41. Winfried Just's Professional Interests to a few particularly simple ones that are called the axioms of set theory. These axiom act as a grand unifying principle for all of mathematics A http://www.math.ohiou.edu/~just/resint.html

My professional interests Whatever interests, is interesting. William Hazlitt My original area of expertise was set theory , especially the forcing method and combinatorial principles Currently, I work on applications of mathematics to biology, in particular, on game-theoretic models of animal interactions and on the multiple alignment problem which is of crucial importance in the new science of bioinformatics I love expository writing and teaching To my great surprise, I found out during my sabbatical in 1997/98 that I like computer programming It is often said that set theory is the foundation of modern mathematics. The meaning of this somewhat pompous phrase is that all the structures studied in various branches of mathematics can be interpreted as sets, and hence all mathematical theorems can, at least in principle, be derived from the rules governing the formation of sets. These rules can be reduced to a few particularly simple ones that are called the axioms of set theory. These axiom act as a grand unifying principle for all of mathematics: A mathematical statement is a theorem if and only if it ultimately follows from the axioms of set theory. This is the case regardless of whether the statement is about algebraic structures, differential equations, or probability distributions. The method for establishing that a mathematical statement is indeed a theorem is to give a

42. Foundations Of Mathematics Presumably, this is the only anchor page one needs to navigate all math foundations axioms of logic and arithmetic, discussion of various approaches to http://sakharov.net/foundation_rt.html

Home Alexander Sakharov Irina Tim Projects Resources Sport Photos Median Logic Math Foundations Badminton Clubs Trip Photos ... Downloads Foundations of Mathematics - Textbook / Reference - with contributions by Bhupinder Anand Harvey Friedman Haim Gaifman Vladik Kreinovich ... Stephen Simpson featured in the Computers/Mathematics section of Science Magazine NetWatch This is an online resource center for materials that relate to foundations of mathematics (FOM). It is intended to be a textbook for studying the subject and a comprehensive reference. As a result of this encyclopedic focus, materials devoted to advanced research topics are not included. The author has made his best effort to select quality materials on www. This reference center is organized as a book as opposed to an encyclopedia dictionary directory , or link collection . This page represents book's contents page. One can use this page to study the foundations of mathematics by reading topics following the links in their order or jumping over certain chapters. Where appropriate, topics covered in the referred web resource are listed under the link. In particular, it is done if the resource covers more than the respective section heading and title suggest. Presumably, this is the only anchor page one needs to navigate all math foundations topics. I believe you can even save some $$ because the materials listed here should be sufficient, and you do not have to buy a book or two. The links below are marked in order to indicate the type of material:

43. 8 Modern Instruction - Where It Is Going. Further, in the exposition of mathematics, rules or axioms for the treatment of units in Modern math instruction has split the discipline into two, http://whyslopes.com/etc/MathCurriculumNotes/ch08.html

www.whyslopes.com Volume 1B, Mathematics Curriculum Notes Visit a site area or page Foreword Introducing Site Books Pattern Based Reason - R. and R. from Mathematics Algebra revisited - check and rebuild comprehension Learn to solve equations and master fractions too WebVideos on primes, GCD, LCD and fractions (Euclidean) Geometry - reasoning without coordinates (Analytic) Geometry - reasoning with coordinates A Calculus intro or preview with the essence of later ideas Algebraic eval of limits for derivatives of y = x**2 (with webvideos) More on Limits - epsilon delta alternative. Decimal Proofs of Theorems in Calculus or Real Analysis. order this book Site uploaded with SmartFTP. Favourite Sites BBC News and the Mathematics portion of English National Curriculum Back Volume Entrance Next 8 Modern Instruction Chapters 1 to 12 (except 11) Volume Entrance Inductive Principles 1 Introduction 1 Two Barriers ... 7 Geometry, 2 Ways 8 Modern Instruction 9 The Two Ends 10 The Transition 10 Explaining Logic 10 Explaining Algebra ... PS Math Ed Revisited All of Chapter 11 11 Primary Math 11 Cue Cards 11 Counting 11 Decimals - Addition ... 11 Numbers, +ve or -ve

