html (1264 words)

	Limit ordinal (Site not responding. Last check: 2007-10-31)
		A limit ordinal is an ordinal number which is not a successor ordinal.
		The classes of successor ordinals and limit ordinals (and if you insist on limit ordinals being infinite, zero) exhaust the entire class of ordinals, so these cases are often used in proofs by transfinite induction or definitions by transfinite recursion.
		Limit ordinals are usually a kind of "turning point" in which we have to use limiting operations such as taking the union over all preceding ordinals (technically we could do anything at limit ordinals, but taking the union is continuous in the order topology and usually this is what we want).
	www.serebella.com /encyclopedia/article-Limit_ordinal.html (525 words)

	Encyclopedia :: encyclopedia : Ordinal number (Site not responding. Last check: 2007-10-31)
		Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc., whereas a cardinal number says "how many there are": one, two, three, four, etc. (See How to name numbers.)
		An ordinal scale defines a total preorder of objects; the scale values themselves have a total order; names may be used like "bad", "medium", "good"; if numbers are used they are only relevant up to strictly monotonically increasing transformations (order isomorphism).
		So in general, raise an ordinal S to the power of an ordinal T, we write down copies of the well-ordered set T, and then replace each element with some element of S, with the restriction that all but a finite number of elements of the sequence must be the first element of S.
	www.hallencyclopedia.com /Ordinal_number (1843 words)

	Limit ordinal (Site not responding. Last check: 2007-10-31)
		Because the class of ordinal numbers is well-ordered, there is a smallest infinite limit ordinal; and we denote this by ω.ω is also the smallest infinite ordinal (forgetting the limit), as it is the least upper bound of the naturalnumbers.
		The classes of successor ordinals and limit ordinals (and if you insist on limit ordinals being infinite, zero) exhaust theentire class of ordinals, so these cases are often used in proofs by transfinite induction or definitions by transfinite recursion.
		Limit ordinals are usually a kind of "turning point" in which we have to uselimiting operations such as taking the union over all preceding ordinals (technically we could do anything at limit ordinals, buttaking the union is continuous in the order topology and usually this is whatwe want).
	www.therfcc.org /limit-ordinal-176837.html (444 words)

	Limit cardinal - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-31)
		In mathematics, limit cardinals are a type of cardinal number.
		With the cardinal successor operation defined, we can define a limit cardinal in analogy to that for limit ordinals: λ is a (weak) limit cardinal if and only if λ is neither a successor cardinal nor zero, i.e.
		Despite the similarity in terminology and concept with limit ordinal, being a limit cardinal is a much stronger condition, because the cardinal successor operation is much more powerful, in the infinite case, than the ordinal successor operation (so we're not just defining something synonymous).
	www.sciencedaily.com /encyclopedia/limit_cardinal (578 words)

	Aleph number - Wikipedia, the free encyclopedia
		Alephs measure the sizes of sets; infinity, on the other hand, is commonly defined as an extreme limit of the real number line, or an extremal point of the extended real number line.
		That it is consistent with ZFC was demonstrated by Kurt Gödel in 1940; that it is independent of ZFC was demonstrated by Paul Cohen in 1963.
		There are, however, some limit ordinals which are fixed points of the aleph function.
	en.wikipedia.org /wiki/Aleph_number (851 words)

	Ordinal number (Site not responding. Last check: 2007-10-31)
		Ordinal numbers or ordinals for short are numbers used to the position in an ordered sequence: first third fourth etc. (See How to name numbers.)
		Indeed since every ordinal contains only ordinals it follows that every member of collection of all ordinals is also its Thus if that collection were a set would have to be an ordinal itself definition; then it would be its own which contradicts the axiom of regularity.
		Ordinals which don't have an immediate can always be written as a limit a net of other ordinals (but not necessarily the limit of a sequence i.e.
	www.freeglossary.com /Ordinal (1608 words)

	[No title]
		We prove that the empty set is an ordinal, and that the members of an ordinal and the successor of an ordinal are ordinals.
		The Successor of an Ordinal is an Ordinal
		Expanding the definition of ordinal and making use of transitivity enables us to infer that members of an ordinals are subsets and permits application of the previous result to obtain connectedness.
	www.rbjones.com /rbjpub/pp/gst/ordinals-m.html (1167 words)

