41. LMS JCM (6) 198-248 The relative consistency of the axiom of choice mechanized using Isabelle/ZF. Lawrence C. Paulson. Abstract The proof of the relative consistency of the http://www.lms.ac.uk/jcm/6/lms2003-001/

The LMS JCM Published 13 Oct 2003. First received 07 Jan 2003. The relative consistency of the axiom of choice mechanized using Isabelle/ZF Lawrence C. Paulson Abstract: The proof of the relative consistency of the axiom of choice has been mechanized using Isabelle/ZF, building on a previous mechanization of the reflection theorem. The heavy reliance on metatheory in the original proof makes the formalization unusually long, and not entirely satisfactory: two parts of the proof do not fit together. It seems impossible to solve these problems without formalizing the metatheory. However, the present development follows a standard textbook, Kenneth Kunen's Set theory: an introduction to independence proofs , and could support the formalization of further material from that book. It also serves as an example of what to expect when deep mathematics is formalized. This paper is available as (320 KB) here for details. In addition to the paper, the following electronic appendices are available to subscribers : Appendix A This appendix was automatically generated by Isabelle/ZF, and presents the full mechanical development.

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43. Axiom Of Choice | Musings That proof also uses the axiom of choice, but doesn’t mention the We accept the axiom of choice, not because it is true, but because it is useful. http://golem.ph.utexas.edu/~distler/blog/archives/000283.html

@import url("/~distler/blog/styles-site.css"); Musings Thoughts on Science, Computing, and Life on Earth. Enough, already! Skip to the content. Note: These pages make extensive use of the latest XHTML and CSS Standards only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser. :hover Craft Main January 08, 2004 Axiom of Choice To choose one sock from each of infinitely many pairs of socks requires the Axiom of Choice, but for shoes the Axiom is not needed. Bertrand Russell latest post Axiom of Choice Homepage . The latter discusses the AC , and a whole range of related topics. Here, for instance, is his discussion of the Banach-Tarski Paradox: Banach and Tarski used the Axiom of Choice to prove that it is possible to take the 3-dimensional closed unit ball, B x y z x y z and partition it into finitely many pieces, and move those pieces in rigid motions (i.e., rotations and translations, with pieces permitted to move through one another) and reassemble them to form two copies of

44. Topological Curiosities Both theorems are proven with what is known as axiom of choice whose usage (a clear intuitive appeal notwithstanding) was questioned by the stream of http://www.cut-the-knot.org/do_you_know/banach.shtml

Username: Password: Sites for teachers Sites for parents Awards Interactive Activities ... Sites for parents Tarski-Banach Decompositions Two theorems I am going to state are mind boggling results associated with the names of F. Hausdorff, A. Tarski, S. Banach, J. von Neumann and R. M. Robinson. References are given in both books by Gelbaum and Olmsted. (The second of which actually proves the first theorem below.) Both theorems use the notion of a rigid motion . A rigid motion of a space is a transformation that does not change the ( euclidean ) distance between two points. In the theorems below, B r R dist (x,0) Tarski-Banach Theorem 1 There exists a decomposition of B into 5 pairwise disjoint sets A ,...,A of which the last is a single point such that there exist rigid motions R ,...,R with B = R (A R (A ) and B = R (A R (A R (A where all unions are disjoint. This means breaking a ball into five pieces such that it's possible to combine these pieces into two balls equal in size to the original one. No seams are visible after the operation. No cavities are created under the surface. I just wonder what prevents me from taking up this occupation professionally. Could have taken a few bowling balls' manufacturers out of business. The problem is one can't be sure that dropping a ball will break it into five pieces, let alone into the right pieces of which one is a point.

