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1. OUP: UK General Catalogue
Nonstandard Models of arithmetic and set theory Other articles in the book present results related to nonstandard Models in arithmetic and set theory,

2. Bibliography: Set Theory With A Universal Set
in Nonstandard Models of arithmetic and set theory, (Enayat, A. and Kossak, R., eds.), Contemporary Mathematics, vol. 361, American Mathematical Society.
Bibliography: Set Theory with a Universal Set
This is a comprehensive bibliography on axiomatic set theories which have a universal set. (Zermelo-Fraenkel set theory, the most widely studied set theory, does not have a universal set.) This field presently includes three main areas of study: "New Foundations", a set theory devised by W. van Orman Quine , the positive set theory of Helen Skala , and model-based extensions of Zermelo-Fraenkel set theory, initiated by Alonzo Church . Recent papers by Holmes (and the original papers of Andrzej Kisielwicz) on "double extension set theory" are referenced in the main body of the bibliography but not under "recent work"; the jury is still out on this system (two versions of which have been shown to be inconsistent) but if the surviving version is consistent, it must be admitted that it is a set theory with universal set. For those unfamiliar with the field, two places to start are the New Foundations Home Page and Thomas Forster's book Set Theory with a Universal Set . A new option is afforded by the recent appearance of Holmes's elementary text Comments, corrections, and information about new publications should be sent to

3. JSTOR On Recursively Enumerable And Arithmetic Models Of Set Theory.
He points out that the Godel completeness theorem guarantees the existence of count able Models of set theory which are standard with respect to arithmetic<167:OREAAM>2.0.CO;2-1

4. 03: Mathematical Logic And Foundations
The implicit dependence on set theory and the inability to determine a Likewise, nonstandard Models of arithmetic open a branch of Number theory
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03: Mathematical logic and foundations
Mathematical Logic is the study of the processes used in mathematical deduction. The subject has origins in philosophy, and indeed it is only by nonmathematical argument that one can show the usual rules for inference and deduction (law of excluded middle; cut rule; etc.) are valid. It is also a legacy from philosophy that we can distinguish semantic reasoning ("what is true?") from syntactic reasoning ("what can be shown?"). The first leads to Model Theory, the second, to Proof Theory. Students encounter elementary (sentential) logic early in their mathematical training. This includes techniques using truth tables, symbolic logic with only "and", "or", and "not" in the language, and various equivalences among methods of proof (e.g. proof by contradiction is a proof of the contrapositive). This material includes somewhat deeper results such as the existence of disjunctive normal forms for statements. Also fairly straightforward is elementary first-order logic, which adds quantifiers ("for all" and "there exists") to the language. The corresponding normal form is prenex normal form. In second-order logic, the quantifiers are allowed to apply to relations and functions to subsets as well as elements of a set. (For example, the well-ordering axiom of the integers is a second-order statement). So how can we characterize the set of theorems for the theory? The theorems are defined in a purely procedural way, yet they should be related to those statements which are (semantically) "true", that is, statements which are valid in every model of those axioms. With a suitable (and reasonably natural) set of rules of inference, the two notions coincide for any theory in first-order logic: the Soundness Theorem assures that what is provable is true, and the Completeness Theorem assures that what is true is provable. It follows that the set of true first-order statements is effectively enumerable, and decidable: one can deduce in a finite number of steps whether or not such a statement follows from the axioms. So, for example, one could make a countable list of all statements which are true for all groups.

5. University Of Chicago Press - Cookie Absent
Euclidean set theory also yields a novel approach to non standard Models of arithmetic, related to work by Edward Nelson and Jan Mycielski (391).

