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Topic: Well-ordering theorem

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Ernst Zermelo

Mathematical induction

Well-ordering theorem - Wikipedia, the free encyclopedia 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. 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 en.wikipedia.org /wiki/Well-ordering_theorem   (182 words)

PlanetMath: proof of Zermelo's well-ordering theorem This is version 5 of proof of Zermelo's well-ordering theorem, born on 2002-08-25, modified 2002-09-28. "proof of Zermelo's well-ordering theorem" is owned by Henry. Cross-references: ordering, simple, well-ordering, bijection, injective, Burali-Forti paradox, contain, axiom, least element, ordinals, class, function, choice function planetmath.org /encyclopedia/ProofOfZermelosWellOrderingTheorem.html   (141 words)

Theorem 8.3 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. Theorem 8.3: Let a and b be two nonzero natural numbers. www.sonoma.edu /users/w/wilsonst/papers/finite/8/t8-3.html   (409 words)

PlanetMath: equivalence of Hausdorff's maximum principle, Zorn's lemma and the well-ordering theorem 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. "Zorn's lemma and the well-ordering theorem equivalence of Hausdorff's maximum principle" is owned by mathcam. It is easy to see that this defines a partial order relation on planetmath.org /encyclopedia/ZornsLemmaAndTheWellOrderingTheoremEquivalenceOfHaudorffsMaximumPrinciple.html   (227 words)

From Frege To Godel: von Heijenoort 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. 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. www.andrew.cmu.edu /~cebrown/notes/vonHeijenoort.html   (8419 words)

Well-ordering principle - Wikipedia, the free encyclopedia Sometimes the phrase well-ordering principle is taken to be synonymous with "well-ordering theorem". On other occasions the phrase is taken to mean the proposition that the set of natural numbers {1, 2, 3,....} is well-ordered, i.e., each of its non-empty subsets has a smallest member. www.wikipedia.org /wiki/Well-ordering_principle   (190 words)

Math Notes - Well-Ordering Principle Theorem 2: The number 1 is the least positive integer. 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’. home.att.net /~p.konieczko/mathwell.html   (315 words)

Theorem 8.1 so there exists a smallest element s in S by Theorem 5.1, the Well Ordering Principle. Theorem 8.1: (The Division Algorithm) Let a and b be natural numbers with b not zero. 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)

Metamath Proof Explorer - mmtheorems38 This theorem is proved assuming the Axiom of Infinity and in fact is equivalent to it. 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. metamath.planetmirror.com /mpegif/mmtheorems38.html   (1695 words)

Georgia Tech Graph Theory Seminar 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. 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. 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. www.math.gatech.edu /~thomas/seminar.html   (335 words)

Math Induction and Well Ordering In order to prove the latter implication, we assume that the Well Ordering Principle is false and show that the Math Induction Principle doesn't hold. The documents summaries some of the ideas around the observation that the concept of Mathematical Induction and logically equivalent to the concept of Well Ordering A set with an order relationship (You should know that there is a technical definition of "order relationship". mathserv.monmouth.edu /coursenotes/kuntz/math410/WellOrdering.htm   (506 words)

Construction of sets and Peano's Axioms 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. 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. mcraeclan.com /MathHelp/BasicSetConstruction.htm   (632 words)

Metamath Proof Explorer - weth This theorem was proved from axioms: ax-1 3 ax-2 4 us.metamath.org /mpegif/weth.html   (19 words)

Set Theory And Logic at the University of Zimbabwe 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. 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. www.uz.ac.zw /science/maths/courses/hmth037.htm   (321 words)

mock savvy » a priori induction? brilliant! In fact, the Axiom of Choice is controversial in no small part because of its equivalence to the well-ordering theorem (the Axiom of Choice was originally introduced by Zermelo to prove the well-ordering theorem); after all, the well-ordering theorem just seems false. Now, I suspect what you were thinking about was what’s usually called the well-ordering theorem, which says that every set can be well-ordered. Despite being called a theorem, this is equivalent to the Axiom of Choice (and thus, of course, Zorn’s Lemma) and so is just as controversial as the Axiom of Choice. mocksavvy.net /?p=4   (1058 words)

Math 446 class summaries Suppose that N is the family of sets that are similar to a well ordered set, A. Then N is called an ordinal number. By the adequacy theorem we can therefore determine whether an wf is a theorem of L without actually producing its proof. A theorem is the last statement of a proof. www.humboldt.edu /~mef2/Courses/m446s.html   (6324 words)

