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1. Set Theory - Wikipedia, The Free Encyclopedia Set Theory is the mathematical Theory of Sets, which represent collections of abstract objects. It encompasses the everyday notions, introduced in primary http://en.wikipedia.org/wiki/Set_theory

var wgNotice = ""; var wgNoticeLocal = ""; var wgNoticeLang = "en"; var wgNoticeProject = "wikipedia"; Set theory From Wikipedia, the free encyclopedia Jump to: navigation search Set theory is the mathematical theory of sets , which represent collections of abstract objects . It encompasses the everyday notions, introduced in primary school , often as Venn diagrams , of collections of objects, and the elements of, and membership in, such collections. In most modern mathematical formalisms, set theory provides the language in which mathematical objects are described. Along with logic and the predicate calculus , it is one of the axiomatic foundations for mathematics , allowing mathematical objects to be constructed formally from the undefined terms of "set" and "set membership". It is in its own right a branch of mathematics and an active field of mathematical research. In naive set theory , sets are introduced and understood using what is taken to be the self-evident concept of sets as collections of objects considered as a whole. In axiomatic set theory , the concepts of sets and set membership are defined indirectly by first postulating certain axioms which specify their properties. In this conception, sets and set membership are fundamental concepts like

2. Set Theory (Stanford Encyclopedia Of Philosophy) Set Theory is the mathematical science of the infinite. It studies properties of Sets, abstract objects that pervade the whole of modern mathematics. http://plato.stanford.edu/entries/set-theory/

Cite this entry Search the SEP Advanced Search Tools ... Please Read How You Can Help Keep the Encyclopedia Free Set Theory First published Thu 11 Jul, 2002 1. The Essence of Set Theory 2. Origins of Set Theory 3. The Continuum Hypothesis 4. Axiomatic Set Theory ... Related Entries 1. The Essence of Set Theory The objects of study of Set Theory are sets . As sets are fundamental objects that can be used to define all other concepts in mathematics, they are not defined in terms of more fundamental concepts. Rather, sets are introduced either informally, and are understood as something self-evident, or, as is now standard in modern mathematics, axiomatically, and their properties are postulated by the appropriate formal axioms. The language of set theory is based on a single fundamental relation, called membership . We say that A is a member of B (in symbols A B ), or that the set B contains A as its element. The understanding is that a set is determined by its elements; in other words, two sets are deemed equal if they have exactly the same elements. In practice, one considers sets of numbers, sets of points, sets of functions, sets of some other sets and so on. In theory, it is not necessary to distinguish between objects that are members and objects that contain members the only objects one needs for the theory are sets. See the supplement Basic Set Theory for further discussion.

3. Set Theory The history of Set Theory is rather different from the history of most other areas of mathematics. For most areas a long process can usually be traced in http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Beginnings_of_set_theory.

A history of set theory Algebra index History Topics Index Version for printing The history of set theory is rather different from the history of most other areas of mathematics. For most areas a long process can usually be traced in which ideas evolve until an ultimate flash of inspiration, often by a number of mathematicians almost simultaneously, produces a discovery of major importance. Set theory however is rather different. It is the creation of one person, Georg Cantor . Before we take up the main story of Cantor 's development of the theory, we first examine some early contributions. The idea of infinity had been the subject of deep thought from the time of the Greeks. Zeno of Elea , in around 450 BC, with his problems on the infinite, made an early major contribution. By the Middle Ages discussion of the infinite had led to comparison of infinite sets. For example Albert of Saxony , in Questiones subtilissime in libros de celo et mundi, proves that a beam of infinite length has the same volume as 3-space. He proves this by sawing the beam into imaginary pieces which he then assembles into successive concentric shells which fill space. Bolzano was a philosopher and mathematician of great depth of thought. In 1847 he considered sets with the following definition

4. Set Theory Page Collection of links related to Set Theory. http://www.cis.syr.edu/~sanchis/setory.html

SET THEORY PAGE There are many sites on the Web that contain material explicitly or implicitly related to set theory. This page is intended to collect as many of these sites as possible, and provide links to them. Please submit your URL if you think your page is consistent with this program: email to sanchis@top.cis.syr.edu Akman:Barwise Situated Set Theory Sanchis:Operational Set Theory Bounded Set Theory Holmes:New Foundations Home Page ... Logic Links

5. Set Theory Homepages Directory of Set theorists, maintained by Jean A. Larson. http://www.math.ufl.edu/~jal/set_theory.html

Set Theory This list of homepages of set theorists was inspired by Computability Theory , maintained by Peter Cholak with encouragement from Ted Slaman. It is complemented by Andrzej Roslanowski's sleek list , and by Herb Enderton's ASL-web , links to ASL members. See also some special topics and a few links to history of set theory . Also consider Bonn's Logic around the world Note: Institutional links point to departments and/or institutes. Send comments, corrections, additions to jal@math.ufl.edu. Select the first letter of the last name: A B C D ... XYZ A Abraham, Uri Ben Gurion University Set theory, model theory, concurrency Hungarian Mathematics Institute Argyros, Spiros University of Athens Banach spaces, set theory Andretta, Alessandro University of Torino Apter, Arthur CUNY Baruch Mathematical Logic, specifically Set Theory: Large Cardinals and Forcing Aharoni, Ron Technion Combinatorics, both finite and infinite, topological methods in matching theory Set theory, logic, computability, especially logic programming, automatic theorem proving, artificial intelligence Arhangelski, Alexander

6. PlanetMath: Set Theory Set Theory is special among mathematical theories, in two ways It plays a central role in putting mathematics on a reliable axiomatic foundation, http://planetmath.org/encyclopedia/SetTheory.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... RSS Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About set theory (Topic) Set theory is special among mathematical theories , in two ways: It plays a central role in putting mathematics on a reliable axiomatic foundation , and it provides the basic language and apparatus in which most of mathematics is expressed. Axiomatic set theory I will informally list the undefined notions, the axioms lines of Bourbaki's equivalent ZFC model. (But some of the axioms are identical to some in ZFC; see the entry ZermeloFraenkelAxioms.) The intention here is just to give an idea of the level and scope of these fundamental things. There are three undefined notions: 1. the relation of equality of two sets 2. the relation of membership of one set in another ( 3. the notion of an

7. 03E: Set Theory Naive Set Theory considers elementary properties of the union and intersection operators Venn diagrams, the DeMorgan laws, elementary counting techniques http://www.math.niu.edu/~rusin/known-math/index/03EXX.html

Search Subject Index MathMap Tour ... Help! ABOUT: Introduction History Related areas Subfields POINTERS: Texts Software Web links Selected topics here 03E: Set theory Introduction Naive set theory considers elementary properties of the union and intersection operators Venn diagrams, the DeMorgan laws, elementary counting techniques such as the inclusion-exclusion principle, partially ordered sets, and so on. This is perhaps as much of set theory as the typical mathematician uses. Indeed, one may "construct" the natural numbers, real numbers, and so on in this framework. However, situations such as Russell's paradox show that some care must be taken to define what, precisely, is a set. However, results in mathematical logic imply it is impossible to determine whether or not these axioms are consistent using only proofs expressed in this language. Assuming they are indeed consistent, there are also statements whose truth or falsity cannot be determined from them. These statements (or their negations!) can be taken as axioms for set theory as well. For example, Cohen's technique of forcing showed that the Axiom of Choice is independent of the other axioms of ZF. (That axiom states that for every collection of nonempty sets, there is a set containing one element from each set in the collection.) This axiom is equivalent to a number of other statements (e.g. Zorn's Lemma) whose assumption allows the proof of surprising even paradoxical results such as the Banach-Tarski sphere decomposition. Thus, some authors are careful to distinguish results which depend on this or other non-ZF axioms; most assume it (that is, they work in ZFC Set Theory).

8. Notes On Set Theory Notes on Set Theory. Notation and Terminology More on Set operations Arbitrary unions and intersections Manipulating Unions and Intersections http://www.math.csusb.edu/notes/sets/sets.html

Next: Notation and Terminology Notes on set theory Notation and Terminology More on set operations Arbitrary unions and intersections Manipulating Unions and Intersections ... About this document ... Dan Rinne Mon Jul 8 15:30:05 PDT 1996

9. Set Theory - Wikibooks, Collection Of Open-content Textbooks Set Theory is concerned with the concept of a Set, essentially a collection of objects that we call elements. Because of its generality, Set Theory forms http://en.wikibooks.org/wiki/Set_Theory

var wgNotice = ""; var wgNoticeLocal = ""; var wgNoticeLang = "en"; var wgNoticeProject = "wikibooks"; Set Theory From Wikibooks, the open-content textbooks collection Jump to: navigation search Contents Introduction Before You Begin How to Use This Book Set Theory ... edit Introduction Set theory is concerned with the concept of a set, essentially a collection of objects that we call elements. Because of its generality, set theory forms the foundation of every other part of mathematics. edit Before You Begin In order to make things easier for you as a reader, as well as for the writers, you will be expected to be familiar with a few topics before beginning. (I hope to have some links to other Wikibooks here soon.) Mathematical Logic Mathematics is all about proofs. One of the goals of this book is to improve your skills in doing proofs, but you will not learn any of the basics here. Many constructions in set theory are simply generalizations of constructions in mathematical logic, and therefore logic is a necessity of learning set theory. edit How to Use This Book A Wikibook is very different from a standard textbook, and this is simultaneously a great strength and a great weakness.