44. Sci.math FAQ: Relevance Of AC Subject sci.math FAQ Relevance of AC; From alopezo@neumann.uwaterloo.ca (Alex Gordon and Breach, 1969. Maddy, Believing the axioms, I , J. Symb. http://www.uni-giessen.de/faq/archiv/sci-math-faq.ac.relevance/msg00000.html

Index sci.math FAQ: Relevance of AC Subject : sci.math FAQ: Relevance of AC From alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Date : Fri, 17 Nov 1995 17:15:53 GMT Newsgroups sci.math sci.answers news.answers Sender news@undergrad.math.uwaterloo.ca (news spool owner) Summary : Part 26 of many, New version, http://www.jazzie.com/ii/math/index.html http://www.jazzie.com/ii/math/index.html Index: Index sci-math-faq.ac.relevance

45. FOM: Does Mathematics Need New Axioms? John Steel steel at math.berkeley.edu Thu May 18 142226 EDT 2000 Steve seems to be saying that the axioms for all of mathematics do not themselves http://www.cs.nyu.edu/pipermail/fom/2000-May/003974.html

FOM: Does Mathematics Need New Axioms? John Steel steel at math.berkeley.edu Thu May 18 14:22:26 EDT 2000 Previous message: FOM: Does Mathematics Need New Axioms? Next message: FOM: Does Mathematics Need New Axioms? Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] axioms are needed in only a small and obscure corner of mathematics, then that is a very different scenario from what Harvey envisions, where the need for large cardinal axioms would be pervasive throughout core mathematics (number theory, geometry, differential equations, etc). part of mathematics proper. Recall that the recent reorganization of the Mathematics Subject Classification scheme abolished set theory as a separate classification (04XX) and merged it into logic and foundations (03XX). I don't see how the fact that set theory has been put one place or another in the Mathematics (!!) Subject Classification scheme is evidence that it is not part of mathematics. (Or "mathematics proper", whatever that is.) > The goal of this work [set theory] is to answer the age-old foundational question: What are the appropriate axioms for *all* of mathematics?

46. FOM: Does Mathematics Need New Axioms? and If set theory needs new axioms, then mathematics needs new axioms. Steel s argument seems to be that math *obviously* needs new axioms, http://www.cs.nyu.edu/pipermail/fom/2000-May/003967.html

FOM: Does Mathematics Need New Axioms? John Steel steel at math.berkeley.edu Mon May 15 21:44:17 EDT 2000 Previous message: FOM: Does Mathematics Need New Axioms? Next message: FOM: Does Mathematics Need New Axioms? Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] This is a quick reply to John Steel's posting of earlier today. Steel said: and Steel's argument seems to be that math *obviously* needs new axioms, because set theory does, and set theory is (an important?) part of math. I would comment that there is a distinction between set theory qua branch of math versus set theory qua f.o.m. As a branch of mathematics, set theory is relatively small and unimportant compared to other branches such as geometry, number theory, and differential equations. Think of the history of these branches. The main importance of set theory derives from its contemporary foundational role, as a proposed foundation for *all* of mathematics. Yes, set theory has special importance because of its foundational role. ( "Proposed" understates the truth here.) One should expect the need for new axioms, and the new axioms needed, to show up here first. In this > sense, set theory, like the rest of logic and f.o.m., is not really

47. Automated Reasoning The usual axioms for Boolean algebra, in terms of 0, 1, union, intersection Am. math. Soc., vol. 5 (1904), 288390 and (our beginning point) New sets of http://www.math.princeton.edu/~nelson/ar.html