6. Harvey Friedman
Working with Nonstandard Models, in Nonstandard Models of arithmetic and set theory, American Mathematical Society, ed. Enayat and Kossak, 7186, 2004.
Degrees and Employment History Distinctions
Others about Friedman ...
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Model Theory
  • Beth's Theorem in Cardinality Logics, Israel J. Math., Vol. 14, No. 2, (1973), pp. 205-212.
    Countable Models of Set Theories, Lecture Notes in Mathematics, Vol. 337, Springer-Verlag, (1973), pp. 539-573.
    On Existence Proofs of Hanf Numbers, J. of Symbolic Logic, Vol. 39, No. 2, (1974), pp. 318-324.
    Adding Propositional Connectives to Countable Infinitary Logic, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 77, No. 1, (1975), pp. 1-6.
    On Decidability of Equational Theories, J. of Pure and Applied Algebra, Vol. 7, (1976), pp. 1-3.
    The Complexity of Explicit Definitions, Advances in Mathematics, Vol. 20, No. 1, (1976), pp. 18-29.
    On the Naturalness of Definable Operations, Houston J. Math., Vol. 5, No. 3, (1979), pp. 325-330.
    (with L. Stanley), A Borel Reducibility Theory for Classes of Countable Structures, J. of Symbolic Logic, Vol. 54, No. 3, September 1989, pp. 894-914. (with Akos Seress), Decidability in Elementary Analysis I, Advances in Math., Vol. 76, No. 1, July 1989, pp. 94-115.
  • 7. MathNet-Mathematical Subject Classification
    03C62, Models of arithmetic and set theory See also 03Hxx. 03C64, Model theory of ordered structures; ominimality. 03C65, Models of other mathematical

    8. The Homepage Of The Helsinki Logic Group
    Taneli Huuskonen , docent, model theory, set theory, logic and analysis Juliette Kennedy, docent, Models of arithmetic, philosophy of mathematics
    The Helsinki Logic Group
    University of Helsinki
    Logiikan opetus

    Logic Colloquium 2003: Group photo and lecture materials Members Research Publications ... Contact Info
    Members - Research Publications Links Contact Info ... Aapo Halko , Ph.D., descriptive set theory Alex Hellsten , Ph.D., set theory Taneli Huuskonen , docent, model theory, set theory, logic and analysis Tapani Hyttinen , docent, stability theory, infinitary logic Juliette Kennedy , docent, models of arithmetic, philosophy of mathematics Meeri Kesälä , Ph.D., model theory Juha Kontinen , Ph.D., finite model theory Kerkko Luosto , docent, finite and infinite model theory, abstract model theory Juha Oikkonen , university lecturer, infinitary logic, nonstandard analysis Matti Pauna , Ph.D. Juha Ruokolainen , Ph.D. , professor, finite model theory, abstract model theory, set theory
    Ph.D. students:
    Tapio Eerola , M.Sc. , Ph.L. Jarmo Kontinen , M.Sc. Hannu Niemistö , Ph.L., finite model theory Ville Nurmi , M.Sc. Ryan Siders , M.Sc. Former members of the group can be found in the list of Ph.Ds

    9. Nonstandard Models Of Arithmetic And Set Theory : AMS Special Session Nonstandar
    Nonstandard Models of arithmetic and set theory AMS Special Session Nonstandard Models of arithmetic and set theory, January 1516, 2003, Baltimore,

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    Algebraic Geometry, Hirzebruch 70 : Proceedings of an Algebraic Geometry Conference in Honor of F. Hirzebruch's 70th Birthday, May 11-16, 1998, Stefan Banach International Mathematical Center Warsaw, Poland