45. Axiom Of Choice Definition Of Axiom Of Choice In Computing Dictionary - By The F Computer term of axiom of choice in the Computing Dictionary and Thesaurus. Meaning of axiom of choice computer term. What does axiom of choice mean? http://computing-dictionary.thefreedictionary.com/Axiom of Choice

Domain='thefreedictionary.com' word='Axiom of Choice' Your help is needed: American Red Cross The Salvation Army join mailing list webmaster tools Word (phrase): Word Starts with Ends with Definition subscription: Dictionary/ thesaurus Computing dictionary Medical dictionary Legal dictionary Financial dictionary Acronyms Columbia encyclopedia Wikipedia encyclopedia Hutchinson encyclopedia Axiom of Choice Also found in: Wikipedia 0.02 sec. Page tools Printer friendly Cite / link Email Feedback (mathematics) Axiom of Choice - (AC, or "Choice") An axiom of set theory If X is a set of sets, and S is the union of all the elements of X, then there exists a function f:X -> S such that for all non-empty x in X, f(x) is an element of x. In other words, we can always choose an element from each set in a set of sets, simultaneously. Function f is a "choice function" for X - for each x in X, it chooses an element of x. Most people's reaction to AC is: "But of course that's true! From each set, just take the element that's biggest, stupidest, closest to the North Pole, or whatever". Indeed, for any finite set of sets, we can simply consider each set in turn and pick an arbitrary element in some such way. We can also construct a choice function for most simple infinite sets of sets if they are generated in some regular way. However, there are some infinite sets for which the construction or specification of such a choice function would never end because we would have to consider an infinite number of separate cases.

46. From The Axiom Of Choice To Choice Sequences Zermelo gave the standard argument that the axiom of choice implies the Here we have a link between the axiom of choice and the theory of choice http://www.hf.uio.no/filosofi/njpl/vol1no1/choice/choice.html

Next: References From the Axiom of Choice to Choice Sequences Herman R. Jervell Department of Linguistics University of Oslo, Norway herman.jervell@ilf.uio.no To make your own printed copy of this article, download one of the following files: Postscript: choice.ps (209005 bytes) Postscript, compressed: choice-ps.zip (46754 bytes) Adobe Acrobat: choice.pdf (232386 bytes) TeX DVI: choice.dvi (14692 bytes) TeX DVI, compressed: choice-dvi.zip (7578 bytes) The theory of choice sequences is usually considered to be far from the mainstream of mathematics. In this note we show that it did not start that way. There is a continuous development from discussions around the use of axiom of choice to Brouwer's introduction of choice sequences. We have tried to trace this development starting in 1904 and ending in 1914. In his book on choice sequences, Troelstra (1977) gives the development after 1914, but does not indicate where Brouwer got his concept. This note is a first attempt at an answer. Our story starts in August 1904, with Zermelo writing a long letter to Hilbert, who thinks part of the letter deserves a wider audience. So he publishes it directly in Mathematische Annalen Zermelo 1904 The leisurely style is clear from the title, ``Proof that every set can be well-ordered, (from a letter sent to Mr. Hilbert)'', and the first sentence:

47. Set TheoryZorn S Lemma And The Axiom Of Choice - Wikibooks That is, given Zorn s Lemma, one can derive the axiom of choice and vice versa. The axiom of choice is named as such because it is independent from http://en.wikibooks.org/wiki/Set_Theory:Zorn's_Lemma_and_the_Axiom_of_Choice

48. Heine Continuity Implies Cauchy Continuity Without The Axiom Of Choice - Apronus On this page we state and prove that every Heine continuous real function is also Cauchy continuous. In our proof we do not use the axiom of choice. http://www.apronus.com/math/cauchyheine.htm

Apronus Home Mathematics Play Piano Online Heine continuity implies Cauchy continuity without the Axiom of Choice On this page we state and prove that every Heine continuous real function is also Cauchy continuous. In our proof we do not use the Axiom of Choice. Two Definitions of Continuity f is Cauchy continuous at x if and only if f is Heine continuous at x if and only if Notation Help Introduction Now, in order to prove the converse (if f is Heine continuous at x then f is Cauchy continuous at x) one needs to employ the Axiom of Choice. Again the proof is very easy: suppose that f is not Cauchy continuous and by the Axiom of Choice obtain a sequence whose existence contradicts Heine continuity. However, if we suppose that f is Heine continuous on some neighborhood of x (= Heine continuous at each point of this neighborhood) then we can derive Cauchy continuity at x without employing the Axiom of Choice. Statement of the Theorem and Proof Fact 1. There exists (by construction - without the Axiom of Choice) a choice function for the set of all rational numbers (denoted Q). Formally: Proof.