10. Set Theory --  Britannica Online Encyclopedia Britannica online encyclopedia article on Set Theory branch of mathematics that deals with the properties of welldefined collections of objects, http://www.britannica.com/eb/article-9109532/set-theory

var britAdCategory = "other"; Already a member? LOGIN Encyclopædia Britannica - the Online Encyclopedia Home Blog Advocacy Board ... Free Trial Britannica Online Content Related to this Topic This Article's Table of Contents Expand all Collapse all Introduction Introduction to naive set theory Fundamental set concepts Operations on sets Relations in set theory Essential features of Cantorian set theory ... Print this Table of Contents Shopping Revised, updated, and still unrivaled. 2008 Britannica Ultimate DVD/CD-ROM The world's premier software reference source. Great Books of the Western World The greatest written works in one magnificent collection. Visit Britannica Store set theory Page 1 of 17 set theory... (75 of 8542 words) To read the full article, activate your FREE Trial Close Enable free complete viewings of Britannica premium articles when linked from your website or blog-post. Now readers of your website, blog-post, or any other web content can enjoy full access to this article on set theory , or any Britannica premium article for free, even those readers without a premium membership. Just copy the HTML code fragment provided below to create the link and then paste it within your web content. For more details about this feature, visit our Webmaster and Blogger Tools page Copy and paste this code into your page var dc_UnitID = 14; var dc_PublisherID = 15588; var dc_AdLinkColor = '009900'; var dc_adprod='ADL'; var dc_open_new_win = 'yes'; var dc_isBoldActive= 'no';

11. Set Theory: Cantor Cantor s last two papers on Set Theory, Contributions to the foundations of infinite Set Theory, 1895/1897, give his most polished study of cardinal and http://www.math.uwaterloo.ca/~snburris/htdocs/scav/cantor/cantor.html

Previous: Dedekind Next: Frege Up: Supplementary Text Topics Set Theory: Cantor typing . Another, more popular solution would be introduced by Zermelo. But first let us say a few words about the achievements of Cantor. We include Cantor in our historical overview, not because of his direct contribution to logic and the formalization of mathematics, but rather because he initiated the study of infinite sets and numbers which have provided such fascinating material, and difficulties, for logicians. After all, a natural foundation for mathematics would need to talk about sets of real numbers, etc., and any reasonably expressive system should be able to cope with one-to-one correspondences and well-orderings. Cantor started his career by working in algebraic and analytic number theory. Indeed his PhD thesis, his Habilitation, and five papers between 1867 and 1880 were devoted to this area. At Halle, where he was employed after finishing his studies, Heine persuaded him to look at the subject of trigonometric series, leading to eight papers in analysis. In two papers 1870/1872 Cantor studied when the sequence converges to 0. Riemann had proved in 1867 that if this happened on an interval and the coefficients were Fourier coefficients then the coefficients converge to as well. Consequently a Fourier series converging on an interval must have coefficients converging to 0. Cantor first was able to drop the condition that the coefficients be Fourier coefficients consequently any trigonometric series convergent on an interval had coefficients converging to 0. Then in 1872 he was able to show the same if the trigonometric series converged on

12. Mathematics Archives - Topics In Mathematics - Logic & Set Theory KEYWORDS Textbook, Platonism, intuition and the nature of mathematics, Axiomatic Set Theory, First order arithmetic, Hilbert s Tenth problem, http://archives.math.utk.edu/topics/logic.html

Topics in Mathematics The Alonzo Church Archive ADD. KEYWORDS: Manuscripts, Photographs AMS's Materials Organized by Mathematical Subject Classification ADD. KEYWORDS: Electronic Journals, Preprints, Web Sites, Databases AMS's Materials Organized by Mathematical Subject Classification ADD. KEYWORDS: Electronic Journals, Preprints, Web Sites, Databases Around Goedel's theorem ADD. KEYWORDS: Textbook, Platonism, intuition and the nature of mathematics, Axiomatic set theory, First order arithmetic, Hilbert's Tenth problem, Incompleteness theorems, Around Goedel's theorem, About model theory Association for Symbolic Logic ADD. KEYWORDS: Society, Meetings Axiome du Choix ADD. KEYWORDS: Quelques variantes de l'axiome du choix. The Beginnings of Set Theory ADD. KEYWORDS: History, Cantor, Bolzano, Dedekind, Russell, Peano, Zermelo, Bibliography The Bulletin of Symbolic Logic ADD. KEYWORDS: Journal CMS Preprints: Set theory The Continuum Hypothesis ADD. KEYWORDS: Draft of an article A Crash Course in the Mathematics Of Infinite Sets ADD. KEYWORDS:

13. Appalachian Set Theory This new series of workshops on Set Theory for the Appalachian region is supported by the National Science Foundation (NSF 0631446). http://www.math.cmu.edu/~eschimme/Appalachian/Index.html

Appalachian set theory General description Instructions on how to apply for funds to attend a workshop Next workshop (click for details) February 9, 2008, at Carnegie Mellon University Ilijas Farah will give a workshop on set theory and operator algebras Past workshops November 17, 2007, at Ohio University in Athens OH The fourth workshop, which was led by Simon Thomas, was on countable Borel equivalence relations. Click for details. List of participants in this workshop June 2, 2007, at the James Madison University in Harrisonburg VA The third workshop, which was led by Matthew Foreman, was on generic embeddings. Click for details. List of participants in this workshop January 27, 2007, at the University of North Carolina at Charlotte The second workshop, which was led by Stevo Todorcevic, was on coherent sequences. Click for details. Lecture notes from this workshop List of participants in this workshop September 9, 2006, at Carnegie Mellon University in Pittsburgh PA The inaugural workshop, which was led by Paul Larson, was an introduction to P max forcing.

14. MathPages: Set Theory And Foundations Set Theory and Foundations Dialogue on the Foundations of String Theory Controversies Over the Equivalence Principle http://www.mathpages.com/home/ifoundat.htm

Set Theory and Foundations Representing Sets of Pure Order Color Space, Physical Space, and Fourier Transforms Global Reversibility of Cellular Automata Guessing Faster Than Light ... Math Pages Main Menu

15. Set Theory Preprints Set Theory Preprint Sites. This list of preprintpages run by Set theorists was derived from Set theorists, maintained by Jean Larson, and by Andrzej http://www.ucl.ac.uk/~ucahcjm/stp.html

Set Theory Preprint Sites This list of preprint-pages run by set theorists was derived from Set theorists , maintained by Jean Larson, and by Andrzej Roslanowski's sleek list , each of which gives a list of home-pages of set theorists. If you are interested in this page you may be interested as well in Boris Piwinger's Abstract Server for Mathematical Logic and in the Mathematical Logic around the world website. A B C D ... Martin Zeman

16. BOUNDED SET THEORY: Home Page A weak version of ordinary Set Theory using bounded quantification. Papers and software. http://www.botik.ru/~logic/bst/bst.html

BOUNDED SET THEORY: Home Page Bounded Set Theory (BST) is a weak version of the ordinary set theory. Its main feature is paying main attention to using bounded quantification (as in the ordinary everyday mathematical practice) and other analogous bounded constructs. The language of this set theory, called DELTA, allows to define both set-theoretic (bounded) formulas and operations. The main universe of sets for BST consists of hereditarily-finite sets and is called HF. However, the universe for Zermelo-Frenkel set theory does as well. Moreover, it is considered also so called antifounded version of HF, called HFA, which satisfies finite version of Aczel's Antifoundation Axiom . Antifounded sets are also called hypersets . This correlates very happily with the concept of HyperText used for creating World-Wide Web (WWW)-pages with hyperlincs between them. Antifounded version of BST is called BSTA. There exists corresponding version of the language DELTA. This language may be naturally interpreted as query language for nested databases with very arbitrary and flexible structure. Such kind of databases may be considered also as

17. Jay's Set Theory Calculator If you don t know what Musical Set Theory is, please read all about it. (Includes help on using the applet.) Turn your speakers up! http://www.jaytomlin.com/music/settheory/

Java Set Theory Machine by Jay Tomlin Sorry, this browser does not support Java. If you don't know what Musical Set Theory is, please read all about it . (Includes help on using the applet.) Turn your speakers up! The applet lets you hear the sets you create. You can hear individual pitches by clicking on the keyboard. Click numbers on the clockface to toggle set members on/off. To define a new set, use the "Define Set..." button or just type a comma-separated list of numbers in the "Pitch Class Set:" field and press Enter (Mac users press Return) Source code: SetTheory.java PitchSet.java IntervalVector.java ForteNumber.java ... ClickableCanvas.java Download for offline use: SetTheoryCalculator.zip (122 KB) Extract contents to a folder and open default.htm.

18. Settheory's Podcast I ve looked around for some new blood to hype after the great reaction to Noel Neiman s appearance on the 3rd Set Theory show. The response has been good http://www.set-theory.net/

Categories general podcasts Links Set Theory Forum ArsTechnica Something Awful DJForums ... Konstrukt Podcast Archives Apr 2006 May 2006 Jun 2006 Jul 2006 ... Dec 2006 December 2007 S M T W T F S Syndication Sat, 16 December 2006 Show Number Four - 2006 Year End Edition Number four brings us a Top 10 for December, as well as another edition of Gear Lust. Playlist- Monosurround - Cocked, Locked, Ready to Rock (Malente Remix) Greg Churchill - Shock Rocket (Original Mix) DJ Delicious - Let It Drop (Henrik B Remix) Robot Needs Oil - Volta (Olivier Giacomotto Remix) Funkagenda and Tikk Takk - Good To Me (Original Mix) **Gear Lust segment** Isabel Guzman - Lovesong (Morgan Page Remix) Wally Montana - 45 Flat Theme (Benson N Hedges Dub) Greg Kobe - Flow (Original Mix) 909 District - Run, Jump And Fly (Original Mix) DJ Fist and Valentti - Stay (Mario Ochoa Remix) Direct download: Category: podcasts posted at: 10:35 PM Comments[3] Sat, 4 November 2006 Set Theory Top 5 New Releases - 11/3/2006 Having a bit of fun with this week's top 5. Hope you like it! BSOD - This Is The Hook (Original Mix) Roman Salzger - Neo Dancer (Original mix) Sidney Samson - Shake And Rock This (Club Mix) Lex - You Came (Dub Mix) Rivaz ft. Ronin - Stop and Play (Original Mix)

19. Earliest Uses Of Symbols Of Set Theory And Logic Explains the early introduction of notations used in logic and Set Theory as we know it today. Includes reference links to key people in this area. http://members.aol.com/jeff570/set.html