Automated Reasoning This is material related to MAT 504, Topics in Logic, for the Spring term of 1997 (MW 1:30-3:00, Fine 214), on automated reasoning. These are not full lecture notes for the course, and sometimes there is only a list of topics presented at the blackboard or computer terminal. Here is the relevant directory index , and this is my home page Blurb After the media hype about the computer solution of the Robbins problem is discounted, what remains? This question can be answered only by looking into what was done. In my opinion, it is a significant achievement. I am skeptical about the role that automated reasoning can play in mathematics in the next few centuries, but there are interesting and non-trivial results. They lead us into an odd mixture of computer science, logic, universal algebra, pragmatic experimentation, and (so help me) elliptic curves. The goals of this course are to describe the methodology of the field, to have fun playing with the programs Otter, Mace, and EQP that led to the solution of the Robbins problem, to look at some other programs that have been developed, to explore some of the corners of mathematics in which automatic reasoning has proved useful, to help anyone seriously wishing to apply these tools to problems of interest, and to try to estimate the promises and limitations of automatic reasoning. Although I have worked on a proof-checking program, which I hope to have in shape to publish this year, I am not a member of the AR (automated reasoning) community and have no expertise in the field. There may well be some boners here and if so I hope that more knowledgeable people will set me (

48. Poster Project, What Is Scientific Truth Poster In any good piece of math there is always a tension between intuition and One of these axioms is called the Law of Excluded Middle, which states that http://www.math.sunysb.edu/posterproject/www/materials/truth/truth.html

Visualizing Women in Science, Mathematics and Engineering Home Posters Materials for Study Biographies ... The Design Team What Is Scientific Truth? What do you think of the following "theorems"? Theorem One Proof Suppose a b and neither is zero. Then, a b a b a b Also, a b a ab So, a ab a b a b a a b a b a b a a b a a a a as required! Theorem Two Movement is impossible! Proof This is a proof by contradiction. I will assume that movement is possible and derive a contradiciton, thereby proving movement to be impossible. So suppose movement is possible. In this case, the great Greek warrior Achilles can have a foot race with the very slow tortoise. Now, Achilles is the greatest warrior ever (imagine Rambo and the Terminator combined, and that dude wouldn't last ten seconds against Achilles. Achilles could flatten Goldberg with one arm tied behind his back. In fact, Achilles was so awesome that this Greek guy named Homer wrote a cool book about him called the Iliad .) It isn't really a fair race. I mean Achilles could smoke the tortoise who is slow and old and has to carry her house on her back. In the interests of fairness, we had better give the tortoise a little bit of a head start, let's say twenty feet. Now we're ready to race. There's Achilles (the greatest warrior ever) and twenty feet ahead of him is the old tortoise. Bang! The starter's gun goes off and they're racing.

49. KURT GODEL He proved the incompleteness of axioms for arithmetic (his most famous result), There remained the problem of analyzing the axioms of set theory. http://www.usna.edu/Users/math/meh/godel.html

Principia Mathematica , Russell and Whitehead built the foundations of mathematics on a set of axioms for set theory; they needed hundreds of preliminary results before proving that 1 + 1 = 2. Habilitationsschrift (probationary essay), and in 1933 he was confirmed as a Privatdozent : this was not a salaried position, but a certificate that gave him the right to lecture and collect fees from students. He taught his first course in the summer of 1933, and that fall he began a year-long appointment at the newly formed Institute for Advanced Study (IAS) in Princeton, New Jersey. if the axioms other than the axiom of choice are consistent, then home math bios document.write(" Last modified:"+document.lastModified+"");

50. Jef Raskin - Effectiveness Of Mathematics And no doubt that is part of the reason that math and nature correspond; It may well be that the axioms for particular mathematical structures (eg http://jef.raskincenter.org/unpublished/effectiveness_mathematics.html

Effectiveness of Mathematics A reply to Eugene Wigner’s paper, "The unreasonable effectiveness of mathematics in the Natural Sciences" and Hamming’s essay "The unreasonable effectiveness of mathematics." Jef Raskin 1998 [edit of 19 Jan 2001] In physics we often describe phenomena in terms of mathematical relationships between quantities that represent observable attributes of the natural world: Double the tension on a spring and the amount it extends doubles; the intensity of light from a point source changes precisely in inverse proportion to the square of the distance from the source. Even quite abstract mathematical constructs which were created without any reference to physics or the physical world turn out later (sometimes much later) to be excellent representations of newly-discovered experimental data. For example, group theory proved to be eminently useful in crystallography and in understanding the organization of elementary particles. Why should one physical parameter be a mathematical function of some others? This is the philosophical problem that Nobel physicist Eugene Wigner called "the unreasonable effectiveness of mathematics in the natural science" in his 1960 paper of that name in Communications of Pure and Applied Mathematics. Albert Einstein put the problem this way in his Sidelights on Relativity, "How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?"