    10. Richard Kaye's Publications
    On interpretations of arithmetic and set theory. By Richard Kaye and Tin Lok `Automorphisms of recursively saturated Models of arithmetic , by Kaye,
    Richard Kaye's research papers
    This page lists research papers (including some web or electronic papers). More substantial publications including books and other major web-based projects are listed on a separate page. Preprints or web-versions of papers are available for some documents listed here. Generally these are papers which haven't yet appeared yet. Some draft documents, or other papers which are not completed yet may be available by following this link email or write to me to see if I can help further. For some of these publications I have further notes and/or a list of errata. These are available in (La)TeX format or postscript format as indicated. Where the papers are only available in XHTML+MathML format, the instructions for setting up your browser are available HERE
  • Generic cuts in models of arithmetic To appear in Mathematical Logic Quarterly. Web resources, including a preliminary version of the paper and other addenda are available from this link. Algumas configura§µes do Minesweeper (Some Minesweeper Configurations) pp 181-189, Boletim Sociedade Portuguesea de Mathem¡tica, Janeiro 2007 (Nºmero especial), Lisbon. ISSN 0872-3672. (
  • 11. Abstracts 2006-2007
    14.0015.00 Kaye Interpretations of arithmetic and set theory 15.00-15.30 TEA 15.30-16.30 Engstrom Transplendent Models Omitting types in expansions
    September 29 Sebastiaan Terwijn : Intervals in the Medvedev lattice
    To main page October 13 Yuri Gurevich : Play to test
    What's testing? Well, from some point of view, it is a game. Testing tasks can be viewed (and organized!) as games against nature. We introduce and study reachability games. Such games are ubiquitous. A single industrial test suite may involve many instances of a reachability game; hence the importance of optimal or near optimal strategies for reachability games. We find out when exactly optimal strategies exist for a given reachability game, and how to construct them.
    To main page October 27 : Dependence Logic
    n 1961 Henkin suggested a game theoretic semantics for first order logic and its extension by so called partially ordered quantifiers. It remained an open problem whether this extension and other similar logics could be given a compositional semantics. In 1997 Wilfrid Hodges made a breakthrough in this area by giving a compositional semantics for these logics. In his semantics satisfaction is defined as a relation between formulas and sets of assignments, rather than as a relation between formulas and individual assignments, as is customary in first order logic.
    Based on this idea, we introduce Dependence Logic. This is an extension of first order logic, in which dependence of variables on each other is a basic atomic concept. We give an overview of this logic, its properties, and its applications, from database theory to set theory.

    12. 1st European Set Theory Meeting In Będlewo, July 9 - 13, 2007 | SPEAKERS
    B dlewo, July 9 13, 2007. 1st European set theory Meeting Washnigton, D.C., USA - set theory and Models of arithmetic - (abstract) (slides)
    1st European Set Theory Meeting Home Registration Participants ... Contact
    The following participants presented a talk in the conference :

    13. Group In Logic And The Methodology Of Science -
    Degree Complexity of Models of arithmetic and Connections with Influence of set theory on Model theory The Syntax/Semantics Tangle in set theory
    Past Colloquia
    Thomas Scanlon , Associate Professor of Mathematics, University of California, Berkeley John Krueger , Morrey Assistant Professor of Mathematics, University of California, Berkeley Some Results on Internal Approachability Sherrilyn Roush , Associate Professor of Philosophy, University of California, Berkeley Knowledge of Logical Truth Dana S. Scott , University Professor Emeritus, Carnegie Mellon University; Visiting Scholar in Logic and the Methodology of Science, University of California, Berkeley Duality in Projective Geometry Assaf Sharon , Visiting Assistant Professor of Mathematics, University of California, Irvine Some Consistency Results in Singular Cardinal Combinatorics Alexander Usvyatsov , Adjunct Assistant Professor of Mathematics and Fellow of the Logic Center, University of California, Los Angeles Model Theory of Metric Structures: An Overview Jose Ferreiros , Associate Professor of History and Philosophy of Science, University of Sevilla, Spain; Visiting Scholar in Philosophy, University of California, Berkeley Is the Concept of Set Intuitive?