49. Axiom Of Choice And Its Equivalents - Apronus.com axiom of choice (AC) If I,Y are sets, AI Y and /\(x-I) A(x) ! axiom of choice (AC ) If X is a set, I = P(X)\{O} then there exists a function fI- X http://www.apronus.com/provenmath/choice.htm

Apronus Home ProvenMath Set Theory Play Piano Online AXIOM OF CHOICE AND ITS EQUIVALENTS Axiom of choice (AC) u(Y) Axiom of choice (AC') Well-ordering principle (WO) If X is a set then there exists E c XxX such that (X,E) is a well ordered set. Definition S.AC.1 - Chain. Let (X,E) be a partially ordered set Definition S.AC.2 - Maximal element. Let (X,E) be a partially ordered set Definition S.AC.3 Let (X,E) be a partially ordered set. L is a maximal chain in (X,E) if and only if for every chain K c X if L c K then K = L. Hausdorff's maximal principle (HMP) Zorn's Lemma (ZL) partially ordered Theorem S.AC.4 If we assume Axioms ZF1, ZF2, ZF3, ZF4, ZF5, ZF6 then AC, AC', WO, HMP and ZL are equivalent. Proof Assume (AC). u(I) We have shown (AC') Assume(AC'). well oredered initial segment Theorem S.O.8 Theorem S.O.7 Thus (WO) is shown. Assume (WO) Recursion Principle Thus (HMP) is shown. Assume (HPM) Thus (LZ) is shown. Assume (LZ) Theorem S.C.14 (AC) is shown. Apronus Home Contact Page ProvenMath Notation document.write("");

50. Sci.math FAQ: The Axiom Of Choice Subject sci.math FAQ The axiom of choice; From alopezo@neumann.uwaterloo.ca (Alex Lopez-Ortiz); Date 17 Feb 2000 225552 GMT; Newsgroups sci.math, http://www.uni-giessen.de/faq/archiv/sci-math-faq.axiomchoice/msg00000.html

Index sci.math FAQ: The Axiom of Choice Subject : sci.math FAQ: The Axiom of Choice From alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Date : 17 Feb 2000 22:55:52 GMT Newsgroups sci.math news.answers sci.answers Summary : Part 23 of 31, New version http://www.jazzie.com/ii/math/index.html http://www.jazzie.com/ii/math/index.html Alex Lopez-Ortiz alopez-o@unb.ca http://www.cs.unb.ca/~alopez-o Assistant Professor Faculty of Computer Science University of New Brunswick Index: Index sci-math-faq.axiomchoice

51. Tea Leaves: Axiom Of Choice axiom of choice. by psu. I have been fortunate enough to get my hands on an iPod Shuffle. I was mostly seduced by the look of the item, but wasn t sure how http://www.tgr.com/weblog/archives/000299.html

Tea Leaves Main February 17, 2005 Axiom of Choice by psu I have been fortunate enough to get my hands on an iPod Shuffle. I was mostly seduced by the look of the item, but wasn't sure how the screenless shuffle-only interface would really work out in practice. Surprisingly, the Shuffle is by far my favorite iPod device for day to day use. In particular, its shuffle play is much more enjoyable in the car than shuffling with the normal iPod. This seems odd, since on the face of it there should be no difference between playing songs at random on a Shuffle and doing the same a normal iPod. However, it turns out that they are different in subtle and important ways, and therein lies the reason I find the Shuffle to be more enjoyable. First, let's review the major differences between the two players. My 40GB iPod basically can hold everything that I currently have ripped on my current iMac. This is around 13GB of music right now, which works out to about 2500 songs or 8 days worth of music. As I add new material into the iMac, I just transfer all of it to the iPod. So, when I want to be able to pick and choose from everything that I own, the 40GB iPod is the thing to take. I love it on planes and other long trips where I have the time to scroll around in the interface and pick what I want to play. It's less than ideal in the car however, where interacting with the iPod is annoying and dangerous. The iPod Shuffle is exactly the opposite. It is not large enough to hold everything I own, and the player has almost no user interface at all. However, iTunes has a system called "Autofill" for randomly downloading songs from a smart playlist into its 1GB of flash memory. The fact that it does this from an iTunes smart playlist is critical because it allows for a valuable level of selective filtering. You can filter the songs by genre, ratings, length and so on, so you avoid the problem of shuffle playing a whole audio book, or movements from a symphony. The autofill mechanism will also weight its choices by your own ratings, so you can make it pick from just your favorites. Once the player is loaded, the interface is simplicity itself. Hit play, hit pause, hit next, adust the volume. That's it.