51. Siegfried/In Search Of A Formula math cannot be caged. It has to evolve, adding new principles and axioms if necessary, Dr. Chaitin declares in his book, titled Meta math! http://www.cs.auckland.ac.nz/CDMTCS/chaitin/siegfried2.html

The Dallas Morning News, Monday, April 5, 2004 DISCOVERIES Dashing a dream Mathematicians used to dream that all truth could be deduced from a system of simple propositions, or axioms. But discoveries of the past century have shown that truth is more elusive, and math contains randomness that no system of axioms can explain. Gottfried Wilhelm von Leibniz, inventor of the binary (0 and 1) system of numbers, observes that simple laws can describe regularities underlying the complexity of the world, but random information requires a lengthier description. David Hilbert proposes the idea of a system of axioms that could generate all the theorems of mathematics and check to see that they were correct. shows that any system of mathematical axioms will be unable to prove some true statements about the math, and will therefore be incomplete. Hilbert's dream was therefore shown to be impossible. Alan Turing proves that there is no way to compute ahead of time whether a computer program given no commands from the outside will ever stop running. Thus answers to some questions are uncomputable, a further demonstration of the hopelessness of Hilbert's dream. Gregory Chaitin proposes length of a computer program as a measure of randomness (or measure of intrinsic complexity of the program), rediscovering the insight of Leibniz.

52. In A Few Words... Ja far Mohammed ibn al Khowarizmi (circa 825) who wrote a math textbook. To define it rigorously we may need a set of axioms, like those proposed by http://www.cut-the-knot.org/do_you_know/few_words.shtml

Username: Password: Sites for teachers Sites for parents Awards Interactive Activities ... Sites for parents In a few words... While putting together these pages I sometimes feel a need to refer to a term without straying from the topic at hand. Oftentimes it's possible to locate a resource on Internet with a necessary definition but cumbersome to specify the reference. In short, I decided to maintain a page of very short topical descriptions which, if and when a need or inspiration induce me to, I'd be able to expand upon. Many of these have been mentioned on the Did you know... page where, as a group, they, I hope, provided some entertainment. As I had neither immediate need for nor intention to describe them, some terms from that page have been left dangling without any reference or definition. Hence the current page. Absolute value The absolute value is defined for real Algorithm The word algorithm comes from the name of a Persian author, Abu Ja'far Mohammed ibn al Khowarizmi (circa 825) who wrote a math textbook. The word refers to a precise prescription (given by a step-by-step description) of a solution to a problem. Braids Theory Braids Theory was invented by Emil Artin and is a part of the Knot Theory.

53. Hilbert S Axioms For Geometry (Archimedes Axiom) If AB and CD are any segments, then there exists a number n such that n copies of CD RI Campbell, campbell@math.umbc.edu. 10 Feb, 2002. http://www.math.umbc.edu/~campbell/Math306Spr02/Axioms/Hilbert.html

Hilbert's Axioms Undefined Terms Points Lines Planes Lie on, contains Between Congruent Axioms Axioms of Incidence Postulate I.1. For every two points A B there exists a line a that contains each of the points A B Postulate I.2. For every two points A B there exists no more than one line that contains each of the points A B Postulate I.3. There exists at least two points on a line. There exist at least three points that do not lie on a line. Postulate I.4. For any three points A B C A B C . For every plane there exists a point which it contains. Postulate I.5. For any three points A B C that do not lie on one and the same line there exists no more than one plane that contains each of the three points A B C Postulate I.6. If two points A B of a line a a Postulate I.7. A in common then they have at least one more point B in common. Postulate I.8. There exist at least four points which do not lie in a plane. Axioms of Order Postulate II.1. If a point B lies between a point A and a point C then the points A B C are three distinct points of a line, and