Mudd Math Fun Facts: Ordinal Numbers The Well-Ordering Theorem (on which the principle of transfinite induction is based) is equivalent to the Axiom of Choice. 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. www.math.hmc.edu /funfacts/ffiles/30003.8.shtml   (590 words)

The Twelf Meta-Theorem Prover 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. A theorem in Twelf is, properly speaking, a meta-theorem: it expresses a property of objects constructed over a fixed LF signature. www.cs.cmu.edu /~twelf/alpha/twelf-prover.html   (1027 words)

Station Information - Cantor-Bernstein-Schroeder theorem 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. This is obviously a very desirable feature of the ordering of cardinal numbers. In set theory, the Cantor-Bernstein-Schroeder theorem is the theorem that for if there exist injective functions www.stationinformation.com /encyclopedia/c/ca/cantor_bernstein_schroeder_theorem.html   (132 words)

Well-ordering principle The fine Doric capitals are well time, seem to support with difficulty their noble. www.termsdefined.net /we/well-ordering-principle.html   (359 words)

Cardinal numbers The Well-ordering Theorem is equivalent to the Axiom of Choice. 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?). people.cs.uchicago.edu /~laci/reu03/n2_7/node3.html   (121 words)

PACM Student Seminar 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. 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. www.math.princeton.edu /~skryazhi/StudentSeminar/archive/Fall2004.html   (1017 words)

The Ultimate Talk:Well-ordering theorem Dog Breeds Information Guide and Reference 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. The Well-Ordering Axiom, on the other hand, states that any set can be well-ordered, and that is what this article is about. It is a theorem of ZFC, in that it is logically entailed by the axioms of ZFC. www.dogluvers.com /dog_breeds/Talk:Well-ordering_theorem   (333 words)

Amazon.com: Books: Graph Coloring Problems To give you an idea of the level of the discussion in the text, here is an excerpt from page 1: After a terse definition of vertex coloring and "chromatic number", the authors state that "The existence of the chromatic number follows from the Well-Ordering Theorem of set theory... The book consists of many short sections, each one often less than a page, formally stating a theorem or a conjecture, then briefly summarizing what is known about it, and where the results have been published. This is a highly technical book that gathers together in one medium-sized volume (less than 300 pages) hundreds of new and classical theorems and conjectures on every conceivable type of graph coloring problem. www.amazon.com /exec/obidos/tg/detail/-/0471028657?v=glance   (1053 words)

The Well Ordering Principle In fact the well ordering principle and the axiom of choice are equivalent. The "well ordering principle" says yes, but it really depends on the axiom of choice. 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)

The Principles of Induction and Well-Ordering. Theorem 7 (The Well-Ordering Principle) Every nonempty subset of natural numbers contains a smallest member. Theorem 1 (The Principle of "Weak" Induction) Let S be a subset of natural numbers Theorem 4 (The Principle of Strong Induction) Let S be a subset of natural numbers www.math.psu.edu /elkin/math/497a-Fa02/induction/induction   (163 words)

Krzysztof Ciesielski: Set Theory for the Working Mathematician; a review of the book by K.P. Hart The full power of the well-ordering theorem is brought to bear on the construction of weird subsets of and weird functions on the reals. Both books give proofs of the Hahn-Banach theorem and the theorem that every vector space has a base as well as a construction of a discontinuous additive function. For example, Cantor's theorem on the uniqueness of rationals as an ordered set is proved using the partially ordered set of finite order-preserving maps. at.yorku.ca /t/o/p/c/62.htm   (1175 words)

pal.courses.S00 The main theorem of this course is Morley's theorem. We will prove Godel's completeness theorem, which says that a theory is consistent in the proof theoretic sense iff it has a model. The highlight of the course will be Godel's 1931 incompleteness theorem, which in a specific sense says that no ``reasonable'' axiomatic system for mathematics is sufficient to derive all truths. www.math.cmu.edu /~rami/pal.courses.S00   (1709 words)

Ordering Online Well-ordering theorem 2: ng theorem''' (not to be confused with the well-ordering axiom) states that every set can be well- 6: ring theorem. It turned out though, that the well-ordering theorem is equivalent to the axiom of choice, in 8: See also well-ordering principle. Path-ordering 1: (or a meta-operator $\{\backslash mathcal\; P\}$) of ordering a product of many operators according to the valu 11: parameter $\backslash sigma$ that determines the ordering is a parameter describing the contour, and be 13: == Time ordering == 15: therefore this type of ordering is called '''time ordering '''. www.witchware.com /File/36251-Ordering.Online.Html   (494 words)

well-ordering theorem the theorem of set theory that every set can be made a well-ordered set. www.infoplease.com /ipd/A0734890.html   (31 words)

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