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Topic: Successor ordinal

Related Topics

ordinal number

limit ordinals

successor ordinal

Ordinal directions

Use of ordinals by monarchs

ordinal indicator

name ordinal of state

Patriarch title ordinal of name of the place

name ordinal of location

name ordinal if necessary of state

ordinal ranking of preferences

name ordinal of a place

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	mmtheorems29 - Metamath Proof Explorer
		An ordinal class is a subclass of the successor of its union.
		A successor ordinal is the successor of its union.
		An ordinal equal to its union is not a successor.
	metamath.planetmirror.com /mpegif/mmtheorems29.html (976 words)

	Ordinal number - Wikipedia, the free encyclopedia
		Ordinals are an extension of the natural numbers different from integers and from cardinals.
		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.
	en.wikipedia.org /wiki/Ordinal_number (3800 words)

	PlanetMath: von Neumann ordinal
		The von Neumann ordinal is a method of defining ordinals in set theory.
		The set of finite von Neumann ordinals is known as the von Neumann integers.
		This is version 7 of von Neumann ordinal, born on 2002-03-10, modified 2006-10-15.
	planetmath.org /encyclopedia/Ordinal2.html (158 words)

	Ordinals
		For any nice ordinal N whose members are all clean and any n in N, we have: `n in N' and `n not in n' so N has a member which n lacks, enabling us to distinguish n from N. Thus N isn't equal to any of its members, so N is clean.
		If a nice ordinal is the successor of anything, then that thing is a member of (and subsumed by) our nice ordinal and, consequently, a nice ordinal (but note that empty, among others, is not the successor of anything).
		Since we're uniting successors, any member x of n permits us to ignore each t+m with t in x, because its successor will still be in the union by virtue of t+m being a member of x+m: likewise for any given y in m we can ignore n+s with s in y.
	www.chaos.org.uk /~eddy/math/ordinal.html (3751 words)

	Limit cardinal (Site not responding. Last check: 2007-10-13)
		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 λ is not a successor cardinal, i.e.
		is a successor cardinal if and only if α is a successor ordinal, hence also a limit cardinal if and only if α is a limit ordinal.
		The preceding defines a notion of "inaccessibility": we are dealing with cases where it is no longer enough to do finitely many iterations of the successor and powerset operations; hence the phrase "cannot be reached" in both of the intuitive definitions above.
	bopedia.com /en/wikipedia/l/li/limit_cardinal.html (444 words)

	The Ordinals (Site not responding. Last check: 2007-10-13)
		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
		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)

	[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.
		Not necessarily big things, for example, I couldn't use his definition of successor ordinal which he pretty much admits himself is what we nerds call a kludge.
		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)

	Mudd Math Fun Facts: Ordinal Numbers
		Ordinal numbers even have an interesting arithmetic: we can add two ordinals by concatenating their order types, and considering the ordinal that represents the new order type.
		Thus ordinal multiplication is not necessarily commutative, either, because (2)(omega) is (omega) which is not the same order time as (omega)(2).
		Ordinal numbers form the basis of transfinite induction which is a generalization of the principle of induction.
	www.math.hmc.edu /funfacts/ffiles/30003.8.shtml (586 words)

	df-suc - Metamath Proof Explorer
		Although it is not conventional to use it with proper classes, it has no affect on a proper class (sucprc 2769), so that the successor of any ordinal class is still an ordinal class (ordsuc 2790), simplifying certain proofs.
		Some authors denote the successor operation with a prime (apostrophe-like) symbol, such as Definition 6 of [Suppes] p.
		246 (who uses the symbol "Suc" as a predicate to mean "is a successor ordinal").
	us.metamath.org /mpegif/df-suc.html (114 words)