    14. List KWIC DDC22 510 And MSC+ZDM E-N Lexical Connection
    set theory extremal 05D05 set theory fuzzy 03E72 set theory games involving topology or 91A44 set theory Models of arithmetic and 03C62
    series, etc.) # analytic approximation solutions (perturbation methods, asymptotic methods,
    series, etc.) # approximation to limiting values (summation of
    series, over-convergence # boundary behavior of power
    series, periods of modular forms, cohomology, modular symbols # special values of automorphic $L$-
    series, power series. convergence, summability (infinite products, integrals) # sequences and
    series, series of functions # power
    series, singular integrals # conjugate functions, conjugate
    series, summability # sequences, 40-XX
    series, transformations, transforms, operational calculus, etc. # analytical theory:
    series. convergence, summability (infinite products, integrals) # sequences and series, power
    series; Weil representation # theta Serre spectral sequences service # queues and sesquilinear, multilinear) # forms (blilinear, set (change of topology, comparison of topologies, lattices of topologies) # several topologies on one set algebra set algebra) # other classical set theory (including functions, relations, and set contractions, etc.) # nonexpansive mappings, and their generalizations (ultimately compact mappings, measures of noncompactness and condensing mappings, $A$-proper mappings, $K$-

    15. Alex M. McAllister's Home Page
    enumerations; Models and completions of Peano arithmetic, set theory and other theories with a certain richness ; Scott sets and weak Scott sets
    Alex M. McAllister
    Contact Info
    Teaching Research Publications ... Some Links Contact Information I am an Associate Professor of Mathematics and the Chair of the Mathematics Program at Centre College. I teach courses in the Mathematics, the Computer Science, and the Philosophy Program.
    I can be reached via e-mail at or in my office in 117 Olin Hall during my office hours . You can also reach me via any of the following:
    Math Department
    Centre College
    600 West Walnut Street
    Danville, KY 40422-1394 (office)
    (fax) (home)
    Teaching I love teaching - that's one of the main reasons came to Centre College. You can reach course web pages via the following links: Term Course Fall, 2006 MAT 170: Calculus I MAT 380: Real Analysis Spring, 2005 MAT 171: Calculus II MAT 407: Mathematical Logic I am also a big proponent of reading and, in particular, of reading mathematics textbooks. And not just the exercises in the text, but all the ideas and examples and definitions and theorems that come before the exercises. Some of my thoughts and reflections on this topics can be found at: Reading Your Mathematics Textbook.

    16. Wiley::Introduction To Modern Set Theory
    Cardinal arithmetic. Cofinality. Infinite Operations and More Exponentiation. Counting. TWO Models OF set theory. A set Model for ZFC.,descCd-tableOfCont
    United States Change Location

    17. - Nonstandard Models Of Arithmetic And Set Theory
    Features the proceedings of the AMS session on nonstandard Models of arithmetic and set theory held at the Joint Mathematics Meetings in Baltimore (MD).

    18. Transactions Of The American Mathematical Society
    in set theory of the Continuum (H. Judah, W. Just and H. Woodin, eds. I cofinal equivalence of Models of arithmetic, Notre Dame Journal of Formal

    ISSN 1088-6850(e) ISSN 0002-9947(p) Previous issue Table of contents Next issue
    Articles in press
    ... Next Article Ultrafilters on -their ideals and their cardinal characteristics Author(s):
    Journal: Trans. Amer. Math. Soc.
    MSC (1991): Primary 03E05, 03E35
    Posted: March 8, 1999
    Retrieve article in: PDF DVI PostScript
    This article is available free of charge Abstract References Similar articles Additional information Abstract: For a free ultrafilter on we study several cardinal characteristics which describe part of the combinatorial structure of . We provide various consistency results; e.g. we show how to force simultaneously many characters and many -characters. We also investigate two ideals on the Baire space naturally related to and calculate cardinal coefficients of these ideals in terms of cardinal characteristics of the underlying ultrafilter. References:
    B. Balcar and P. Simon, On minimal -character of points in extremally disconnected compact spaces

    Importacao de Publicacoes Tecnicas sob Demanda Livros Revista Normas - NONSTandARD Models OF arithmetic and set theory This is the proceedings of the AMS

    20. Springer Online Reference Works
    For the construction of Models of the set theory in which the negation of model theory is occupied by studies on nonstandard Models of arithmetic and

