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Topic: Well ordering theorem

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	Well-order - Wikipedia, the free encyclopedia
		The standard ordering ≤ of the integers is not a well-ordering, since, for example, the set of negative integers does not contain a least element.
		The standard ordering ≤ of the positive real numbers is not a well-ordering, since, for example, the open interval (0, 1) does not contain a least element.
		There exists proofs depending on the axiom of choice that it is possible to well order the real numbers, but these proofs are non-constructive and no one has yet shown a method to well order the real numbers.
	en.wikipedia.org /wiki/Well-order (476 words)

	Well-ordering theorem - Wikipedia, the free encyclopedia
		The well-ordering theorem (not to be confused with the well-ordering axiom) states that every set can be well-ordered.
		Georg Cantor considered the well-ordering theorem to be a "fundamental principle of thought." Most mathematicians however find it difficult to visualize a well-ordering of, for example, the set
		It turned out though, that the well-ordering theorem is equivalent to the axiom of choice, in the sense that either one together with the Zermelo-Fraenkel axioms is sufficient to prove the other.
	en.wikipedia.org /wiki/Well-ordering_theorem (182 words)

	Science Fair Projects - Well-order
		The standard ordering of the integers, is not a well-ordering, since, for example, the set of negative integers does not contain a least element.
		The standard ordering of the positive real numbers, is not a well-ordering, since, for example, the open interval (0, 1) does not contain a least element.
		The well-ordering theorem, which is equivalent to the axiom of choice, states that every set is well-orderable.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Well-order (537 words)

	PlanetMath: proof of Zermelo's well-ordering theorem (Site not responding. Last check: 2007-10-07)
		Since the ordinals are well ordered, there is a least ordinal
		Cross-references: ordering, simple, well-ordering, bijection, injective, Burali-Forti paradox, contain, axiom, least element, ordinals, class, function, choice function
		This is version 5 of proof of Zermelo's well-ordering theorem, born on 2002-08-25, modified 2002-09-28.
	planetmath.org /encyclopedia/ProofOfZermelosWellOrderingTheorem.html (141 words)

	Theorem 8.3 (Site not responding. Last check: 2007-10-07)
		Theorem 8.3: Let a and b be two nonzero natural numbers.
		By Theorem 5.1, the Well Ordering Principle, there is a smallest element c in this set which will then be the smallest nonzero natural number that can be written as a sum or difference of multiples of a and b.
		By Theorem 6.8, Theorem 6.10 and Theorem 7.5, the Associative and Distributive Properties of Multiplication, d will divide any natural number which is a sum or difference or multiples of a and b including c.
	www.sonoma.edu /users/w/wilsonst/papers/finite/8/t8-3.html (409 words)

	Georgia Tech Graph Theory Seminar (Site not responding. Last check: 2007-10-07)
		The proof is very similar to proofs of well-quarsi-ordering theorem of graphs by Robertson and Seymour and well-quasi-ordering theorem of binary matroids (or representable over a fixed finite field) by Geelen, Gerards, and Whittle.
		We can think of a partially ordered set as an interrupted sorting process- it is a natural question to ask how much of the work has already been done.
		This paper presents a relationship between number of linear extensions of a poset and its entropy, as well as a nice characterization of graph entropy in the context of a poset.
	www.math.gatech.edu /~thomas/seminar.html (335 words)

	From Frege To Godel: von Heijenoort (Site not responding. Last check: 2007-10-07)
		A new proof of Lowenheim's theorem is given, filling in what Quine [1955] called the "law of infinite conjunction" without using the axiom of choice (Konig's lemma is often used for this purpose today).
		Skolem notes that the theorem implies there must be countable models of set theory and points out the Lowenheim-Skolem "paradox." Skolem points out that separation is not enough to imply the existence of "large" sets such as aleph_omega and proposes the axiom of replacement.
		The theorems of the theory of functions, such as the theory of conformal mapping and the fundamental theorems in the theory of partial differential equations or of Fourier series--to single out only a few examples from our science--are merely ideal propositions in my sense and require the logical epsilon-axiom for their development.
	www.andrew.cmu.edu /~cebrown/notes/vonHeijenoort.html (8419 words)

	Metamath Proof Explorer - mmtheorems38
		This theorem is applicable to a topology, which (among other axioms) is closed under finite intersections.
		Theorem for alternate representation of ordered pairs, requiring Regularity.
		This theorem is proved assuming the Axiom of Infinity and in fact is equivalent to it.
	metamath.planetmirror.com /mpegif/mmtheorems38.html (1695 words)

	PlanetMath: Zermelo's well-ordering theorem (Site not responding. Last check: 2007-10-07)
		The well-ordering theorem is equivalent to the Axiom of Choice.
		See Also: Hausdorff's maximum principle, Zorn's lemma and the well-ordering theorem equivalence of Hausdorff's maximum principle, every vector space has a basis, Kuratowski's lemma, axiom of choice
		This is version 2 of Zermelo's well-ordering theorem, born on 2002-08-25, modified 2002-08-25.
	planetmath.org /encyclopedia/ZermelosWellOrderingTheorem.html (107 words)