	PlanetMath: ordinal arithmetic
		Ordinal arithmetic is the extension of normal arithmetic to the transfinite ordinal numbers.
		Cross-references: argument, functions, equivalent, commutative, differences, infinite, sum, finite, limit, ordinals, operation, successor, ordinal numbers, arithmetic, normal, extension
		This is version 3 of ordinal arithmetic, born on 2003-02-23, modified 2005-01-10.
	planetmath.org /encyclopedia/OrdinalArithmetic.html (137 words)

	2.2 Ordinals
		Ordinals are defined in a way that captures the essence of well-ordered sets.
		Ordinals were created to abstract the order structure from the natural numbers, and apply that structure to larger infinite sets, so as one might suspect
		As the successor function for natural numbers forms the basis for their arithmetic, it does also for ordinal arithmetic.
	www.u.arizona.edu /~miller/thesis/node7.html (575 words)

	Ordinals and cardinals
		Ordinals and cardinals form the backbone of mathematics.
		Ordinals can be thought of as general way of representing induction or iteration.
		Recursive iteration is characterized by the recursive ordinals but there is no recursive algorithm to describe the structure of all recursive ordinal s although there is such an algorithm for any recursive ordinal.
	www.mtnmath.com /book/node54.html (387 words)

	[No title]
		Note that for the limit ordinal 0 we have that F[0] is the trivial (one-element) subspace.
		And note that for all other limit ordinals, 'a, we have that F['a] is the closure of the union of the previous values.
		We have reduced to the case that the structure map on the initial algebra is the identity map which means that the naturally isomorphic co-algebra is given by the same identity map.
	www.mta.ca /~cat-dist/catlist/1999/algebraically-compact (1546 words)

	Successor ordinal - Wikipedia, the free encyclopedia
		When defining the ordinal numbers, an absolutely fundamental operation that we can perform on them is a successor operation S to get the next higher one.
		An ordinal number which is S(β) for some ordinal β is called a successor ordinal.
		Ordinals which are neither zero nor successors are called limit ordinals.
	en.wikipedia.org /wiki/Successor_ordinal (154 words)

	Ordinals in a recursive hierarchy
		The notation generated is a function on the integers that outputs a notation for the nth successor to x for integer input n.
		If the ordinal type is that of the integers then one typically constructs a functional on the integers that iterates some process up to the value of the integer parameter.
		If the type is that of recursive ordinal then one defines a functional that does different operations for 0, a successor ordinal and a limit ordinal.
	www.mtnmath.com /book/node64.html (432 words)

	Infinite Ink: Cardinal Numbers
		Since all well-ordered cardinals are ordinals, sometimes I use ordinal notation for a cardinal or vice versa.
		1 is the ordinal successor of a, is the cardinal successor of aleph
		Another way to say this is to say that it is not possible to represent k as the supremum of fewer than k smaller ordinals, or k cannot be written as the sum of fewer smaller cardinals.
	www.ii.com /math/cardinals (1276 words)

	2.3 Ordinal Arithmetic
		Therefore, arithmetic is not done within some set of all ordinals, but rather within some ordinal large enough to contain all the ordinals on which the arithmetic is to be performed.
		Likewise, to prove that some property holds for all ordinals, it is necessary and sufficient that it be proved for all ordinals less than some arbitrary ordinal.
		The definition of addition for ordinals is done using transfinite recursion, and the proof of the theorem below shows how much this definition relies on Theorem 2.18.
	www.u.arizona.edu /~miller/thesis/node8.html (600 words)

	[No title]
		we call these "successors"; - m is not the successor of someone else, nonetheless it is bigger than a whole lot of other members; e.g., w, w+w,...
		We may as well pick one of them and call it "canonical", "representative", etc. An ordinal is a "canonical" well-ordered set.
		In general, there is 0, there are successor ordinals x+1 = x U {x} such as 1 and w+1, and there are limit ordinals (non-successors) such as w and w+w.
	www.cs.toronto.edu /~trebla/transfinite.txt (434 words)