Earliest Uses of Symbols of Set Theory and Logic Last updated: Sept. 29, 2007 The study of logic goes back more than two thousand years and in that time many symbols and diagrams have been devised. Around 300 BC Aristotle introduced letters as term-variables, a "new and epoch-making device in logical technique." (W. & M. Kneale The Development of Logic (1962, p. 61). The modern era of mathematical notation in logic began with George Boole (1815-1864), although none of his notation survives. Set theory came into being in the late 19 th and early 20 th centuries, largely a creation of Georg Cantor (1845-1918). See MacTutor's A history of set theory or, for more detail, Set theory from the Stanford Encyclopedia of Philosophy Most of the basic symbols of logic and set theory in use today were introduced between 1880 and 1920. The main contributors were Ernst Schröder Giuseppe Peano Alfred North Whitehead (1861-1947) and Bertrand Russell (1872-1970). Peano had a strong influence on Whitehead and Russell and their joint work, Principia Mathematica (1910-1913), was itself very influential. Today

20. Algebraic Set Theory Algebraic Set Theory uses the methods of category Theory to study elementary Set Theory. The purpose of this website is to link together current research in http://www.phil.cmu.edu/projects/ast/

Algebraic Set Theory Algebraic set theory uses the methods of category theory to study elementary set theory. The purpose of this website is to link together current research in algebraic set theory and make it easily available. It is hoped that this will encourage and facilitate further development of the subject. Why do you call it "algebraic set theory"? Researchers in Algebraic Set Theory Steve Awodey (Carnegie Mellon University, Pittsburgh) Benno van den Berg (Darmstadt University of Technology) Carsten Butz (IT University, Copenhagen) Henrik Forssell (Carnegie Mellon University, Pittsburgh) Nicola Gambino Claire Kouwenhoven-Gentil (Utrecht University) F. William Lawvere (University at Buffalo, SUNY) Ieke Moerdijk (Utrecht University) Erik Palmgren (Uppsala University) Ivar Rummelhoff (University of Oslo) Dana Scott (Carnegie Mellon University, Pittsburgh) Alex Simpson (University of Edinburgh) Thomas Streicher (Darmstadt University of Technology) Michael A. Warren (Carnegie Mellon University, Pittsburgh) Bibliography The following is a brief survey of the current literature on algebraic set theory. The bibliography is ordered chronologically by year of publication and then alphabetically by the author's surname (by the first author's surname in the case of works with multiple authors). Draft and preprint versions of papers are listed as they become available. Upon publication preprints are removed and papers are listed under year of publication. In most cases preprints are still available on individual author homepages or on the arXiv.

21. Set Theory In Set Theory there is one primitive object, the empty Set , and one relationship, Set membership . All of mathematics can be modeled with these primitives. http://www.mtnmath.com/book/node51.html

New version of this book Next: Notation Up: Einstein's Revenge Previous: Who am I Set theory In mathematics a formal system is a set of axiom s and the rules of logic for deriving theorems from those axioms. It can be thought of as a computer program for outputting theorems. We can write a program to output all the theorems for any formal system. The axioms say what primitive objects and relationships exist and how new objects can be constructed. In set theory there is one primitive object, the empty set , and one relationship, set membership . All of mathematics can be modeled with these primitives. For example the integer 1 is defined as the set that contains the empty set. The integer two is the set containing 1 and the empty set. Integer N is the set containing the empty set and all integers less than N The most powerful generally accepted formal system is Zermelo Fraenkel (ZF ) set theory . We will list the axioms of ZF adapted from Cohen(50)[ ]. First we need to explain the notation. Sometimes we refer to arbitrary statements in the language of ZF with upper case letters. A refers to any valid statement in the language of ZF.

22. Special Session On Set Theory (AMS-DMV-OeMG Meeting In Mainz) Special Session on Set Theory organized by Joel D. Hamkins (New York NY), Peter Koepke (Bonn) and Benedikt Löwe (Amsterdam) Speakers http://staff.science.uva.nl/~bloewe/AMSDMV/

Mainz (Germany) June 16 - 19, 2005 Special Session on Set Theory organized by Joel D. Hamkins (New York NY), Peter Koepke (Bonn) and (Amsterdam) Speakers: Matt Foreman , Irvine CA (50 minute general audience talk) Classifying Automorphisms of the Measure Algebra Joan Bagaria , Barcelona: Recent results on the generic absoluteness programme Natasha Dobrinen , Vienna: Stationary subsets of P kappa lambda with respect to the ground model Mirna Dzamonja , Norwich: Constructions related to a problem of Efimov Bernhard Irrgang , Bonn: Morasses and Finite Support Iterations Heike Mildenberger , Vienna: Specializing Aronszajn trees without adding reals and preserving some weak diamonds Ralf Schindler Projective equivalence relations and inner model theory Agatha Walczak-Typke , Leeds: Dedekind-finite Structures Philip Welch , Bristol: Mutual Stationarity Saturday, June 18 Sunday, June 19 Natasha Dobrinen Matt Foreman Heike Mildenberger Agatha Walczak-Typke Philip Welch Bernhard Irrgang Joan Bagaria Mirna Dzamonja Ralf-Dieter Schindler Last changed: April 8th, 2005

23. The Restored Eye Website Devoted to Highfalutin Intellectual issues. Set Theory as a programming language, Logic proof verifiers, Topolgy. Visual Psychophysics. http://www.settheory.com/

The Restored Eye A Website Devoted to High-falutin Intellectual issues, and other Fun Things J E M Mechanics Draft Educational Module Notes on quantum mechanics Bell's inequality and Bell figures Proof Verification Introductory material for A set-theoretically based proof verifier and its application to the basic theorems of analysis Course syllabus for 2002 Computational Logic Course Educational multimedia: preliminary drafts The role of computer-based interactives in education Comments on design of interactive educational materials Middle-school materials Sketch plan for middle-school mathematics interactive Eggs-in-boxes, recursion, and prime numbers Course syllabus for 2004 Bioinformatics Course Sample code items for 2004 Bioinformatics Course A bit of topology An elementary knot invariant related to edge-colorings by elements of groups A strange vanity item. Jack Schwartz' brain, with eyeballs and a bit of nose Illusion-based artwork Programming in SETL. (Draft in Progress) Chapter 2 - Elementary Types Chapter 3 - Operations Chapter 4 - Control structures Chapter 5 - Procedures ... Chapter 12 - Data bases Studies in visual psychophysics Study I - Some observations on the psychophysics of Glass patterns and related visual phenomena Study II - Motion perception in various settings.

24. The Math Forum - Math Library - Set Theory The Math Forum s Internet Math Library is a comprehensive catalog of Web sites and Web pages relating to the study of mathematics. This page contains sites http://mathforum.org/library/topics/set_theory/

Browse and Search the Library Home Math Topics Logic/Foundations : Set Theory Library Home Search Full Table of Contents Suggest a Link ... Library Help Selected Sites (see also All Sites in this category The Beginnings of Set Theory - MacTutor Math History Archives Linked essay describing the rise of set theory from Cantor (with discussion of earlier contributions) through the first half of the 20th century, with another web site and 25 references (books/articles). more>> Interactive Basic Math Sets - Martin Selditch A tutorial on sets, convering the definition of sets and their elements, union, intersection, subsets, and sets of numbers. more>> Set Theory - Dave Rusin; The Mathematical Atlas more>> All Sites - 70 items found, showing 1 to 50 Around the Goedel's Theorem - Karlis Podnieks A draft translation of Podnieks' book, published in 1992 in Russian. Contents include: Platonism, intuition and the nature of mathematics; Axiomatic set theory; First order arithmetic; Hilbert's Tenth problem; Incompleteness theorems; Around the Goedel's ...more>> Bell Package - Jacek Kisynski This package provides functions which are useful while dealing with set partitions. We provide (hopefully) fast methods for sets of size up to 15 and methods with no set size restrictions which use BigInteger objects. The later ones are constrained

25. Sets, Logic And Categories Venn diagrams by Frank Ruskey; Beginnings of Set Theory (on MacTutor History of Mathematics site at St Andrews); British Logic Colloquium (with many further http://www.maths.qmul.ac.uk/~pjc/slc/

Peter J. Cameron Sets, Logic and Categories This book is published by Springer-Verlag , in the Springer Undergraduate Mathematics Series , in February 1999. Another book in the series is Geoff Smith's Introductory Mathematics: Algebra and Analysis A PDF file of the preface and table of contents is available. Solutions to the exercises (PDF files): Chapter 1: Naive set theory Chapter 2: Ordinal numbers Chapter 3: Logic Chapter 4: First-order logic Others to be added! Here is a list of known misprints, together with comments and improvements from various readers. From the review by A. M. Coyne in The text is clearly written. It would make an excellent first course in foundational issues in mathematics at the undergraduate level. Further references Sheaves in Geometry and Logic: A first introduction to topos theory , Springer 1990. (Suggested by Steve Awodey A computer scientist's view: Paul Taylor, Practical Foundations of Mathematics , Cambridge University Press, 1999. A book about how our brains are wired to do mathematics: Brian Butterworth, The Mathematical Brain , Macmillan, London, 1999.

26. Martin Flashman's Logic And Set Theory Web Surfing Page MATHS 315 Mathematical Logic (University of Aukland) Handouts etc. for logic and Set Theory that parallel Hamilton s textbook Logic for Mathematicians. http://www.humboldt.edu/~mef2/logicsites.html

Martin Flashman's Logic and Set Theory Web Surfing Page Curently under construction. 1/12/98 WEB SURFING TOOLS AND SITES-CAVEAT: Use at your own risk :) Latest major changes: Jan. 12, 1998 Logic Set Theory Turing Machines Major References, Searches (Math) ... Return to Math 446 Logic and Set Theory Please send suggestions for improvements and comments to Martin Flashman E-Mail: flashman@axe.humboldt.edu Logic: Mathematical Logic around the world Association for Symbolic Logic LogicAL: An excellent resource for links. Gateway to Logic Some excellent logic software on-line from Germany A. Selby's home page:much interesting stuff on education, logic, algebra, and more. Situation Puzzles Proofs in Mathematics (YES!!) Mathematics as a language ... Mathematical Logic. Around Goedel's Theorem. By K.Podnieks two hyper-textbooks in mathematical logic including an Introduction to Mathematical Logic. ... MATHS 315 Mathematical Logic (University of Aukland) Handouts etc. for logic and set theory that parallel Hamilton's textbook Logic for Mathematicians.

