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Topic: Well-ordering principle

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PlanetMath: well-ordering principle for natural numbers This is version 6 of well-ordering principle for natural numbers, born on 2001-10-16, modified 2004-02-27. Maybe we should be specific, and say "well-ordering principle for natural numbers" and "general well-ordering principle", because when you say it, it seems to mean "the naturals can be well-ordered", and when I say it, it means "all sets can be well-ordered." I don't know which is the standard definition. For example, the positive integers are a well-ordered set under the standard order. www.planetmath.org /encyclopedia/WellOrderingPrinciple.html   (200 words)

Principle Principle of relativity In general, the principle of relativity is the requirement that the frame of reference of an imp... Pauli exclusion principle The Pauli exclusion principle is a 1925. Divine Principle Divine Principle is the main theological textbook of the Unification Church, held to have the status of... www.brainyencyclopedia.com /topics/principle.html   (1107 words)

Zorn Well similar maximum principles had been proposed earlier in different contexts by several mathematicians, for example Hausdorff, Kuratowski and Brouwer. Well at the end of the 1935 paper he did say that these three are all equivalent and promised a proof in a future paper. His maximum principle asserted that if a collection of sets is closed, then it must contain a maximal member, that is a set which is not a proper subset of some other in the collection. www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Zorn.html   (1490 words)

Well-order - Wikipedia The well-ordering principle, which is equivalent to the axiom of choice, states that every set can be well-ordered. For example, the standard ordering of the natural numbers is a well-ordering, but neither the standard ordering of the integers nor the standard ordering of the positive real numbers is a well-ordering. A well-order (or well-ordering) on a set S is a total order on S with the property that every nonempty subset of S has a least element in this ordering. nostalgia.wikipedia.org /wiki/Well-order   (313 words)

Talk:Well-ordering theorem - Wikipedia, the free encyclopedia I have always learned that the Well-Ordering Principle is simply that the Naturals are well-ordered, and that the Well-Ordering Principle is equivalent to the Principle of Mathematical Induction. The Well-Ordering Principle certainly doesn't deserve to be called an axiom, and although the Well-Ordering Theorem can be proved from AC, it can also receive axiom status indifferently - it is independent of ZF. The Well-Ordering Axiom, on the other hand, states that any set can be well-ordered, and that is what this article is about. en.wikipedia.org /wiki/Talk:Well-ordering_theorem   (348 words)

math_class: NumberTheory101 (#1): Modular Arithmetic The Well-Ordering Principle says that there is a smallest element in the set. For the moment, let me just say a bit about how the Well-Ordering Principle is typically used in proofs. The Well-Ordering Principle appears frequently in mathematical proofs. www.livejournal.com /talkread.bml?journal=math_class&itemid=1334   (2317 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)

math_class: Number Theory 101 (Chinese Remainder Theorem) It shouldn't come as a surprise to you that the proof of this takes advantage of the Well-Ordering Principle. Well, it turns out to be useful in lots of things to know that the Greatest Common Divisor of two numbers is one. We'll get back to how we knew to mutliply by three in a moment. www.csh.rit.edu /~pat/math/series/nt/20020926   (2315 words)

Well-ordering principle Sometimes the phrase well-ordering principle (or the axiom of choice) is taken to be synonymous with well-ordering theorem ". I found the 2e of "Principles of Biochemistry" by A. Lehninger, one of the most exciting books written in the field of Biochemistry. On other occasions the phrase is taken mean the proposition that the set of numbers {1 2 3....} is well-ordered i.e. www.freeglossary.com /Well-ordering_principle   (482 words)

tr-abstracts.txt As well as being simpler and quicker than their predecessors, our protocols also have slightly stronger security properties: in particular, they make no cryptographic use of s and so impose no subtle restrictions upon the use which is made of s by other protocols. Both these are well understood, and we present a summary of existing work, laying a foundation for the central body of the report where the sub-process of reconstruction is studied. In particular, the proofs confirm the explicitness principle of Abadi and Needham. www.cl.cam.ac.uk /TechReports/tr-abstracts.txt   (20634 words)