    Encyclopaedia of Mathematics
    Article referred from
    Article refers to
    Mathematical logic,
    symbolic logic The branch of mathematics concerned with the study of mathematical proofs and questions in the foundation of mathematics.
    Historical sketch.
    The idea of constructing a universal language for the whole of mathematics, and of the formalization of proofs on the basis of such a language, was suggested in the 17th century by G. Leibniz . But not until the middle of the 19th century did there appear the first scientific work on the algebraization of Aristotelean logic ( G. Boole , A. de Morgan ). After G. Frege ) and C. Peirce ) put the logic of predicates, variables and quantifiers into the language of algebra, it became possible to apply this language to questions in the foundations of mathematics. On the other hand, the creation of non-Euclidean geometry in the 19th century K. Weierstrass R. Dedekind and G. Cantor , and G. Peano ). In this connection, Peano created a more suitable symbolic representation for the language of logic. Afterwards, this language was perfected in the joint work of B. Russell

    21. The Consistency Of "P = NP" And Related Problems With Fragments Of Number Theory
    22 P.J. Cohen, set theory and the Continuum Hypothesis, Benjamin, 1966. 23 A. Ehrenfeucht and G. Kreisel, “Strong Models of arithmetic”, Mathematics and

    22. Nonstandard Models Of Arithmetic And Set Theory : AMS Special Session Nonstandar
    Nonstandard Models of arithmetic and set theory AMS Special Session Nonstandard Models of arithmetic and set theory, January 1516, 2003, Baltimore,
    Your Web browser is not enabled for Javascript. Some features of WorldCat will not be available. Home WorldCat Home About WorldCat Help ... Search for Contacts You are not signed in Sign In to WorldCat or Register Search for items: Advanced Search
    Nonstandard models of arithmetic and set theory : AMS Special Session Nonstandard Models of Arithmetic and Set Theory, January 15-16, 2003, Baltimore, Maryland
    by Ali Enayat Roman Kossak
    Type: Book Language: English Publisher: Editions: 2 Editions ISBN: OCLC: Related Subjects: Set theory. Ensembles, Théorie des. Citations: Cite this Item Export to EndNote Export to RefWorks
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    There will be a more serious use of set theory than needed for model to study formal systems of arithmetic, including primitive recursive arithmetic,
    CARNEGIE MELLON UNIVERSITY PROGRAM IN PURE AND APPLIED LOGIC LOGIC AND LOGIC-RELATED COURSES AND SEMINARS FOR FALL 2004 21-600 Mathematical Logic I Instructor: Peter Andrews MWF 11:30am-12:20pm Baker Hall A53 12 Units Description: The study of formal logical systems which model the reasoning of mathematics, scientific disciplines, and everyday discourse. Propositional calculus and first-order logic. Syntax, axiomatic treatment, derived rules of inference, proof techniques, computer-assisted formal proofs, normal forms, consistency, independence, semantics, soundness, completeness, the Lowenheim-Skolem Theorem, compactness, equality. 21-602 Set theory I INSTRUCTOR: Ernest Schimmerling TuTh 3:00-4:20 pm OSC 201 12 units DESCRIPTION: First semester graduate level set theory. The main topics are ZFC, infinitary combinatorics, relative consistency, constructibility, and descriptive set theory. TEXTBOOK: Kenneth Kunen, "Set Theory : An Introduction to Independence Proofs" COMMENT: Students should have a background in undergraduate level set theory (e.g., 21-229) and logic, which includes a working knowledge of basic ordinal and cardinal arithmetic, G¶del's completeness theorem, and the downward L¶wenheim-Skolem theorem. An understanding of the statement (but not the proof) of G¶del's theorem on consistency proofs will also be assumed. Those without the required background should meet with the instructor as soon as possible to discuss their options, which may include doing some reading over the summer. Set Theory I is a prerequisite for Set Theory II (21-702), which will be taught by Professor Uri Abraham in Spring, 2005. 80-310/610 Logic and Computation Instructor: Jeremy Avigad TuTh 3:00-4:20 pm Hamerschlag Hall B103 12 units Description: Among the most significant developments in logic in the twentieth century is the formal analysis of the notions of provability and semantic consequence. For first-order logic, the two are related by the soundness and completeness theorems: a sentence is provable if and only if it is true in every interpretation. This course begins with a formal description of first-order logic, and proofs of the soundness and completeness theorems. Other topics may include: compactness, the Lowenheim-Skolem theorems, nonstandard models of arithmetic, definability, other logics, and automated deduction. Prerequisites: 80-210, 80-211, or equivalent background in first-order logic. 