27. Zermelo-Fraenkel Set Theory -- From Wolfram MathWorld A version of Set Theory which is a formal system expressed in firstorder predicate logic. Zermelo-Fraenkel Set Theory is based on the Zermelo-Fraenkel http://mathworld.wolfram.com/Zermelo-FraenkelSetTheory.html

Search Site Algebra Applied Mathematics Calculus and Analysis ... General Set Theory Zermelo-Fraenkel Set Theory A version of set theory which is a formal system expressed in first-order predicate logic . Zermelo-Fraenkel set theory is based on the Zermelo-Fraenkel axioms Zermelo-Fraenkel set theory is not finitely axiomatized. For example, the axiom of replacement is not really a single axiom, but an infinite family of axioms, since it is preceded by the stipulation that it is true "For any set-theoretic formula ." Montague (1961) proved that Zermelo-Fraenkel set theory is not finitely axiomatizable, i.e., there is no finite set of axioms which is logically equivalent to the infinite set of Zermelo-Fraenkel axioms provides an equivalent finitely axiomized system. SEE ALSO: Logic Set Theory Zermelo-Fraenkel Axioms Zermelo Set Theory ... [Pages Linking Here] REFERENCES: Montague, R. "Semantic Closure and Non-Finite Axiomatizability. I." In Infinitistic Methods, Proceedings of the Symposium on Foundations of Mathematics, (Warsaw, 2-9 September 1959). Oxford, England: Pergamon, pp. 45-69, 1961.

28. Definition Of Set Theory - Merriam-Webster Online Dictionary Definition of Set Theory from the MerriamWebster Online Dictionary with audio pronunciations, thesaurus, Word of the Day, and word games. http://www.m-w.com/dictionary/set theory

Home Visit Our Sites Unabridged Dictionary Learner's Dictionary ... Contact Us Also Visit: Unabridged Encyclopedia Britannica Visual ESL: Learner's for Kids: Spell It! Word Central Dictionary Thesaurus Spanish/English Medical Search "set theory" in: Thesaurus Spanish/English Medical Dictionary Open Dictionary Browse words next to: set theory Browse the Dictionary: A B C D ... Z set theory One entry found. set theory Main Entry: set theory Function: noun Date:  a branch of mathematics or of symbolic logic that deals with the nature and relations of sets set theoretic adjective Learn more about "set theory" and related topics at Britannica.com See a map of "set theory" in the Visual Thesaurus Pronunciation Symbols Products Premium Services ... Advertising Info

29. Set Theory: Scientific American Set Theory. PATRICK MERRELL. hink of this crossword as a Venn diagram consisting of two Setsone right side up, the other rotated 180 degrees. http://www.sciam.com/article.cfm?chanID=sa006&colID=1&articleID=C9D2CB39-E7F2-99

30. Meeting On Set Theory And Analysis, 10-12 July 2006 In Torino | PRESENTATION Set Theory grew out of analysis through Georg Cantor s work on Sets of uniquness of trigonometric series. Ever since, the two subjects have had a special http://www.logique.jussieu.fr/~boban/STA/

P R E S E N T A T I O N Organized by the University and the Politechnic of Torino the Meeting on Set Theory and Analysis will be held July 10-12, 2006 Dipartimento di Matematica via Carlo Alberto 10 10123 Torino map/itinerary View/Download Conference Poster jpg format [ 689K ] Photo credit : Enrico Aliberti Set theory grew out of analysis through Georg Cantor's work on sets of uniquness of trigonometric series. Ever since, the two subjects have had a special relationship, at times close and at others distant, but there has always existed a possibility of interaction with the potential of enriching both subjects. In recent years there has been a number of important developments on the confluence of the two subjects. Examples include Gowers' dichotomy theorem in functional analysis which was motivated by the Galvin Prikry partition theorem in infinitary combinators, the theorem of Harrington, Kechris, Louveau which extends the well known Glimm Effros dichotomy on the structure of the orbit spaces of a transformation group and which gave rise to a rich theory of Borel equivalence relations, the recent solution of Talagrand of the famous Control Measure Problem which has numerous ramifications for the theory of Boolean algebras and forcing, etc. The goal of this conference is to investigate this interactions between the two subjects by bringing together a number researchers from set theory and analysis. The intention is that the talk should be accessible to the specialists from the other subjects. We are particularly interested in exploring new connections and we plan to hold a special problem session the last day of the conference.

31. OUP: UK General Catalogue Set Theory. Felix Hausdorff. Price £22.25 (Hardback) ISBN13 978-0-8218-3835-8 Publication date 30 March 2006 American Mathematical Society 352 pages, mm http://www.oup.com/uk/catalogue/?ci=9780821838358

32. Set Theory Set Theory. A Set is a group of objects. Each object is known as a member of the Set. A Set can be represented using curly brackets. So a Set containing the http://www.mathsrevision.net/alevel/pure/set_theory.php

mathsrevision.net alevel pure MathsRevision.net A-Level Home Revision Guides Discussion Forum Links ... Maths Notation Pure Section Algebra Quadratic Equations Simultaneous Equations Indices Surds ... Reduction to Linear Form Calculus Differentiation from First Principles Differentation Tangents and Normals Uses of Differentation ... Differential Equations Trigonometry Sine and Cosine Formulae Radians Sin, Cos and Tan Trigonometric Identities ... Solving Trigonometric Equations Geometry Coordinate Geometry Curve Sketching Vectors Circles ... Contact the Author Set Theory E E Common Sets Some sets are commonly used and so have special notation: Other Notation Subsets B ( means is a subset of). Number of Members If A A ) = 4. This is because n( A ) means the number of members in set A The Universal Set The universal set is the set of all sets. All sets are therefore subsets of the universal set. Venn Diagrams Venn diagrams are used to represent sets. Here, the set A A is a subset of the universal set and so it is within the rectangle. The complement of A, written A', contains all events in the sample space which are not members of A. A and A' together cover every possible eventuality. A B means the union of sets A and B and contains all of the elements of both A and B. This can be represented on a Venn Diagram as follows:

33. New Set Theory Extends the language of Set Theory through restricted selfreference and through certain large cardinals. Also discusses higher order Set Theory and http://web.mit.edu/dmytro/www/NewSetTheory.htm

Dmytro Taranovsky April 20, 2005 Extending the Language of Set Theory Abstract: We discuss the problems of incompleteness and inexpressibility. We introduce almost self-referential formulas, use them to extend set theory, and relate their expressive power to that of infinitary logic. We discuss the nature of proper classes. Finally, we introduce and axiomatize a powerful extension to set theory. List of Sections: Introduction and Outline Incompleteness and Inexpressibility Almost Self-Referential Formulas Expressive Power of the Extensions ... Higher Order Set Theory Introduction and Outline Despite vast advances in set theory and mathematics in general, the language of set theory, which is also the language of mathematics, has remained the same since the beginning of modern set theory and first order logic. That formal language has served us well, but it is of necessity limited, and does not adequately deal with properties for which there is no set of all objects satisfying the property. The purpose of this paper is to address the deficiencies in the expressive power of ordinary set theory. The second section discusses incompleteness and inexpressibility in general. Much of the paper discusses logic that allows some self-reference but with limitations to prevent loops and infinite regress. The logic is more expressive than (a formulation of) infinitary logic. We show that levels of infinitary logic correspond to levels of almost self-referential logic and levels of constructible hierarchy above V. We use strong logics and weak set theories to clarify the strengths of inaccessible and Mahlo cardinals. We also use large cardinals to get reflection principles, and use reflection principles to axiomatize extensions to the language of set theory.

34. Synopses Of Topics - Set Theory math.usask.ca/emr/Sett.html Similar pages Set Theory in KobeResearch in our group can be broadly described as the investigation of the combinatorial structure of the reals from the point of view of Set Theory. http://math.usask.ca/emr/sett.html

Set Theory instructions on how to get the symbol font working. It also has some advice for Macintosh users. Symbol Name the empty set element of not an element of contained in contains not contained in intersection union Terminology A set is any collection of objects. Examples The empty set is the set containing no elements, denoted by If A is a set and x is a member of A , we say x is an element of A and denote this by x A If A and B are sets and every element of A is also an element of B (that is, x A implies x B then we say A is a subset of B or A is contained in B and we denote this by A B The intersection The union Notation: As you may have surmised from the terminology section, the following conventions for notation are used in set theory: A, B, X: capital letters denote sets a, b, x: lowercase letters denote elements of sets Venn Diagrams When working with sets, it is often easier to visualise concepts with a picture. We can do this with Venn Diagrams . A set is drawn as a geometric area ( e.g. circle, rectangle) and shading is used to indicate a specific portion of the set or sets. Here are some Venn Diagrams that illustrate the concepts we have already discussed: Element Subset Intersection Union x A A X, B

35. Fields Institute - Set Theory And Analysis Semester long program at the Fields Institute, Toronto, Ontario, along with some shorter workshops. September through December, 2002. http://www.fields.utoronto.ca/programs/scientific/02-03/set_theory/

Home About Us NPCDS/PNSDC Mathematics Education ... Search THEMATIC PROGRAMS December 24, 2007 Thematic Program on Set Theory and Analysis September - December 2002 Attendee List Seminars Workshops Lectures ... Coxeter Lecture Mail List To receive on-going information about this program please subscribe to the mail list Organizers: Alan Dow, University of North Carolina Alexander Kechris, California Institute of Technology Miklos Laczkovich, Etvos Lorand University Claude Laflamme, University Of Calgary Juris Steprans York University Stevo Todorcevic, C.N.R.S., Paris and University of Toronto Overview: From its very beginnings, set theory has enjoyed a relationship with analysis which, while at times close and at others distant, has always allowed for the possibility of symbiosis. During the Fall of 2002 the Fields Institute will hosts a thematic program devoted to fostering the interaction between these two areas. Internationally recognized experts from both disciplines will be on site from September 2002 through to December 2002. The format of the program will include at least two short, but intense thematic workshops. One will focus on set theoretic techniques in the theory of