Well-ordering principle - Wikipedia The well-ordering principle is equivalent to the axiom of choice, in the sense that either one together with the Zermelo-Fraenkel axioms (see set theory) is sufficient to prove the other. The well-ordering principle states that every set can be well-ordered. This is important because it makes every set susceptible to the powerful technique of transfinite induction. nostalgia.wikipedia.org /wiki/Well-ordering_principle   (99 words)

PlanetMath: principle of finite induction proven from well-ordering principle This is version 4 of principle of finite induction proven from well-ordering principle, born on 2001-10-18, modified 2003-07-26. "principle of finite induction proven from well-ordering principle" is owned by KimJ. principle of finite induction proven from well-ordering principle planetmath.org /encyclopedia/PrincipleOfFiniteInductionProvenFromWellOrderingPrinciple.html   (87 words)

Encyclopedia: Axiom of Choice This is a joke that although the axiom of choice, the well-ordering principle, and Zorn's lemma are mathematically equivalent, most mathematicians find the axiom of choice to be intuitive, the well-ordering principle to be counterintuitive, and Zorn's lemma to be too complex for any intuition. In fact, Zermelo initially introduced the axiom of choice in order to formalize his proof of the well-ordering principle. Perhaps if we were clever we might say, "Even though the usual ordering of the real numbers does not work, it may be possible to find a different ordering of the real numbers which is a well-ordering. www.nationmaster.com /encyclopedia/Axiom-of-Choice   (4632 words)

Zorn's lemma - Wikipedia, the free encyclopedia Since T is totally ordered, we know that J is a subset of K or vice versa. For every totally ordered subset T we may then define a bigger element b(T), because T has an upper bound, and that upper bound has a bigger element. Then there exists a partially ordered set, or poset, P such that every totally ordered subset has an upper bound, and every element has a bigger one. en.wikipedia.org /wiki/Zorn%27s_lemma   (830 words)

PlanetMath: Zermelo's well-ordering theorem 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 www.planetmath.org /encyclopedia/ZermelosWellOrderingTheorem.html   (107 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)

Mathematics, Metamathematics and Computer Science Well ordering principle: Every set X can be well ordered; that is, there is a relation < which well orders X. Zermelo's axiom of choice: Let C be any collection of non-empty sets. Mathematical induction and the well-ordering principle are equivalent. Hausdorff's maximal principle: Every partially ordered set includes a maximal chain, that is, a chain which is not a proper subset of any other chain. cs.wwc.edu /~aabyan/CII/MetaMath.html   (1374 words)

Re: Well-ordering principle Well I have to say I was surprised to read that. However, with WOP, a roster notation of the reals is possible, so >take >> >that roster and list it. I was expecting this thread to start veering towards the dark side... www.usenet.com /newsgroups/sci.math/msg15123.html   (338 words)

Rational Mathematics: Order The validity of the method of mathematical induction for the nonfinite 'set of all numbers' cannot be proved without assuming the well-ordering principle, that any set of numbers has a least member. For the properties of the order relations to be true we require that a set cannot be placed in one-to-one correspondence with a proper subset of itself (since this would mean we could have s < s). We can continue to define symbols for numbers for as far as we can count, and theoretically this can 'go on for ever': but there are in fact practical bounds on this. homepage.ntlworld.com /gpjnow/RM3-order.htm   (611 words)

mock savvy » a priori induction? brilliant! The well-ordering principle is not a controversial axiom, other than in the sense that it is the only second-order Peano axiom and we’d prefer to be able to characterize the natural numbers entirely in first-order logic. PS: The funny thing is, what made mathematical induction clear for me was the proof involving the well-ordering principle, which is actually a controversial axiom, though I’m not very familiar with the controversy. As the resident mathematician at selling waves has said before, dealing with stuff like this isn’t exactly intuitive (ok, so it’s not as weird as matrix groups, but infinite sets are still a little hard to ponder). www.mocksavvy.net /?p=4   (1058 words)

nurice: Empirical musings. There is no well ordering of the states of being. Well to be honest, that is the exact opposite of what would happen. Oh well, I have the time, the internet is down and these thoughts have been a stewin' for some time. www.livejournal.com /~nurice/22537.html   (11594 words)