52. Misc: Axiom Of Choice The axiom of choice is a frequent topic at the weekly meetings. A web search of axiom of choice should probably give you all you really wanted to know http://www.utah.edu/utahlogic/misc/choice.html

Misc: Axiom of Choice The axiom of choice is a frequent topic at the weekly meetings. Some folk think that it is obviously true, some think it is obvously false, and some have other views ("I think it is true, but I don't see how it could be.") There are a number of different equivalent statements of the axiom, but a rough one is: "Given an infinite number of sets with an infinite number of elements in them, can you form a set that has one representative from each set?" (Note that a set may have *uncountably* infinitely many members.) While this is a vaguely religious topic, everyone generally accepts that this axiom is formally independent of the other axioms of set theory, and that its truth has to be decided on other grounds. However there is the view of some that it *IS* an axiom of any *reasonable* set theory. This view is, of course, disputed by those that think the axiom false. A web search of "Axiom of Choice" should probably give you all you really wanted to know about the topic. Go to ...

53. AoPS Math Forum :: View Topic - Axiom Of Choice? Post Posted Wed Jun 01, 2005 310 pm Post subject axiom of choice? I have heard the axiom of choice described as obviously true by someone who http://www.artofproblemsolving.com/Forum/topic-39610.html

WOOT - Olympiad training with many top students worldwide. Sponsored by D. E. Shaw group and Google Click here for more information. Font Size: The time now is Fri Sep 16, 2005 4:11 pm All times are GMT - 7 Hours Axiom of Choice? Mark the topic unread Math Forum College Playground ... Theorems, Formulas and Other Questions Moderators: amfulger fedja Kent Merryfield Moubinool ... Myth Page of [2 Posts] Printer View Tell a Friend View previous topic View next topic Author Message Press alt f4 Poincare Conjecture Joined: 01 Feb 2005 Posts: 112 Posted: Wed Jun 01, 2005 3:10 pm Post subject: Axiom of Choice? Back to top Kent Merryfield Navier-Stokes Equations Joined: 11 Jun 2004 Posts: 1682 Posted: Wed Jun 01, 2005 3:31 pm Post subject: This does not belong here in "Intermediate Topics". Here's what the Axiom of Choice actually says: Given a collection of nonempty sets where belongs to some set of indices is it possible to find a set which selects one element from each Another way to state this is that it is possible to find a function such that for each Yet another paraphrase is that it is possible to find an element of the cartesian product of all of the nonempty sets You must understand that no restriction is being placed on the index set - it can be infinite, even (especially) uncountable.

54. About "The Axiom Of Choice (AC)" The axiom of choice (AC). _ implications of perhaps the last great controversy of mathematics, the axiom of choice, http://mathforum.org/library/view/5189.html

The Axiom of Choice (AC) Library Home Full Table of Contents Suggest a Link Library Help Visit this site: http://math.vanderbilt.edu/~schectex/ccc/choice.html Author: Eric Schechter; Vanderbilt University Description: An introduction to and some implications of perhaps the last great controversy of mathematics, the Axiom of Choice, now a basic assumption used in many parts of mathematics. With a collection of annotated links to relevant sites: Introductory/elementary; Especially noteworthy books and/or researchers; Slightly more advanced and specialized topics; Formal logic and/or automatic theorem-proving; and Miscellaneous. Includes The Banach-Tarski Decomposition, the beginnings of set theory, Godel, Sierpinski, Zermelo, and Zorn. Levels: College Research Languages: English Resource Types: Link Listings Books Bibliographies Math Topics: Logic/Foundations Home The Math Library Quick Reference ... Help http://mathforum.org/