36. Semantics Beyond Set Theory Semantics Beyond Set Theory. Jeudi 25 octobre 2007 Salle Dussane ENS - 45, rue d Ulm - 75005 Paris Organisé par Alda Mari et David Nicolas http://semantique.free.fr/sip/sip2/

Semantics Beyond Set Theory Jeudi 25 octobre 2007 Salle Dussane - ENS - 45, rue d'Ulm - 75005 Paris Programme Peter Simons Categories, combinations, and constructions: semantics for grown-ups Sets, properties, and semantics 11:40 Pause David Nicolas Superplurals in English Lucia Tovena A class of pluractional verbs in Italian and French Michael Hegarty The role of processes in dynamic semantics 16:10 Pause Carl Pollard Stone-dual semantics for natural language : Tense and actuality entailment

37. 80.07.04: Logic And Set Theory While in no way does the unit cover the entire fields of Set Theory and Logic, it does, I hope, offer an introduction to the basic concepts, http://www.yale.edu/ynhti/curriculum/units/1980/7/80.07.04.x.html

Yale-New Haven Teachers Institute Home Logic and Set Theory by Richard Canalor Contents of Curriculum Unit 80.07.04: Narrative Pretest Logic Conjunctions and Disjunction’s ... To Guide Entry The following unit is designed to offer teachers and children a chance to explore what may be to them a different area of Finite Mathematics. While in no way does the unit cover the entire fields of Set Theory and Logic, it does, I hope, offer an introduction to the basic concepts, symbols and importance of these two fields of Mathematics. As you will see, Set Theory and Logic are related and have therefore been combined for the content of this unit. The unit is approximately two weeks in length and is intended for grade 6,7, or 8 although both length and grade level may vary. We will begin with a short Pretest. The purpose of the pretest is twofold. On one hand it will give some barometer of success (there will be a posttest) and hopefully the pretest will foster discussion and motivate children to want to hear more. Pretest 1. There were 20 people at a party. Thirteen had coke, 7 had sandwiches, 5 had both. How many did not eat or drink?

38. SET THEORY, QUANTUM SET THEORY & CLIFFORD ALGEBRAS An essay on the algebraic nature of an essentially quantum and probably relativistic universe. http://graham.main.nc.us/~bhammel/QSET/qset1.html

Notes On The Concept of Quantum Set Theory Introduction Classical Set theory: symmetric difference and complementation Duality Maps Boolean Algebras (Rings) ... Quantum Categories Introduction This continues the introductory essay on the physical notions of quantum set theory The idea of quantum set theory, while it sounds to be of a mathematical nature, is necesarily of a physical nature if one means to quantize "points" that comprise sets in such a way that they are treated as physical objects with physical properties. Although there is a fair amount of pure mathematics below, the hard part comes when having to decide on the physics of treating the concept so that it becomes reduced to something which adequately deals with "physical space" and its classically approximate locally Euclidean E Though the expression of quantum theory is highly mathematical, the physical theory is indeed physical, hence more specific than any particular mathematics. A recurring mathematical structure in physics is the Lie algebra su(2), which describes spin, isotopic spin associated with electrical charge, happens to be abstractly identical to the algebra of SO(3) since SU(2) is the universal covering group of SO(3), whose algebra is normally associated with classical and QM angular momentum operators in an E It occurs again in my FCCR(n) The su(2) algebra appears later below, but for now, begining at the begining, I construct a genuine algebra of sets, which act together, with a few possibly interesting mathematical remarks, and get to an algebra to Clifford algebra mapping as quantization centering on the power set, outline some Clifford algebra theory and its various relationships, and finally get to discussing the nature of any quantum set theory.

39. Logic Colloquium 2007: Contributed Talk Schedule Topic, Set Theory, Proof Theory, Computable Model Theory, Modal Logic . On Cardinals in Set Theory Without Choice and Regularity, Assaf Hasson (Oxford) http://www.math.wisc.edu/~lempp/conf/contrib.html

Contributed Talk Schedule List of All Abstracts Alphabetical by Speaker(s) List by Day and Session: Saturday, July 14: Topic Set Theory Proof Theory Computable Model Theory Modal Logic Room/Chair GW/P. Larson SW/Kohlenbach CS-222/Lempp CS-223/Visser Natasha Dobrinen (Vienna): On the Consistency Strength of the Tree Property at the Double Successor of a Measurable Cardinal Joost Joosten (Amsterdam): Interpretability in PRA Alexander Gavryushkin (Novosibirsk): Computable Models Spectra of Ehrenfeucht Theories Finite Reduction Trees in Modal Logic Andrew Brooke-Taylor (Vienna): Definable Well-ordering, the GCH, and Large Cardinals Ryan Young (Leipzig): Asymmetric Systems of Natural Deduction Andrey Frolov (Kazan): n -categorical linear orderings Mikhail Sheremet (London): A Modal Logic of Metric Spaces ,1) Simplified Morasses With Linear Limits Using an Unfoldable Cardinal Elliott Spoors (Leeds): Provably Recursive Functions in Extensions of a Predicative Arithmetic Alexandra Revenko (Novosibirsk): Automatic Linear Orders Zofia Kostrzycka (Opole): On Formulas in One Variable and Logics Determined by Wheel Frames in NEXT( KTB Sunday, July 15:

40. Set Theory Primer In Set Theory intervals are measured by the number of semitones. Thus, CE is not a major third (M3) but 4 semitones, or simply 4. A minor sixth would be 8. http://solomonsmusic.net/setheory.htm

Set Theory Primer for Music Part I. Nonlinear Sets Go to Set Theory Part II. Linear Sets (Serial Theory) Contents Frequently Asked Questions Set Theory Glossary Constructing a Simple Clock Calculator for Music Set Theoretical Analysis Basic Definitions ... Inversional Invariance Basic Definitions Abdo. see pitch number div, or directed-interval vector, also interval string. The distance between successive (ordered) pcs cycling to an octave. A prime div is the div of a prime form. Forte Prime . A generalized version of a set that includes its inversion. Index number . The transposition number, in semitones, above a reference pc. P5 would be a transposition up 5 semitones from P0. Interval. The distance between two pitches. In set theory intervals are measured by the number of semitones. Thus, CE is not a major third (M3) but 4 semitones, or simply 4. A minor sixth would be 8. Interval class ic ). The distance between two pitch classes, measured by the shortest distance. C to G may be the interval of 7, but its interval class is 5. Thus, the largest ic is the tritone (6). Interval String . see div Modulo 12 ). An arithmetic system nearly identical to that of a clock, where 13=1, 14=2 etc. However, in modulo 12 the number 12=0. If we want to know what 2 hours past 11 is (11+2), we say it is one o'clock (1). Thus, in mod12, 11+2=1, and there is no number greater than 11.

41. Set Theory From FOLDOC Many mathematicians use Set Theory as the basis for all other mathematics. Axiomatic Set Theory The study of formal systems whose theorems, on the intended http://www.swif.uniba.it/lei/foldop/foldoc.cgi?set theory

42. Faculty Of Philosophy: Michael Potter Set Theory and its Philosophy A Critical Introduction , Oxford University Press, A presentation of Set Theory intended for beginning graduate students. http://www.phil.cam.ac.uk/teaching_staff/potter/potter_index.html

Faculty of Philosophy University of Cambridge Faculty of Philosophy Teaching Staff Michael Potter MICHAEL POTTER is Reader in the Philosophy of Mathematics at Cambridge and has been a Fellow of Fitzwilliam College since 1989. He was previously at Oxford, where he took a D.Phil. in pure mathematics and was a Fellow of Balliol College. He has spent periods of research leave in the Department of Logic and Metaphysics at St Andrews and the Department of Philosophy at Harvard. In 2004 and 2005 he was on research leave from Cambridge as a Senior Research Fellow at Stirling University funded by the AHRC. Lectures Michaelmas Term 2006 Wittgenstein's Tractatus Wittgensteinian Themes Philosophy of Mathematics ... Set theory Lent Term 2007 Non-classical logics (With Dr Smith) Mathematical Logic Seminar Easter Term 2007 Research Current research interests in the following areas: The Tractatus The philosophy of set theory Wittgenstein's later philosophy of arithmetic Constructive philosophies of arithmetic Interests of current and recent research students: Modal ontological arguments for the existence of God Theories of ontology Intuitionism Prospects for neo-Kantian philosophy of mathematics Impredicativity Publications Books (Ed. with Mary Leng and Alexander Paseau)

43. Math 512 Descriptive Set Theory In Descriptive Set Theory we try to avoid these pathologies by concentrating on natural classes of wellbehaved Sets of reals, like Borel Sets or projective http://www.math.uic.edu/~marker/math512/

Math 512: Descriptive Set Theory Fall 2002 MWF 11:00 216 Taft Hall Instructor David Marker Office: 411 SEO Office Phone: (312) 996-3069 Office Hours: M 9-11, W 12-1 and by appointment Fax: (312) 996-1491 e-mail: marker@math.uic.edu web page: http://www.math.uic.edu/~marker course web page: http://www.math.uic.edu/~marker/math512 Description It is well know that when one studies arbitrary subsets of the real numbers one runs into many pathologies (non-measurable sets) and independent problems (the Continuum Hypothesis). In Descriptive Set Theory we try to avoid these pathologies by concentrating on natural classes of well-behaved sets of reals, like Borel sets or projective sets (the smallest class of sets containing Borel sets and closed under projections from higher dimenional spaces). While this is a restricted class of sets it includes most of the sets that arise naturally in mathematical practice. Lately there have been many intersting connections with dynamical systems, through the study of orbit equivalence relations. The first half of the course will be devoted to developing the fundamental results and techniques of descriptive set theory. In the second half we will look at more recent developments. The exact topics covered will depend on the background and interest of the class.