The Prime Glossary: Well-Ordering Principle This process can not continue indefinitely because by the Well-Ordering Principle, the set of positive integers This simple principle of positive integers has many consequences. This factorization is also unique (up to the order of the factors), see the Fundamental Theorem of Arithmetic. primes.utm.edu /glossary/page.php?sort=WellOrdering   (247 words)

Modern Algebra I Lecture Notes, 08/31/98 Well Ordering Principle: Every nonempty set of positive integers has a least element. The Second Principle of Mathematical Induction: Let S be a set of integers containing a. The First Principle of Mathematical Induction: Let S be a set of integers containing a. www.assumption.edu /Alfano/MAT351-FA98/Notes/083198.html   (211 words)

Axiom Of Choice There are also a remarkable number of statements that are equivalent to the axiom of choice, most important among them Zorn's lemma and the well-ordering principle: every set can be well-ordered, and in fact, Zermelo initially introduced the axiom of choice in order to formalize his proof of the well-ordering principle. The principle seems obvious: if you have several boxes lying around with at least one item in each box, the axiom simply states that you can choose one item out of each of them. Several central theorems in various branches of mathematics require the axiom of choice (or one of its weaker versions, such as the ultrafilter lemma, the axiom of countable choice, or the axiom of dependent choice). www.wikiverse.org /axiom-of-choice   (1257 words)

2.3. The Principle of Induction The set of natural numbers, with the usual ordering, is well-ordered, and in addition every element except of 1 has an immediate predecessor. Based on the Induction principle is the principle of Recursive Definition that is used frequently in computer science. A set S is called ordered if it is partially ordered and every pair of elements x and y from the set S can be compared with each other via the partial ordering relation. www.shu.edu /projects/reals/infinity/induct.html   (1088 words)

Clearing up the market cycle... best Principle of Strong Induc... The well-ordering principle is a concept which is equivalent to mathematical induc- tion. In your textbook, there is a proof for how the well-ordering principle implies. The principle of induction says: Suppose P(1), and P(n)... ascot.pl /th/Fourier5/Principle-of-Strong-Induc....htm   (292 words)

A Nerd's Love Page Then by the well-ordering principle, there exists a least uncute girl. O well, I hope love is not a game or probability. It might also be the probability I find a girl friend this year. www.students.bucknell.edu /tphan/mathromance.htm   (175 words)

ASHA, the Ordering Principle of Creation, at zoroastrianism.cc/asha.html Without Asha, scientifically the use of verifiable repeatable experiments to establish scientific theory, laws and facts would not work, for Asha is the Ordering Principle of the Universe. If he does not, he is lying and if Asha is Truth then he is violating Asha." Well, since Asha promotes that which is conducive to refreshing (improving) creation and achieving fulfillment, wholeness and or completeness, the Mazdayasni's duty is clear. The relationship of Ahura Mazda and His Essences, makes it necessary for us to realize that in order to truly understand any given passage in the Gathas which refers to any one of the Essences, one must look very closely, to determine which is most fitting in the relevant context. www.zoroastrianism.cc /asha.html   (966 words)

MACM-101-D3 notes and readings (2004-3: Manuel Zahariev) readings: Section 4.1 (The Well-Ordering Principle: Mathematical Induction), 4.2 (Recursive Definitions) and 4.3 (The Division Algorithm: Prime Numbers) from the textbook. readings: Section 4.1 (The Well-Ordering Principle: Mathematical Induction) from the textbook. - the axiom (principle) of mathematical induction, from Mathworld. www.cs.sfu.ca /CC/101.MACM/manuelz/notes.html   (369 words)

section probs This is a contradiction, which means that T did, in fact, have a smallest member, which establishes the principle of well-ordering. Given a non-empty set of positive integers T without a smallest member, we will prove that this is a contradiction if the first principle of induction holds. Thus, by the first principle of mathematical induction, we have demonstrated that P(n) is true for all www.nku.edu /~longa/classes/2002fall/mat385/days/day14/highlights2.2/probs/probs.html   (395 words)

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