	The Ordinals (Site not responding. Last check: 2007-10-31)
		An ordinal is a set with a particular kind of order relation associated with it.
		However, note that the ordinal "0" contains no elements; that the ordinal "1" contains one element; that the ordinal "2" contains two elements etc. We therefore have a recursive definition of what an ordinal is. In other words we have learnt to count.
		If the only ordinals were successor ordinals then we would be limited to countable ordinals - but since it is possible to take any uncountable set and construct it's "transitive closure" there must be some ordinals which are not successor ordinals.
	www.jboden.demon.co.uk /SetTheory/ordinals.html (432 words)

	Ordinals (Site not responding. Last check: 2007-10-31)
		As described in the last section, the first ordinal is 0, the next ordinal is 1, the next ordinal is 2, and so on, with no sets squeezed in between these successor ordinals.
		If the ordinal s is not 0 and not a successor ordinal, then s is, by definition, a limit ordinal.
		An infinite ordinal is an ordinal that is not finite.
	www.mathreference.com /set-zf,ord.html (571 words)

	Ordinal number (Site not responding. Last check: 2007-10-31)
		Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. (See.) In mathematics, ordinal numbers are an extension of the natural numbers to accommodate infinite sequences, introduced by Georg Cantor in 1897.
		This relies on the axiom of well foundation : every nonempty set S has an element a which is disjoint from S.
		as a limit of countably many smaller ordinals) and are called limit ordinal s; the other ordinals are the successor ordinal s.
	www.serebella.com /encyclopedia/article-Ordinal_number.html (1818 words)

	Practical Foundations of Mathematics
		The ordinals admit a peculiar kind of arithmetic, into which it is often possible to make a crude translation of syntax.
		Although Cantor had developed ordinal arithmetic in 1880, only in 1899 did he formulate a correct definition [Dau79], that an ordinal is an ordered set in which every non-empty subset has a least member: ``there's a first time for everything.'' In other words, a trichotomous (Definition 3.1.3) well founded relation (Proposition 2.5.6).
		The relation on an ordinal is easily seen to be transitive and extensional (Example 6.3.3), so it is useful to identify each ordinal with its set of predecessors, cf 5 = {0,1,2,3,4} and Remark 6.3.5.
	www.cs.man.ac.uk /~pt/Practical_Foundations/html/s67.html (2057 words)

	Finite Induction
		Suppose s is a finite ordinal such that f(s) is false.
		This is a nonempty set of ordinals; let z be the least of these.
		By the above lemma y is a finite ordinal, and f(y) implies f(successor(y)), hence f(z) is true.
	www.mathreference.com /set-zf,ind.html (275 words)

	Transfinite induction (Site not responding. Last check: 2007-10-31)
		It may be regarded as one of three forms of mathematicalinduction.
		The latter step is often broken down into two cases: the case for successor ordinals (ordinals which have an immediate predecessor), where the usual inductive approach canbe applied (show that P(a) implies P(a+1)), and the case for limit ordinals, which have no predecessor, and thus cannot be handled by such an argument.
		Typically, the case for limit ordinals is approached by noting that a limit ordinal b is (by definition) the supremum of all ordinals a < b and using this fact to proveP(b) assuming that P(a) holds true for all a < b.
	www.therfcc.org /transfinite-induction-176834.html (232 words)

	Transfinite induction (Site not responding. Last check: 2007-10-31)
		It may be as one of three forms of mathematical induction.
		The latter step is often broken down two cases: the case for successor ordinals (ordinals which have an immediate predecessor) the usual inductive approach can be applied that P(a) implies P(a +1)) and the case for limit ordinals which have no predecessor and thus be handled by such an argument.
		Typically the case for limit ordinals is by noting that a limit ordinal b is (by definition) the supremum of all ordinals a < b and using this fact to prove b) assuming that P(a) holds true for all a < b.
	www.freeglossary.com /Transfinite_induction (397 words)