80-315/615 Modal Logic Instructor: Horacio Arlo-Costa Wed 1:30-3:50 pm Baker Hall 150 9-12 units. Description: An introduction to first-order modal logic. The course considers several modalities aside from the so-called alethic ones (necessity, possibility). Epistemic, temporal or deontic modalities are studied, as well as computationally motivated modals (like "after the computation terminates"). Several conceptual problems in formal ontology that motivated the field are reviewed, as well as more recent applications in computer science and linguistics. Kripke models are used throughout the course, but we also study recent Kripkean-style systematizations of the modals without using possible worlds. Special attention is devoted to Scott-Montague models of the socalled "classical" modalities. Prerequisites: 80-210, or 80-211, or instructor's permission. 1-703 Model theory II INSTRUCTOR: Rami Grossberg MWF 10:30 Baker Hall 231A 12 units DESCRIPTION: This is a second course in model theory. The main topic of discussion will be classification theory for non-elemntary classes. TEXTBOOK: No official text. DETAILS: I will concentrate in what is the deepest part of pure model theory. Namely non-first order theories. In a typical case we will deal with abstract elementary classes. An AEC is essentially a class K of models all of the same similarity type (or a category of sets) which is closed under direct limits and little more. The aim is to have an analysis of such general classes. Most of the material to be discussed appears in (badly written) papers only. I will start with minimal prerequisites, but will progress quickly to some of the research frontieers of the field. I will emphasize aspects of the theory that may eventually converge to a proof cases of Shelah's categoricity conjecture which is the prominent open problem in the field, it is a parallel to Morley's theorem for L ?1,?, most results will be about more general classes. The common to all these classes is that the compactness theorem fails badly. Hopefully some of the techniques will turn to be usefull also in the study of classes of finite models, but we will concentrate at uncountable models. There will be a more serious use of set theory than needed for model theory of first-order logic. PREREQUISITES: About half of a basic graduate course in set theory and parts of an elementary model theory (about 60-70% of 21-603 ) or permission of the instructor. 80-411/711 Proof Theory Instructor: Jeremy Avigad TuTh 10:30-11:50 am Baker Hall 231A 12 units Description: This course is an introduction to Hilbert-style proof theory, where the goal is to represent mathematical arguments using formal deductive systems, and study those systems in syntactic, constructive, computational, or otherwise explicit terms. In the first part of the course, we will study various types of deductive systems (axiomatic systems, natural deduction, and sequent calculi) for classical, intuitionistic, and minimal logic. We will prove Gentzen's cut-elimination theorem, and use it to prove various theorems about first-order logic, including Herbrand's theorem, the interpolation theorem, the conservativity of Skolem axioms, and the existence and disjunction properties for intuitionistic logic. In the second part of the course, we will use these tools to study formal systems of arithmetic, including primitive recursive arithmetic, Peano arithmetic, and subsystems of second-order arithmetic. In particular, we will try to understand how mathematics can be formalized in these theories, and what types of information can be extracted using metamathematical techniques. A solid understanding of the syntax and semantics of first-order logic, as obtained from courses like 80-310/610 or 21-300/600, is required. A course covering issues topics like primitive recursion and coding, like 80-311/611 or 21-700, would be helpful, but is not essential. 15-814 Type Systems for Programming Languages INSTRUCTOR: Robert Harper TuTh 1:30-2:50pm Wean Hall 5409 12 units. DESCRIPTION: This course is an introduction to the theory and practice of type systems for programming languages. Topics include typed lambda calculus, subtyping, polymorphism, data abstraction, recursive types, and objects. PREREQUISITE: Background equivalent to programming skills and programming language exposure afforded by a typical undergraduate Computer Science degree. TEXTBOOK: Benjamin Pierce "Types and Programming Languages" MIT Press 2002. COMMENT: This course satisfies the CS distribution requirement in programming languages. Enrollment is limited to CS PhD students or permission of the instructor.