	Math Notes - Well-Ordering Principle
		Theorem 1: The least element of a set is unique.
		Corollary to Theorem 1: If b is a least element of set S, and b’ £ b then b = b’.
		Theorem 2: The number 1 is the least positive integer.
	home.att.net /~p.konieczko/mathwell.html (315 words)

	PlanetMath: equivalence of Hausdorff's maximum principle, Zorn's lemma and the well-ordering theorem (Site not responding. Last check: 2007-10-07)
		It is easy to see that this defines a partial order relation on
		"Zorn's lemma and the well-ordering theorem equivalence of Hausdorff's maximum principle" is owned by mathcam.
		This is version 4 of Zorn's lemma and the well-ordering theorem equivalence of Hausdorff's maximum principle, born on 2002-09-29, modified 2003-08-28.
	planetmath.org /encyclopedia/ZornsLemmaAndTheWellOrderingTheoremEquivalenceOfHaudorffsMaximumPrinciple.html (227 words)

	Theorem 8.1 (Site not responding. Last check: 2007-10-07)
		Theorem 8.1: (The Division Algorithm) Let a and b be natural numbers with b not zero.
		so there exists a smallest element s in S by Theorem 5.1, the Well Ordering Principle.
		The uniqueness of such a q follows from the fact that s is uniquely determined by a, and by Theorem 5.7, The Uniqueness of Predecessors, q is uniquely determined.
	www.sonoma.edu /users/w/wilsonst/Papers/finite/8/t8-1.html (253 words)

	The Twelf Meta-Theorem Prover
		A theorem in Twelf is, properly speaking, a meta-theorem: it expresses a property of objects constructed over a fixed LF signature.
		Theorems are stated in the meta-logic M2 (the name might change) whose quantifiers range over LF objects.
		For certain theorems, the theorem prover will not be able to find a proof, even that it should.
	www.cs.cmu.edu /~twelf/alpha/twelf-prover.html (1027 words)

	Set Theory And Logic at the University of Zimbabwe (Site not responding. Last check: 2007-10-07)
		First order languages and first order theories: the tautology theorem, results concerning quantifiers, introduction rule, generalization rule, substitution rule, substitution theorem, distribution theorem, closure theorem, deduction theorem, theorem on constants.
		Natural numbers: Peano axioms, existence, uniqueness, and recursion theorems, establishing the set \Bbb N of natural numbers along with the properties of addition, multiplication and order, well orderedness of \Bbb N. Ordinals and well ordering: definitions (well order, ordinal), examples and elementary results about ordinals, ordinal arithmetic.
		Cardinal Arithmetic: Cardinals, cardinal functions, ordering cardinals, Cantor-Bernstein theorem, the axiom of choice on classes of cardinals, Dedekind infiniteness, cardinal addition, multiplication, exponentiation and properties, continuum hypothesis and generalized continuum hypothesis.
	www.uz.ac.zw /science/maths/courses/hmth037.htm (321 words)

	Math 446 class summaries
		By the adequacy theorem we can therefore determine whether an wf is a theorem of L without actually producing its proof.
		The principle of induction fir arithmetic was compared in its first order expression and in its second (set theoretic) statement which quantifies the pinciple over all sets as opposed to the first order axiom scheme which applies only to the countable number of predicates that can be expressed in the formal first order language.
		Suppose that N is the family of sets that are similar to a well ordered set, A. Then N is called an ordinal number.
	www.humboldt.edu /~mef2/Courses/m446s.html (6324 words)

	PACM Student Seminar
		Although these classic nonlinear diffusion equations are well understood on finite dimensional continuum domains, they have not been studied on general networks.
		Additionally, we present high-quality numerical data on the two discontinuities in the split-second peak of $g_{2}$, and use a shared-neighbor analysis of the graph representing the contact network to study the local particle clusters responsible for the peculiar features.
		The proof is very similar to proofs of well-quasi-ordering theorem of graphs by Robertson and Seymour and well-quasi-ordering theorem of binary matroids (or representable matroids over a fixed finite field) by Geelen, Gerards, and Whittle.
	www.math.princeton.edu /~skryazhi/StudentSeminar/archive/Fall2004.html (1017 words)

	Mudd Math Fun Facts: Ordinal Numbers
		Multiplication of two ordinals A and B can be defined as the ordinal representing the order type of B many copies of A, concatenated.
		Ordinal numbers form the basis of transfinite induction which is a generalization of the principle of induction.
		The Well-Ordering Theorem (on which the principle of transfinite induction is based) is equivalent to the Axiom of Choice.
	www.math.hmc.edu /funfacts/ffiles/30003.8.shtml (590 words)