54. Structural Axioms (from Algebra, Modern) --  Encyclopædia Britannica A set satisfying only axioms 1–7 is called a ring, and if it also satisfies axiom 9 it math Forum Modern Algebra Directory of links on modern algebra. http://www.britannica.com/eb/article?tocId=231058

55. Hilbert's First And Second Problems And The Foundations Of Mathematics By Peter Wiles has shown it follows from the usual ZFC axioms; but does it already follow from K. Gödel, The Consistency of the Continuum Hypothesis, Ann. math. http://at.yorku.ca/t/a/i/c/52.htm

Topology Atlas Document # taic-52 Topology Atlas Invited Contributions vol. 9, no. 3 (2004) 6 pp. Hilbert's first and second problems and the foundations of mathematics Peter J. Nyikos Department of Mathematics University of South Carolina Columbia, SC 29208 USA http://www.math.sc.edu/~nyikos AtlasImage format 6 pages PostScript file (113 Kb) Adobe PDF file (126 Kb) DVI file (21 Kb) LaTeX file (16 Kb) In 1900, David Hilbert gave a seminal lecture in which he spoke about a list of unsolved problems in mathematics that he deemed to be of outstanding importance. The first of these was Cantor's continuum problem, which has to do with infinite numbers with which Cantor revolutionised set theory. The smallest infinite number, , `aleph-nought,' gives the number of positive whole numbers. A set is of this cardinality if it is possible to list its members in an arrangement such that each one is encountered after a finite number (however large) of steps. Cantor's revolutionary discovery was that the points on a line cannot be so listed, and so the number of points on a line is a strictly higher infinite number ( c , `the cardinality of the continuum') than . Hilbert's First Problem asks whether any infinite subset of the real line is of one of these two cardinalities. The axiom that this is indeed the case is known as the Continuum Hypothesis ( CH This problem had unexpected connections with Hilbert's Second Problem (and even with the Tenth, see the article by M. Davis and the comments on the book edited by F. Browder). The Second Problem asked for a proof of the consistency of the foundations of mathematics. Some of the flavor of the urgency of that problem is provided by the following passage from an article by S.G. Simpson in the same volume of JSL as the article by P. Maddy:

56. Platonism, Intuition And The Nature Of Mathematics. Part 2. By K.Podnieks All other assertions of the theory must be proved using the axioms. (See also Devlin s Angle, June 2001, http//arxiv.org/pdf/math. http://www.ltn.lv/~podnieks/gt1a.html

Back to title page Left Adjust your browser window Right 1. Platonism - the Philosophy of Working Mathematicians 2. Investigation of Stable (Self-contained) Models - the Nature of the Mathematical Method 3. Intuition and Axioms The stable character of mathematical models and theories is not always evident - because of our Platonist habits (we are used to treat mathematical objects as specific "world"). Only few people will dispute the stable character of a fully axiomatic theory. All principles of reasoning, allowed in such theories, are presented in axioms explicitly. Thus the principal basis is fixed, and any changes in it will mean explicit changes in axioms. Could we also fix those theories that are not fully axiomatic yet? How could it be possible? For example, all mathematicians are unanimous about the ways of reasoning that allow us to prove theorems about natural numbers (other ways yield only hypotheses or errors). Still, most mathematicians do not use, and even do not know axioms of arithmetic! And even in the theories that seem to be axiomatic (as, for example, geometry in Euclid's "Elements") we can find aspects of reasoning that are commonly acknowledged as correct, yet are not presented in axioms. For example, properties of the relation "the point A is located on a straight line between the points B and C ", are used by Euclid without any justification. Only in XIX century