    24. Logic, Set Theory And Arithmetic (
    Logic, set theory and arithmetic. Show printerfriendly view A model checking approach to query evaluation on XML documents Reasoning and computing
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  • Integrated Security and Privacy in a Networked World (ISTRICE)
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  • Combinatorial optimization, combinatorial algorithms and graph theory
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  • Combinatorial Optimization Current research projects associated with this classification: (the most recent research is placed on top)
  • G¶del’s Philosophy of Mathematics
  • Elections and Coalition Formation
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  • Development of advanced methods for stochastic model validation in satellite gravity modelling ...
  • Arithmetic geometry, motives: computational aspects
  • 25. Set Theory & The Euclidean Model
    set theory Euclidean Model for the codification of mathematics and this in turn implies a set theory basis for arithmetic with whole numbers,
    read math-free logic chapters 1 to 5 Français
    Volume 1A, Pattern Based Reason
    Striving for objectivity, not subjectivity, a reference for critical thinking, law, science, engineering.
    Définition d'une variable
    Algèbre Arithmetique Logique
    14 Set Theory
    Road Safety Message Book Entrance Up Next Study With Others: twiddle this email the site author for an appointment. No charge for inquiries or first session. Start whenever you have a problem. References OnlineVolumes
    Elements of Reason.
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    Pattern Based Reason

    - striving for objectivity, etc Math Curriculum Notes inductive principles etc Three Skills for Algebra - unifying themes + study skills Read Volumes 2 and 3 if you are in or heading for calculus. More Site Areas Help Your Child or Teen Learn Fraction Skills - Sec I to V level Fractions Ratios Rates Proportions Units Euclidean Geometry - Sec IV Analytic Geometry Number Theory More Calculus Complex Numbers Sec II to VI 9. Qc Maths Education Secondary IV(?) math

    26. Peano Axioms - Wikipedia, The Free Encyclopedia
    3.1 Firstorder theory of arithmetic; 3.2 Equivalent axiomatizations; 3.3 Nonstandard Models; 3.4 set-theoretic Models; 3.5 Interpretation in category
    var wgNotice = ""; var wgNoticeLocal = ""; var wgNoticeLang = "en"; var wgNoticeProject = "wikipedia";
    Peano axioms
    From Wikipedia, the free encyclopedia
    (Redirected from Peano arithmetic Jump to: navigation search In mathematical logic , the Peano axioms , also known as the Dedekind-Peano axioms or the Peano postulates , are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano . These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of consistency and completeness of number theory The need for formalism in arithmetic was not well appreciated until the work of Hermann Grassmann , who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction In 1888, Richard Dedekind proposed a collection of axioms about the numbers, and in 1889 Peano published a more precisely formulated version of them as a collection of axioms in his book, The principles of arithmetic presented by a new method Latin rithmetices principa, nova methodo exposita

    27. Instytut Matematyczny PAN - Podstawy Matematyki
    His research is on the border of model theory (of arithmetic) and proof theory. include model theory, its interactions with set theory and algebra,
    Instytut Matematyczny Polskiej Akademii Nauk ENGLISH IMPAN O nas Zak³ady Podstawy Matematyki
    Zak³ad Podstaw Matematyki
    • prof. dr hab. Ryszard Frankiewicz (profesor)
      dr hab. Adam Obtu³owicz (docent)

      pok. 424, tel.: 022 5228 235 prof. dr hab. Czes³aw Ryll-Nardzewski (profesor)

      tel.: 071 320 21 10 mgr Konrad Zdanowski (asystent)

      610a, tel.: 022 5228 224
    O Zak³adzie
    Research of the Section involves a rather wide spectrum of matters which are connected with the foundations of mathematics, such as set theory, elements of functional analysis, elements of real analysis, foundations of arithmetic, cathegorical logic.
    Zofia Adamowicz
    Zofia Adamowicz is working in foundations of arithmetic. Her main research is in "bounded arithmetic" together with its links to computational complexity and the P=NP problem. Recent research concerns the power of the exponential function in arithmetic, e.g. its influence on the existence of end extensions of models.
    Main recent papers:
    • conservativeness problem , Ann. Pure Appl. Logic 61 (1993), 3-48.