44. Set Theory. Zermelo-Fraenkel Axioms. Russell's Paradox. Infinity. By K.Podnieks What is Mathematics? Goedel s Theorem and Around. Textbook for students. Section 2. By K.Podnieks. http://www.ltn.lv/~podnieks/gt2.html

set theory, axioms, Zermelo, Fraenkel, Frankel, infinity, Cantor, Frege, Russell, paradox, formal, axiomatic, Russell paradox, axiom, axiomatic set theory, comprehension, axiom of infinity, ZF, ZFC Back to title page Left Adjust your browser window Right 2. Axiomatic Set Theory The origin of Cantor's set theory Formalization of Cantor's inconsistent set theory Zermelo-Fraenkel axioms Around the continuum problem ... Ackermann's set theory (Church thesis for set theory?) For a general overview and set theory links, see Set Theory by Thomas Jech in Stanford Encyclopedia of Philosophy 2.1. The Origin of Cantor's Set Theory F. A. Medvedev. Development of Set Theory in the XIX Century. Nauka Publishers, Moscow, 1965, 350 pp. (in Russian) F. A. Medvedev. The Early History of the Axiom of Choice. Nauka Publishers, Moscow, 1982, 304 pp. (in Russian) See also: Online paper "A history of set theory" in the MacTutor History of Mathematics archive A. Kanamori . Set Theory from Cantor to Cohen, Bulletin of Symbolic Logic

45. SET THEORY, Part 2 Set Theory, Part 2. Nondiatonic SubSets A trichord or larger pitch Set can include any pitch of the chromatic scale, a collection of all twelve pitches http://jan.ucc.nau.edu/~krr2/settheory/settheory2.html

SET THEORY, Part 2 Nondiatonic Subsets Example 6: a nondiatonic subset Terms Used in The Set Theory of Pitches Octave Equivalence Pitch Class pitch class A There are twelve pitch classes, one for each note in the chromatic octave. Enharmonic Equivalence B = C, E = F , or C = D . Thus, C and D are members of the same pitch class. Pitch Class Numbers do ." This approach is quite convenient under most circumstances. However, relative numbering ( first note is always 0) was adopted in this text in order to demonstrate the direct relationship between set transformation and mod 12 arithmetic. This is like using a "moveable do Both approaches are useful but one should be careful not to mix the two in the same analysis. Example 7: Pitch Class numbers of the C chromatic scale in mod 12 integers Interval Numbers Table 1: dodecimal (mod 12) numbering of intervals INTERVAL Interval number OPERATION Interval number INTERVAL INVERSION m2(+U) m2 (+u) Interval Class Numbers Example 8: Interval numbers, inversion arithmetic Interval Vector Example 9: Interval vector of the major triad IC (interval class) no. of occurrences

46. Hajnal & Hamburger's Set Theory Book Site This is a classical introduction to Set Theory in three parts. The first part gives a general introduction to Set Theory, suitable for undergraduates; http://www.ipfw.edu/math/hamburger/book.html

Set Theory András Hajnal Rutgers University Peter Hamburger Indiana University - Purdue University Fort Wayne Translated from Hungarian to English by Attila Máté Summary: Contents: Part I. Introduction to set theory: Notation, conventions. Definition of equivalence. The concept of cardinality. The Axiom of Choice. Countable cardinal, continuum cardinal. Comparison of cardinals. Operations with sets and cardinals. Examples. Ordered sets. Order types. Ordinals. Properties of wellordered sets. Good sets. The ordinal operation. Transfinite induction and recursion. Some consequences of the Axiom of Choice, the wellordering theorem. Definition of the cardinality operation. Properties of cardinals. The cofinality operation. Properties of the power operation. Hints for solving * problems in Part I. Appendix. An axiomatic development of set theory: The Zermelo-Frankel axiom system of set theory. Definition of concepts; extension of the language. A sketch of the development. Metatheorems. Definitions of simple operations and properties (continued). R (continued).

47. Book's Contents Krzysztof Ciesielski, CUP (1997). Contents and preface. http://www.math.wvu.edu/~kcies/STbook.html

Set Theory for the Working Mathematician by Krzysztof Ciesielski London Math Society Student Texts Cambridge University Press, 1997. Hardback ISBN 0-521-59441-3, price $59.95; paperback ISBN 0-521-59465-0, price $19.95. To order call 1-800-872-7423 or link to Cambridge University Press order page. To see Preface click here Table of Contents Preface vii I Basics of set theory Axiomatic set theory 1.1 Why axiomatic set theory? 1.2 The language and the basic axioms Relations, functions and Cartesian product 2.1 Relations and the axiom of choice 2.2 Functions and the replacement scheme axiom 2.3 Generalized union, intersection and Cartesian product 2.4 Partial and linear order relations Natural numbers, integers, and real numbers 3.1 Natural numbers 3.2 Integers and rational numbers 3.3 Real numbers II Fundamental tools of set theory Well orderings and transfinite induction 4.1 Well-ordered sets and the axiom of foundation 4.2 Ordinal numbers 4.3 Definitions by transfinite induction 4.4 Zorn's Lemma in algebra, analysis and topology Cardinal numbers 5.1 Cardinal numbers and the continuum hypothesis

48. Front: [math.LO/0211397] The Future Of Set Theory Journal reference Haim Judah (editor), Set Theory of the Reals. Israel Mathematical Conference Proceedings, vol. 6, Proceedings of the Winter Institute http://front.math.ucdavis.edu/math.LO/0211397

Front for the arXiv Mon, 24 Dec 2007 Front math LO math.LO/0211397 search register submit journals ... iFAQ math.LO/0211397 Title: The Future of Set Theory Authors: Saharon Shelah Categories: math.LO Logic math.HO History and Overview Report number: Shelah [Sh:E16] Journal reference: Haim Judah (editor), Set Theory of the Reals. Israel Mathematical Conference Proceedings, vol. 6, Proceedings of the Winter Institute held at Bar-Ilan University, Ramat Gan, January 1991 Abstract: Haim Judah has asked me to speak on the future of set theory, so, as the next millennium is coming, to speak on set theory in the next millennium. But we soon cut this down to set theory in the next century. Later on I thought I had better cut it down to dealing with the next decade, but I suspect I will speak on what I hope to try to prove next year, or worse - what I have done in the last year (or twenty). It seems I am not particularly suitable for such a lecture, as I have repeatedly preferred to try to prove another theorem rather than to prepare the lecture (or article); so why did I agree at all to such a doubtful endeavor? Well, under the hypothesis that I had some moral obligation to help Haim in the conference (and the proceedings) and you should not let a friend down, had I been given the choice to help with organizing the dormitories, writing a lengthy well written expository paper or risking making a fool of myself in such a lecture, I definitely prefer the last. Owner: Saharon Shelah's Office Version 1: Tue, 26 Nov 2002 01:44:31 GMT

49. Set Theory Definition - Dictionary - MSN Encarta Search for Set Theory in all of MSN Encarta. Email this entry Blog about this entry on MSN Spaces Download the MSN Encarta Right-Click Dictionary http://encarta.msn.com/dictionary_1861720327/set_theory.html

var s_account="msnportalencarta"; MSN home Mail My MSN Sign in ... more Hotmail Messenger My MSN MSN Directory Air Tickets/Travel Autos City Guides Extra ... More Additional Reference Materials Thesaurus Translations Multimedia Other Resources Education Resources Math Help Foreign Language Help Project Planner ... Help Dictionary Find in Dictionary Thesaurus Translations A B ... See pronunciation key Search for " set theory " in all of MSN Encarta E-mail this entry Blog about this entry on MSN Spaces Download the MSN Encarta Right-Click Dictionary set theory set the·o·ry noun Definition: mathematics of sets: the branch of mathematics that deals with the properties and relationships of sets system of set axioms: the system of axioms for sets Advertisement Also on Encarta 10 tips for building your English vocabulary Secret students What colleges really want Making the most of unemployment ... Coffee break: Recharge your brain Our Partners ApplyWise Online College Admissions Counseling ClassesUSA.com: Compare online degrees Kaplan Test Prep and Admissions The Princeton Review ... Tutor.com: Live Help When You Get Stuck Advertisement Also on MSN Rediscover 'Planet Earth' on the Discovery Channel MSN Shopping: Best books for the season MSN Careers: Nail that job interview Today's news on MSNBC MSN Shopping var fSucceed = false;

50. Set Theory Online Experts Expert Advice, Live. Get immediate help online via live chat, phone, or email from qualified professional advisors in Set Theory and more. http://www.kasamba.com/Categories/ViewCategory.aspx?CatID=10223

51. SET THEORY FOR COMPUTING Set Theory FOR COMPUTING From decision procedures to declarative programming with Sets Decision algorithms for fragments of Set Theory http://www.dipmat.unict.it/~cantone/SetTheoryForComputing/

SET THEORY FOR COMPUTING From decision procedures to declarative programming with sets Domenico Cantone Eugenio G. Omodeo Alberto Policriti Table of contents ... Selected exercises solved Additional exercises Programs SETL2 programs SWI-Prolog programs Metamorpho system Bibliographic reference update Relevant links Decision algorithms for fragments of set theory Logic programming with sets The "Programming with Sets" home page Calculus of dyadic relations COST Action no. 274 (TARSKI) Metamorpho system Nonclassic logics SETL programming systems Monographs in Computer Science Series of Springer-Verlag Errata list (misprints) Please contact us in order to - make comments and suggestions - propose solutions to exercises - raise questions and challenges You can order this book through Last modified Nov 3, 2003

52. EBRSC to EBRSC. What is Rough Set Theory? Animated example! Bibliography. Go to the Bulletin of the International Rough Set Society. http://www.cs.uregina.ca/~roughset

Electronic Bulletin of the Rough Set Community Introduction to EBRSC. What is Rough Set Theory Animated example! Bibliography Conferences ... Data files available for testing. Rough Set software links. Conference Abstracts! The latest Issue! Volume 1 archives. Volume 2 archives. Volume 3 archives. Volume 4 archives. Volume 5 archives. Volume 6 archives. Volume 7 archives. Volume 8 archives. Volume 9 archives. Volume 10 archives. Related pages Mail the Editors a message (roughset@cs.uregina.ca). Subscribe to EBRSC Now! Go to the Bulletin of the International Rough Set Society Go to our host, the University of Regina Department of Computer Science homepage. Last updated: Wed 16-Jul-03 This page has been accessed times. Page author: ( roughset@cs.uregina.ca

53. Downloading "Set Theory" These notes for a graduate course in Set Theory are on their way to becoming a book. They originated as handwritten notes in a course at the University of http://www.math.toronto.edu/~weiss/set_theory.html

The preliminary version of the book Set Theory by William Weiss is available here. You can download the book in PDF format, gzip'ed PDF format, format, format, dvi format, or gzip'ed dvi format. Below is the Preface from the book. Preface These notes for a graduate course in set theory are on their way to becoming a book. They originated as handwritten notes in a course at the University of Toronto given by Prof. William Weiss. Cynthia Church produced the first electronic copy in December 2002. James Talmage Adams produced the copy here in February 2005. Chapters 1 to 9 are close to final form. Chapters 10, 11, and 12 are quite readable, but should not be considered as a final draft. One more chapter will be added.