	Well-ordering theorem -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-07)
		The well-ordering theorem (not to be confused with the (Click link for more info and facts about well-ordering axiom) well-ordering axiom) states that every (A group of things of the same kind that belong together and are so used) set can be (Click link for more info and facts about well-ordered) well-ordered.
		This is important because it makes every set susceptible to the powerful technique of (Click link for more info and facts about transfinite induction) transfinite induction.
		It turned out though, that the well-ordering theorem is equivalent to the axiom of choice, in the sense that either one together with the (Click link for more info and facts about Zermelo-Fraenkel axioms) Zermelo-Fraenkel axioms is sufficient to prove the other.
	www.absoluteastronomy.com /encyclopedia/W/We/Well-ordering_theorem.htm (204 words)

	Station Information - Cantor-Bernstein-Schroeder theorem (Site not responding. Last check: 2007-10-07)
		In set theory, the Cantor-Bernstein-Schroeder theorem is the theorem that for if there exist injective functions
		This is obviously a very desirable feature of the ordering of cardinal numbers.
		An earlier proof by Cantor relied, in effect, on the axiom of choice by inferring the result as a corollary of the well-ordering theorem.
	www.stationinformation.com /encyclopedia/c/ca/cantor_bernstein_schroeder_theorem.html (132 words)

	Cardinal numbers (Site not responding. Last check: 2007-10-07)
		Theorem 0.9 (Well-ordering Theorem: Cantor) Every set can be well-ordered.
		It is immediate that the Axiom of Choice follows from the Well-Ordeing Theorem (why?).
		This theorem is equivalent to the Axiom of Choice.
	people.cs.uchicago.edu /~laci/reu03/n2_7/node3.html (121 words)

	The Ultimate Talk:Well-ordering theorem Dog Breeds Information Guide and Reference (Site not responding. Last check: 2007-10-07)
		Is this technically speaking a theorem (in that it follows from the axiom of choice); or is it a principle (it is equivalent to the axiom of choice)?
		It is a theorem of ZFC, in that it is logically entailed by the axioms of ZFC.
		My point is, it doesn't make sense to say that the Well-Ordering Theorem is "not to be confused with" the Well-Ordering Axiom, when as far as I know, everyone who says "Well-Ordering Axiom" is referring to the Well-Ordering Theorem, not to the Well-Ordering Principle, which is not equivalent to AC.
	www.dogluvers.com /dog_breeds/Talk:Well-ordering_theorem (333 words)

	Archimedes Plutonium (Site not responding. Last check: 2007-10-07)
		Zermelo theorem was that every subset of the Reals can be arranged to have a first element sounded close to the aufbau of the Reals themselves via Dedekind cut.
		For example, in the Banach Tarski theorem the AC was used but the Naturals were not used in Banach Tarski, thus B-T is valid.
		(P> The Banach Tarski theorem is a valid proof for the use of AC is to use AC on Whole Rationals or Rationals in order to manufacture Irrationals.
	www.iw.net /~a_plutonium/File107.html (2837 words)

	Construction of sets and Peano's Axioms
		A ring is ordered if it has a relation of 'greater than', symbolized by ">", with 1>0, b>a if and only if b-a>0, and such that the positive elements (those greater than 0) are closed under addition and multiplication.
		The set of Reals (limits of converging series of Rationals) is usually denoted R. An ordered field is complete if every nonempty set X of its elements that is bounded above (there is some element of the field that is greater than or equal to every element in X) has a smallest upper bound.
		Counting Ordered Pairs of Integers -- An explanation of the "square spiral" that puts the set of natural numbers in one-to-one correspondence with the set of rational numbers.
	mcraeclan.com /MathHelp/BasicSetConstruction.htm (632 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		The direct generalization of Spectors Theorem to the class of linear orderings does not hold; it is not the case that every linear ordering with a hyperarithmetic presentation is isomorphic to a computable one.
		Many properties of linear orderings are invariant under equimorphism, as, for example, extendibility, indecomposability, well foundedness, being scattered, or having a certain Hausdorff rank.
		The structure of equimorphism types of countable linear orderings ordered by embeddability is also an interesting object.
	www.wesleyan.edu /cgi-bin/cdf_manager/template_renderer.cgi?item=21933 (275 words)

	The Well Ordering Principle
		The "well ordering principle" says yes, but it really depends on the axiom of choice.
		In fact the well ordering principle and the axiom of choice are equivalent.
		The axiom of choice is equivalent to the well ordering principle, which asserts a well ordering on every set.
	www.mathreference.com /set-card,wop.html (639 words)

	2.2 Ordinals
		The next few theorems provide this foundation, resulting in Theorem 2.14 and Theorem 2.17, which form the basis for ordinal addition.
		To prove some theorems regarding ordinals, it is necessary to discuss, more generally, well-ordered sets and order isomorphisms.
		The first elements of a well-ordered set, with respect to its ordering, have special importance, and are useful in proving theorems about well-ordered sets.
	www.u.arizona.edu /~miller/thesis/node7.html (575 words)

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