	General Topology - NoiseFactory Science Archives (http://noisefactory.co.uk) (Site not responding. Last check: 2007-10-13)
		We observe, but don't need to go into the detail, that it is possible to define arithmetic on ordinals, much as we define it on integers.
		For finite ordinals there is no distinction between the two interpretations, but things are very different for transfinite ordinals, and we shall generally avoid ordinal arithmetic in our web-pages.
		One final warning before we leave ordinals: It is easy to show that there is no such thing as the "set of all ordinals".
	noisefactory.co.uk /maths/topology.html (4788 words)

	2.4 Ordinal Sequences and Large Sets
		two sequences of ordinals are defined that aid in the proof of Goodstein's Theorem and its independence.
		If is a successor ordinal, then the result is clear.
		is a successor ordinal, then the theorem is immediate.
	www.rdegraaf.nl /mirror/www.u.arizona.edu/~miller/thesis/node9.html (341 words)

	[No title]
		One specialization of the idea of a ~-filtered category is the notion of a ~-* filtered ordinal*.
		First let ~ be any ordinal and we will describe a general embedding method.
		Suppose ff is an ordinal and suppose that we have already constructed X = X0 X1 X2.
	www.math.purdue.edu /research/atopology/Gillespie/sheafproblem.txt (4399 words)

	Searching for Chaos in Cellular Automata: New Tools for Classification
		is used to denote the upper bound of the ordinals family
		The previous definitions are not new but we can extend them in a third way, using transfinite iterations.
		the period grows from one to an ordinal "close" to the cardinality of our configuration space, and the resulting attractor grows from a homogeneous fixed configuration to almost the whole configuration space.
	www.complexity.org.au /ci/vol02/pfgfattr2/pfgfattr2.html (2674 words)

	Continuum Hypothesis. Axiom of Constructibility. Axiom of Determinateness. Ackermann's Set Theory. By K.Podnieks
		The relation bordinals b, c is defined simply as "b in c".
		An ordinal b is called successor ordinal, iff b=c+1 for some c.
		Let us say that an ordinal b is a cardinal number, iff there is no one-to-one correspondence between b and an ordinal less than b.
	linas.org /mirrors/www.ltn.lv/2005.01.29/~podnieks/gt2a.html (5376 words)

	Continuum Hypothesis. Alternative Set Theories
		This formula defines the class of all ordinal numbers (or simply, ordinals), that is denoted traditionally by On.
		The second limit ordinal is w (the set of all natural numbers).
		We know already that if L(s) were a model of ZF containing all ordinals, then L(s)>=L. But, maybe, under these conditions, L(s)=L? Goedel's result of 1938 did not contradict the Cantor's opinion (Cantor died in 1918) that the continuum hypothesis "must be" provable.
	linas.org /mirrors/www.ltn.lv/2001.03.27/~podnieks/gt6_2.html (5306 words)

	Banach space preliminaries
		is a countable successor ordinal, unless Z is already weak-* closed, in which case
		must be either 0 or a successor ordinal.
		It is known that the converse of the previous theorem also holds: For every countable successor ordinal
	www.math.psu.edu /simpson/papers/convex-l/node2.html (999 words)

	The Revision Theory of Truth (Stanford Encyclopedia of Philosophy)
		One way to think of the ordinal numbers is as follows.
		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.
	www.seop.leeds.ac.uk /entries/truth-revision (7149 words)

	[No title]
		Y is a fibration for each successor ordinal fi, then p is also a fibration.
		As long as each A is ~A -small for some ~A, we may choose ~ to be an upper bound for the ordinals ~A.
		Therefore, we may take ~ to be !, and there are no limit ordinals in the construction of W.
	jdc.math.uwo.ca /papers/duality.txt (7642 words)

	Infinite Ink Glossary: Cofinality, Regular, & Singular
		of a well-ordered set X, denoted cf(X), is the least ordinal a such that a can be mapped unboundedly into X.
		Cofinality gives a measure of how "reachable from below" an ordinal is.
		If X is a successor ordinal, i.e., X=a
	www.ii.com /glossary/cofinality_regular_singular (157 words)

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