54. A/Prof N J Wildberger Personal Pages This is the Fall and original sin of Cantor s Set Theory . Most (but not all) of the difficulties of Set Theory arise from the insistence that there http://web.maths.unsw.edu.au/~norman/views2.htm

N J Wildberger Contact: School of Maths UNSW Sydney 2052 Australia n.wildberger@unsw.edu.au Tel:61 (02) 9385 7098 Fax:61 (02) 9385 7123 Set Theory: Should You Believe? N J Wildberger School of Maths UNSW Sydney NSW 2052 Australia webpages: http://web.maths.unsw.edu.au/~norman "I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction." (Gauss) "I don't know what predominates in Cantor's theory - philosophy or theology, but I am sure that there is no mathematics there." (Kronecker) "...classical logic was abstracted from the mathematics of finite sets and their subsets...Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of [Cantor's] set theory ..." (Weyl) Modern mathematics as religion Modern mathematics doesn't make complete sense . The unfortunate consequences include difficulty in deciding what to teach and how to teach it, many papers that are logically flawed, the challenge of recruiting young people to the subject, and an unfortunate teetering on the brink of irrelevance.

55. IRA: 1. Sets And Relations Many results in Set Theory can be illustrated using Venn diagram, So far, we have reviewed a few basic facts from Set Theory, and also got an idea about http://web01.shu.edu/projects/reals/logic/index.html

56. Set Theory Introduction to Set Theory, definitions of Set, subSet, unions, intersections, complements, and properties such as distributive laws and deMorgan s laws. http://www.efunda.com/math/settheory/settheory.cfm

Set Theory About Us Directory Career News ... Ask an Expert Statistics Set Theory Probability Theory Distributions Reliability Least Squares Resources Bibliography Login Free Magazines Design-2-Part ... more Search All Materials Design Center Processes Units and Constants Formulas Mathematics for Home Membership Palm Store Forum ... Math Definition A set is a collection of individual elements in the domain . The universal set contains every element in . The null set contains no element. If is a set in the domain must be a subset of the universal set , denoted as If consists of some but not all elements, is then called a proper subset of , denoted as Top of Page Unions, Intersections, Complements The definition of union or intersection and "), and complement not ") can be illustrated by Venn Diagrams as follows: Suppose that and are two sets in the same domain whose universal set is The union of and consists of all elements which belong to either or , denoted by The intersection of and consists of only elements which belong to both and , denoted by The complement of consists of elements which do not belong to , denoted by Top of Page Important Properties Distributive Laws De Morgan's Laws Top of Page Membership About Us Privacy ... Advertise

57. Thoralf A. Skolem, Abstract Set Theory (Notre Dame, Ind Chapter III Axiomatic Set Theory; Axioms of Zermelo and Fraenkel Chapter XIV The ramified Theory of types. Predicative Set Theory http://projecteuclid.org/euclid.ndml/1175197470

Log in RSS Title Author(s) Abstract Subject Keyword All Fields FullText more options Home Browse Search ... Help Abstract Set Theory Thoralf A. Skolem Notre Dame Mathematical Lectures, Number 8 Notre Dame, Indiana : University of Notre Dame, 1962. 1st 70 pp. Subjects: Set theory Permanent link to this monograph: http://projecteuclid.org/euclid.ndml/1175197470 Mathmatical Reviews number (MathSciNet): i-ii View PDF View Abstract Preface iii View PDF View Abstract Table of Contents v View PDF View Abstract Chapter 1: Historical remarks; Outlines of Cantor's theory View PDF View Abstract Chapter II: Ordered sets; A theorem of Hausdorff View PDF View Abstract Chapter III: Axiomatic set theory; Axioms of Zermelo and Fraenkel View PDF View Abstract Chapter IV: The well-ordering theorem View PDF View Abstract Chapter V: Ordinals and alephs View PDF View Abstract Chapter VI: Some remarks on functions of ordinal numbers View PDF View Abstract Chapter VII: On the exponentiation of alephs View PDF View Abstract Chapter VIII: Set representing ordinals View PDF View Abstract Chapter IX: The notions "finite" and "infinite" View PDF View Abstract Chapter X: The simple infinite sequence; Development of arithmetic

58. Set Theory -- From Eric Weisstein's Encyclopedia Of Scientific Books Set Theory for the Working Mathematician. Cambridge, England Cambridge University Press, 1997. Cohen, Paul J. Set Theory and the Continuum Hypothesis. http://www.ericweisstein.com/encyclopedias/books/SetTheory.html

Set Theory see also Set Theory Bourbaki, Nicolas. Elements of Mathematics: Theory of Sets. Paris, France: Hermann, 1968. 414 p. Breuer, Joseph. Introduction to the Theory of Sets. Englewood Cliffs, NJ: Prentice-Hall, 1958. Ciesielski, Krzysztof. Set Theory for the Working Mathematician. Cambridge, England: Cambridge University Press, 1997. 240 p. $22.95. Cohen, Paul J. Set Theory and the Continuum Hypothesis. New York: W.A. Benjamin, 1966. 154 p. $?. Dauben, Joseph Warren. Georg Cantor: His Mathematics and Philosophy of the Infinite. Princeton, NJ: Princeton University Press, 1990. $29.95. Dedekind, Richard. Was sind und was sollen die Zahlen, 4. unveranderte aufl. Braunschweig, 1918. 58 p. 1918. Devlin, K. The Joy of Sets: Fundamentals of Contemporary Set Theory, 2nd ed. New York: Springer-Verlag, 1993. 192 p. $39.95. Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Halmos, Paul Richard. Naive Set Theory. New York: Springer-Verlag, 1974. 104 p. $37.95. Henle, James M.

59. Free Books > Science > Mathematics > Pure Mathematics > Set Theory Free Books Science Mathematics Pure Mathematics Set Theory. http://2020ok.com/13953.htm

Your browser does not support JavaScript and this site utilizes JavaScript to build content and provide links to additional information. You should either enable JavaScript in your browser settings or use a browser that supports JavaScript in order to take full advantage of this site. Directory of FREE Online Books and FREE eBooks Free Books Science Mathematics Pure Mathematics Set Theory Computability and Incompleteness Fibred Categories Foundations of Cryptography Future of Spatial Data and Society ... Logic Related Tags Advanced Algebra Anwendungen Auf ... Contact Us

60. Index Of /music/general_theory/post-tonal/index.html Introduction to Set Theory An interesting pitchclass Set table Row finder for the 2nd Viennese School, and other good Set-related stuff http://www.listeningarts.com/music/general_theory/post-tonal/

Search this site or the web powered by FreeFind Site search Web search Post-Tonal Theory Links Send mail to Irene Girton All About Set Theory: an excellent page An excellent 12-tone page Set Theory Machine by Jay Tomlin ... Back to ListeningArts home This page last edited

61. Fuzzy Set Theory Fuzzy Set Theory. Fuzzy Set Theory defines Set membership as a possibility distribution. The general rule for this can expressed as. displaymath1725 http://www.cs.cf.ac.uk/Dave/AI2/node99.html

Next: Further Reading Up: Fuzzy Logic Previous: Fuzzy Logic Fuzzy Set Theory Fuzzy set theory defines set membership as a possibility distribution The general rule for this can expressed as: where n some number of possibilities. This basically states that we can take n possible events and us f to generate as single possible outcome. This extends set membership since we could have varying definitions of, say, hot curries. One person might declare that only curries of Vindaloo strength or above are hot whilst another might say madras and above are hot. We could allow for these variations definition by allowing both possibilities in fuzzy definitions. Once set membership has been redefined we can develop new logics based on combining of sets etc. and reason effectively. dave@cs.cf.ac.uk

62. Welcome To The Hotel Infinity Cantor is the founder of the branch of mathematics called Set Theory, which is at the foundation of much of 20th century mathematics. At the heart of Set http://www.c3.lanl.gov/mega-math/workbk/infinity/inbkgd.html

Infinity is for Children-and Mathematicians! How Big is Infinity? Most everyone is familiar with the infinity symbolthe one that looks like the number eight tipped over on its side. The infinite sometimes crops up in everyday speech as a superlative form of the word many . But how many is infinitely many? How far away is "from here to infinity"? How big is infinity? You can't count to infinity. Yet we are comfortable with the idea that there are infinitely many numbers to count with: no matter how big a number you might come up with, someone else can come up with a bigger one: that number plus oneor plus two, or times two. Or times itself. There simply is no biggest number. Is there? Is infinity a number? Is there anything bigger than infinity? How about infinity plus one? What's infinity plus infinity? What about infinity times infinity? Children to whom the concept of infinity is brand new, pose questions like this and don't usually get very satisfactory answers. For adults, these questions don't seem to have very much bearing on daily life, so their unsatisfactory answers don't seem to be a matter of concern. At the turn of the century, in Germany, the Russian-born mathematician Georg Cantor applied the tools of mathematical rigor and logical deduction to questions about infinity in search of satisfactory answers. His conclusions are paradoxical to our everyday experience, yet they are mathematically sound. The world of our everyday experience is finite. We can't exactly say where the boundary line is, but beyond the finite, in the realm of the