	Transfinite induction - TheBestLinks.com - Cardinal number, Mathematical induction, Class (set theory), Ordinal, ...
		The latter step is often broken down into two cases: the case for successor ordinals (ordinals which have an immediate predecessor), where the usual inductive approach can be applied (show that P(a) implies P(a+1)), and the case for limit ordinals, which have no predecessor, and thus cannot be handled by such an argument.
		Typically, the case for limit ordinals is approached by noting that a limit ordinal b is (by definition) the supremum of all ordinals a < b and using this fact to prove P(b) assuming that P(a) holds true for all a < b.
		If P(b) follows from the truth of P(a) for all a < b, then it is simply a special case to say that P(0) is true, since it is vacuously true that P(a) holds for all a < 0.
	www.thebestlinks.com /Transfinite_induction.html (278 words)

	Transfinite induction
		For any ordinal b, if P(a) is true for all ordinals a < b then P(b) is true as well.
		Typically, the case for limit ordinals is approached by noting that a limit ordinal b is (by definition) the union of all ordinals a < b and using this fact to prove P(b) assuming that P(a) holds true for all a < b.
		If P(b) follows from the truth of P(a) for all a < b, then it is simply a special case to say that P(0) is true, since it is vacuously true that P(b) holds for all b < 0.
	www.fastload.org /tr/Transfinite_induction.html (277 words)

	LogBlog: Ordinal Logics - Richard Zach's Logic and Philosophy Blog
		The tricky part is defining the provability predicate for theories in these progressions; you have to use Kleene's recursive ordinals to do this for transfinite ordinals.
		this is because you can only squeeze in a new pi-1 truths at the limit ordinals - the trick is to pick perverse fundamental sequences for the limit ordinals in a way that makes the whole completeness proof utterly irrelevant to any epistemological concerns - and thus you need to go up to omega*omega.
		Fefrman's analysis of predicative acceptability and the determination of Gamma_0 (the Schutte-Feferman ordinal) as the proof theoretical ordinal of predicativity was obtained by means of such autonomous progressions.
	www.ucalgary.ca /~rzach/logblog/2006/03/ordinal-logics.html (912 words)

	The Revision Theory of Truth
		Given a limit ordinal η, a sequence S of objects is an η-long sequence if there is an object S
		Suppose that S is a η-long sequence of hypothesis for some limit ordinal η.
		The constraints are, in some sense local: the first constraint is achieved by putting restrictions on which hypotheses can be used, and the second constraint is achieved by putting restrictions on what happens at limit ordinals.
	plato.stanford.edu /entries/truth-revision (7132 words)

	Hyperstition: TX2.
		While the raw numeracy of TX is most accurately conceived as sub-qabbalistic, due to its indifference to modulus notation (the primary motor of qabbalistic occulturation), its very independence from convention makes it a valuable tool when investigating the basic features of numerical (arithmetical or qabbalistic) codes.
		Within the Anglobal Oecumenon, the most pragmatically prevalent ordinal functions are alphabetical, utilizing the ordering convention of the Neoroman letters to arrange, sort, search and archive on the basis of Alphabetical or Alphanumerical Order, organizing dictionaries, encyclopaedias, lists and indexes 'lexicogrpahically.' The word 'alphabet' itself performs a (Greek) ordinal operation.
		Although it would make much sense to try and work out which was the larger infinity (as you can with ordinals), the two examples are nontheless distict infinities, as the first is a prime number whereas the second is a power of 2.
	www.hyperstition.abstractdynamics.org /archives/005047.html (13012 words)

	Ordinal number : Ordinal
		Another example is the fact that the collection of all ordinals isn't a set.
		One of the distributive laws holds for ordinal arithmetic: R(S+T) = RS + RT.
		Placing this code on your page will help others
	www.fastload.org /or/Ordinal.html (1284 words)

	Limit ordinal - The Jiggies Reference Guide (Site not responding. Last check: 2007-10-31)
		In precise terms, we say λ is a limit ordinal if for any α < λ, S(α) < λ.
		Phrased in yet another way, an ordinal is a limit ordinal if and only if it is equal to the supremum of all the ordinals below it.
		This page was last modified 18:25, Feb 28, 2004.
	www.jiggies.com /reference/Limit_ordinal (476 words)

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