57. What Is Mathematics: Gödel's Theorem And Around. Incompleteness. By K.Podnieks Visit FOM Archives Visit History of mathematics at The math Forum@Drexel From Peano axioms to first order axioms 3.2. How to find arithmetic in other http://www.ltn.lv/~podnieks/gt.html

what is mathematics, logic, mathematics, foundations, incompleteness theorem, mathematical, Gödel, Godel, book, Goedel, tutorial, textbook, methodology, philosophy, nature, theory, formal, axiom, theorem, incompleteness, online, web, free, download, teaching, learning, study, student, Podnieks, Karlis My personal page - click here Any comments are welcome - e-mail to Karlis.Podnieks@mii.lu.lv This web-site presents 100% of a hyper-textbook for students. Try joining the Foundations of Mathematics (FOM) e-mail list Visit FOM Archives Visit History of Mathematics at The Math Forum@Drexel Diploma What is Mathematics: Gödel's Theorem and Around Hyper-textbook for students by Karlis Podnieks, Professor University of Latvia Institute of Mathematics and Computer Science An extended translation of the 2nd edition of my book " Around Goedel's theorem " published in 1992 in Russian. Diploma Left Adjust your browser window Right  This work is licensed under a Creative Commons License Russian original.

58. Godel And Godel's Theorem: Math The string 0=0 is a valid TNT theorem (ie can be derived from axioms). By learning more about the math involved, you can work the proof to ever finer http://www.ncsu.edu/felder-public/kenny/papers/godel.html

by Kenny Felder In the nineteenth and early twentieth centuries, one of the big mathematical goals was to reduce all of number theory to a formal axiomatic system. Like Euclid's Geometry, such a system would start off with a few simple axioms that are almost indisputable, and would provide a mechanical way of deriving theorems from those axioms. It was a very lofty goal. The idea was that this system would represent every statement you could possibly make about natural numbers. So if you made the statement "every even number greater than two is the sum of two primes," you would be able to prove strictly and mechanically, from the axioms, that it is either true or false. For real, die-hard mathematicians, the words "true" and "false" would become shorthand for "provable" or "disprovable" within the system. Russell and Whitehead's Principia Mathematica was the most famous attempt to find such a system, and seemed for a while to be the pinnacle of mathematical rigor. there is always a statement about natural numbers which is true, but which cannot be proven in the system. In other words, mathematics will always have a little fuzziness around the edges: it will never be the rigorous unshakable system that mathematicians dreamed of for millennia.

59. VIGRE Activity Description The axioms for mathematics. In logic, mathematics is viewed as proceeding by rigorous deduction starting with certain axioms for mathematics. http://www.math.ohio-state.edu/vigre/display-activity.php?ID=30&AJ=true

60. Culture Of Science Brown Bag Series If the truth of math derives from a set of axioms, not from the world; if the truth it proves is a chosen, arbitrary one, not accountable to the world, http://serendip.brynmawr.edu/local/scisoc/brownbag0203/tapp.html

Center for Science in Society Bryn Mawr College http://serendip.brynmawr.edu/local/scisoc 2002-2003 Weekly Brown Bag Lunch Discussion "The Culture of Science November 20 Kris Tapp (Keck Fellow in Math) "The Philosophies of Mathematics" Summary Prepared by Anne Dalke Additions, revisions, extensions are encouraged in the Forum Participants As a way of inviting us into a discussion of the culture of mathematics, Kris Tapp assigned "The Ideal Mathematician," from The Mathematical Experience, by Philip Davis and Reuben Hersh (1980). That essay paints a portrait of a narrowly focused super-specialized fellow who is disinterested in the application of his work to any real-life concerns. Although Kris called this a caricature, he also posited that it was an apt description of the ever-increasing specialization in his field. He demonstrated the disheartening effects of such a trajectory with a list of Mathematics Doctoral Recipients awarded in 2000, and a report of the Geometry-Topology Preprints Posted this Weekend. It is difficult, not only across subspecialities, but within them as well, for mathematicians to understand one another's work. The disjunction between common knowledge and specialization becomes apparent when one does not know the history of the literature that lies behind the claims being made. The phenomenon Kris describes is not particular to math; most professional literatures "spiral inward," leading not to general answers but rather to more and more specialized ones. With the revolution in molecular biology, however, shared tools and methodologies allow for common conversations, even when the systems being treated are distant. Can the work of high level theoretical physicists not be evaluated because it has no empirical grounding? Are there fewer questions in nature? Are scientists drawing their questions from a smaller set than mathematicians do? Do mathematicians think that there are physical grounds for their theorems? In what sense do mathematical objects really exist?

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