63. LogBlog: Early Development Of Set Theory | Richard Zach | Philosophy | Universit Another one of the SEP entries commissioned by Paolo and me The Early Development of Set Theory, by José Ferreirós, author of Labyrinth of Thought. http://www.ucalgary.ca/~rzach/logblog/2007/04/early-development-of-set-theory.ht

@import "http://www.ucalgary.ca/templates/styles/uofc-drupal.css"; @import "http://www.ucalgary.ca/templates2/styles/uofc-level-c.css"; @import "http://wcm2.ucalgary.ca/rzach/files/rzach/custom_colors/uofc_c_v7.css"; @import url("http://www.blogger.com/css/blog_controls.css"); @import url("http://www.blogger.com/dyn-css/authorization.css?blogID=7108230"); var BL_backlinkURL = "http://www.blogger.com/dyn-js/backlink_count.js";var BL_blogId = "7108230"; Skip to Headline Skip to Navigation Skip to Content Skip to Sidebar UofC Navigation Prospective Students Current Students Alumni Community Search UofC: IT MY U OF C Contacts LogBlog Logic . Philosophy . Other Fun Stuff Site Navigation Primary links Home Research Teaching CV ... LogBlog Search LogBlog Most Recent Posts Previous Posts How to Capture Escaping Logicians Bookshelf Changes II Changes ... Logician attacked by Alligator at ASL Meeting Links My Homepage Calgary Philosophy Department UC Berkeley Logic Group Association for Symbolic Logic Blogroll Subscribe to Posts [ Atom LogBlog Tuesday, April 10, 2007

64. LINZ2008 - 29th Linz Seminar On Fuzzy Set Theory About this Seminar. Since their inception in 1979 the Linz Seminars on Fuzzy Sets have emphasized the development of mathematical aspects of fuzzy sets by http://www.flll.jku.at/research/linz2008/

Home Program Committees Dates ... Archive About this Seminar Since their inception in 1979 the Linz Seminars on Fuzzy Sets have emphasized the development of mathematical aspects of fuzzy sets by bringing together researchers in fuzzy sets and established mathematicians whose work outside the fuzzy setting can provide direction for further research. The seminar is deliberately kept small and intimate so that informal critical discussion remains central. There are no parallel sessions and during the week there are several round tables to discuss open problems and promising directions for further work. Foundations of Lattice-Valued Mathematics with Applications to Algebra and Topology Accordingly, the topics of the Seminar will include but not be limited to: Categorical and logical approaches to lattice valued algebraic structures, powerset theories, topological structures Lattice valued categories, equivalences, locales, orders, topologies Presheaf and sheaf theoretic approaches to lattice valued structures Programming semantics, semantic domains, topological systems

65. Intute Science, Engineering And Technology - Full Record Details For Set Theory Topics studied in these notes are the axioms of ZermeloFraenkel Set Theory, ordinals, transfinite induction, ordinal arithmetic, well-ordered Sets, http://www.intute.ac.uk/sciences/cgi-bin/fullrecord.pl?handle=20071125-180500

66. Set Theory™ Font Family : MyFonts Set Theory™ font family from Haiku Monkey, prices starting at $19.00. http://www.myfonts.com/fonts/haiku/set-theory/

transitional Help: How to Buy This Font More Fonts Like This ... index Set Theory™ from Haiku Monkey Set Theory™ is a Haiku Monkey font family with 1 style priced from Click the Purchase Options button below to view pricing and availability information. 8 points 10 points 12 points 14 points 18 points 24 points 36 points 48 points 60 points 72 points Set Theory Set Theory Design Credits First seen on MyFonts: September 18th, 2007 Designed by: Alec Julien Designed when: 2007 Design owned by: Haiku Monkey MyFonts Keywords: futuristic round techno suggest Set Theory If you were a superhero, wouldn't you want your unitard to sport Set Theory, big and bold, on the front? Try looking at more fonts like this ‘Set Theory’ is a trademark of Haiku Monkey. About Us Testimonials Sell Your Fonts Become an Affiliate ... Sign In MyFonts.com, Inc. 245 First Street 17 th Floor Cambridge MA 02142 USA MyFonts and MyFonts.com are registered trademarks of MyFonts.com, Inc. WhatTheFont and Starlets are trademarks of MyFonts.com, Inc.

67. Handbook Of Set Theory Unofficial index of online chapters in Handbook of Set Theory (Eds. Foreman, Kanamori, Magidor). Volume I. Chapter name. Author(s). Combinatorics http://www.tau.ac.il/~rinot/host.html

Unofficial index of online chapters in Handbook of Set Theory (Eds. Foreman, Kanamori Magidor Volume I Chapter name Author(s) Combinatorics Stationary sets Thomas Jech Partition Relations ... Todorcevic Ramsey Theory for Banach Spaces Stevo Todorcevic Continuum Borel equivalence relations Greg Hjorth ... Bartoszynski Revised countable support iterations Hans-Dieter Donder , Ulrich Fuchs Constructibility Constructibility and Class Forcing Sy D. Friedman ... Philip Welch Volume II Chapter name Author(s) Elementary Embeddings Elementary embeddings and algebra Patrick Dehornoy Iterated Forcing and Elementary Embeddings James Cummings Ideals and Generic Elementary Embeddings Matthew Foreman Singular Cardinals Cardinal Arithmetic Uri Abraham Menachem Magidor Prikry -type Forcings Moti ... Eisworth Volume III Chapter name Author(s) Inner Model Theory Beginning Inner Model Theory William J. Mitchell The Covering Lemma William J. Mitchell ... Schimmerling Determinacy Structural Consequences of AD Stephen C. Jackson Determinacy in L(R) Itay ... Neeman Large Cardinals from Determinacy Peter Koellner W. Hugh

68. Tarski Grothendieck Set Theory It includes the axioms of the Tarski Grothendieck Set Theory. They are the axiom stating that everything is a Set, the extensionality axiom, http://www.cs.ualberta.ca/~piotr/Mizar/mirror/htdocs/JFM/Axiomatics/tarski.html

Journal of Formalized Mathematics Axiomatics, 1989 University of Bialystok Association of Mizar Users Tarski Grothendieck Set Theory Andrzej Trybulec Warsaw University Bialystok Supported by RPBP.III-24.B1. Summary. This is the first part of the axiomatics of the Mizar system. It includes the axioms of the Tarski Grothendieck set theory. They are: the axiom stating that everything is a set, the extensionality axiom, the definitional axiom of the singleton, the definitional axiom of the pair, the definitional axiom of the union of a family of sets, the definitional axiom of the boolean (the power set) of a set, the regularity axiom, the definitional axiom of the ordered pair, the Tarski's axiom~A introduced in [ ] (see also [ MML Identifier: TARSKI Contents Bibliography Received January 1, 1989 Download a postscript version MML identifier index Mizar home page

69. Set Theory: Blogs, Photos, Videos And More On Technorati When last we left off, I d used Set Theory to show how to construct the natural numbers; and then from the natural numbers, I showed how to construct the http://technorati.com/tag/set theory

Get the buzz on film, TV, music, and celebs now in Entertainment Join Sign in Popular ... Blogger Central 95 posts tagged set theory Subscribe Posts Blogs Videos Photos search in entire post tags only of blogs with any authority a little authority some authority a lot of authority in language all languages Arabic (العربية) Chinese (中文) Dutch (Nederlands) English French (Fran§ais) German (Deutsch) Greek (Ελληνικά) Hebrew (עברית) Italian (Italiano) Japanese (日本語) Korean (한국어) Norwegian (Norsk) Persian (فارسی) Polish (Polski) Portuguese (Portuguªs) Russian (Русский) Spanish (Espa±ol) Swedish (Svenska) Turkish (T¼rk§e) Vietnamese (Tiếng Việt) From the politics channel http://digbysblog.blogspot.com/ 2007/ 12/ dogwhistling-into-hell-by-digby.html Dogwhistling Into Hell by digby According to Steve Benen, Ron Paul doesn't believe in evolution. (Excuse me, Dr. Ron Paul, the obstetrician.) This means that four of the original GOP candidates (five if you count Alan Keyes) don't believe in evolution. But really, we shouldn't be surprised by this. ... 12 hours ago in Hullabaloo Tags: set theory Difficult Brain Teaser: Combinatorics Problem on Selection with Replacement Problem Statement: Assume that there are n items (numbered from 1 to n) in an urn. We select b items from the urn and record their numbers. We return the selected b items into the urn and perform another selection. We do in total m such selections. At the end of the m selections we check the recorded numbers.

70. An Elementary Introduction To Logic And Set Theory: Table Of Contents An Elementary Introduction to Logic and Set Theory. I. Overview II. Sentential Logic V. Naïve Set Theory Notions, Notations and Axioms http://faculty.matcmadison.edu/alehnen/weblogic/logcont.htm

An Elementary Introduction to Logic and Set Theory I. Overview II. Sentential Logic Propositions Negation ... Madison Area Technical College 3550 Anderson Street Madison, WI 53704 alehnen@madison.tec.wi.us

71. Set Theory: Should You Believe? ~ Stephen's Web ~ By Stephen Downes Set Theory Should You Believe? I have papers like this in my own notes from my university days. My doubt in mathematics was created when, one day, http://www.downes.ca/cgi-bin/page.cgi?post=42552

72. Venn Diagram How it works Students can move around the events A and B and see how the Sets and probabilities to the right change. The Set which is being shaded may be http://www.stat.tamu.edu/~west/applets/Venn.html

Venn Diagram Applet How it works: Students can move around the events A and B and see how the sets and probabilities to the right change. The set which is being shaded may be selected to the right as well. west@stat.sc.edu

73. Sets Welcome to the Set tutorial. To find out how to use this tutorial click here (Note by clicking that link, you will be opening a new window) http://www.geocities.com/basicmathsets/

Welcome to the set tutorial. To find out how to use this tutorial click here (Note: by clicking that link, you will be opening a new window) Below you will see a list of this unit's page headings and the subjects discussed in each of these pages. Click on any one of them in order to get started. GETTING STARTED WITH SETS What is a set? Elements Union ... Click here to practice your arithmetic skills. c2002 Martin Selditch