55. Axiom Of Choice Hilbert calls the axiom of choice the most attacked up to the present in using the axiom of choice. As time permits, I ll talk about important http://facweb.cs.depaul.edu/research/TheorySeminar/abstract100804.htm

Axiom of Choice Hilbert calls the Axiom of Choice the "most attacked up to the present in mathematical literature," Fraenkel characterizes it as "the most discussed axiom of mathematics second only to Euclid 's axiom of parallels," and Kanamori sees it as the beginning of "abstract set theory." I'll start will a brief historical overview and then trace Zermelo's 1904 proof of the Well-Ordering Theorem If PowerSet(X) has a choice function, then X can be well ordered. using the Axiom of Choice. As time permits, I'll talk about important consistency results and offer examples of implicit use of AC before Zermelo's 1904 formulation. Also, I have proof sketches of various equivalences (e.g., the well-ordering theorem and the multiplicative axiom, AC and the theorem that any set has a denumberable subset). Other topics include set-theoretic implications of AC and the link between AC and von Neumann's theory of ordinals. I'll conclude with some points on the history and controversy surrounding AC.

56. Axiom Of Choice axiom of choice from MathWorld Report this link as dead Report this link as dead pop-up. track image. –+ Bulgaria National anthems –+ Numbers http://www.uni-bonn.de/~manfear/mathdict-entry.php?term=axiom of choice&lang=en&

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59. Zermelo Theorem And Axiom Of Choice which well orders it proposition (26)) and axiom of choice (for every It is result of the Tarski s axiom A introduced in 5 and repeated in 6. http://mizar.uwb.edu.pl/JFM/Vol1/wellord2.html

Journal of Formalized Mathematics Volume 1, 1989 University of Bialystok Association of Mizar Users Zermelo Theorem and Axiom of Choice Grzegorz Bancerek Warsaw University, Bialystok Summary. The article is continuation of [ ] and [ ], and the goal of it is show that Zermelo theorem (every set has a relation which well orders it - proposition (26)) and axiom of choice (for every non-empty family of non-empty and separate sets there is set which has exactly one common element with arbitrary family member - proposition (27)) are true. It is result of the Tarski's axiom A introduced in [ ] and repeated in [ ]. Inclusion as a settheoretical binary relation is introduced, the correspondence of well ordering relations to ordinal numbers is shown, and basic properties of equinumerosity are presented. Some facts are based on [ MML Identifier: The terminology and notation used in this paper have been introduced in the following articles [ Contents (PDF format) Bibliography 1] Grzegorz Bancerek. The ordinal numbers Journal of Formalized Mathematics 2] Grzegorz Bancerek. The well ordering relations Journal of Formalized Mathematics 3] Czeslaw Bylinski.

60. Read About Axiom Of Choice At WorldVillage Encyclopedia. Research Axiom Of Choic axiom of choice. Everything you wanted to know about axiom of choice but had no clue how to find it.. Learn about axiom of choice here! http://encyclopedia.worldvillage.com/s/b/Axiom_of_choice

Culture Geography History Life ... WorldVillage Axiom of choice From Wikipedia, the free encyclopedia. In mathematics , the axiom of choice is an axiom of set theory . It was formulated in by Ernst Zermelo . While it was originally controversial, it is now accepted and used casually by most mathematicians. However, there are still schools of mathematical thought, primarily within set theory, which either reject the axiom of choice, or at the least investigate consequences of its negation. The axiom of choice is typically abbreviated AC, or C as a suffix. Contents 1 Statement 2 Usage 3 Independence of AC 4 Weaker versions of choice ... edit Statement The axiom of choice states: Let X be a set of non-empty sets. Then we can choose a member from each set in X Stated more formally: Let X be a set of non-empty sets. Then there exists a choice function f defined on X . In other words, there exists a function f defined on X , such that for each set S in X f S ) is an element of S Another formulation of the axiom of choice states: Given any set of mutually disjoint non-empty sets, there exists at least one set that contains exactly one element in common with each of the non-empty sets. edit Usage Until the late 19th century, the axiom of choice was often used implicitly. For example, after having established that the set

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