    28. The Math Forum - Math Library - Set Theory
    Contents include Platonism, intuition and the nature of mathematics; Axiomatic set theory; First order arithmetic; Hilbert s Tenth problem;
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    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

    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
  • 29. Logic And Computation
    Peano arithmetic; Skolem s Nonstandard Model for arithmetic; Gödel s First Incompleteness Theorem. ZermeloFraenkel set theory. Zermelo-Fraenkel set theory
    Module 3LC for third year mathematics MAM3.
    Vasco Brattka
    University of Cape Town
    Time Table (2007)
    The lectures take place in at (fourth period) on Tuesdays Fridays and the following Wednesdays
    • 28 Feb, 14 Mar, 18 Apr, 25 Apr, 2 May, 16 May.
    The tutorial takes place in ZOO 3 at (sixth period), each Tuesday
    Course Description
    In particular, we will cover the following topics:
    • distinction between syntax and semantics,
    • propositional logic,
    • first-order logic,
    • theories and models,
    • arithmetic.
    Part A: Propositional Logic
  • Introduction
    • Introduction
    • Historical Background
    • Aristotle's Analytics
    • Hilbert's Program
    • Berry's Paradox
    • Syntax and Semantics
    • Different Logics
  • Syntax of Propositional Logic
    • Propositional Logic
    • Symbols of Propositional Logic
    • Syntax of Propositional Logic
    • Formation Trees
    • Recursive Definitions and Structural Induction
  • Semantics of Propositional Logic
    • Semantics of Propositional Logic
    • Truth Tables
    • Evaluation of Formulas with Trees
    • Satisfiability and Tautologies
    • Coincidence Lemma
    • The Truth Table Method
  • Logical Implication and Equivalence
    • Logical Implication and Equivalence
    • Syntactical Implications and Logical Consequences
    • Realizability of Boolean Functions
    • Disjunctive and Conjunctive Normal Form
    • Complete Sets of Connectives
    • Digital Circuits
  • The Compactness Theorem
    • Satisfiability and Logical Consequences of Sets of Formulas
    • The Compactness Theorem
  • Computability Notions for Subsets
    • Notions of Computability
    • Computability and Computable Enumerability
    • The Truth Table Method
  • 30. Atlas: On An Arithmetic In A Set Theory Within Lukasiewicz Logic By Shunsuke Yat
    A significance of the set theory with the comprehension principle is to The arithmetic in H is somehow similar to one in nonstandard Models of PA.
    August 5-9, 2007
    St Anne's College, University of Oxford
    Oxford, England Organizers
    Mai Gehrke and Hilary Priestley View Abstracts
    Conference Homepage
    On an arithmetic in a set theory within ukasiewicz logic
    Shunsuke Yatabe
    Kobe University
    On an arithmetic in a set theory within ukasiewicz logic
    Shunsuke Yatabe
    A significance of the set theory with the comprehension principle is to allow a general form of the recursive definition []: For any formula j j (x, ..., z)] within CFLew , i.e. we can define a set z by using a parameter z itself. This allows us to represent any partial recursive function on w Let H is a set theory with the comprehension principle within ukasiewicz infinite-valued predicate logic with its standard semantics. It has been conjectured that H is enough strong to develop an arithmetic because the recursive definition on w can be used in place of mathematical induction: We review about this. The arithmetic in H is somehow similar to one in "non-standard models" of PA . For example, we can prove an

    31. CiteULike: Consequences Of Arithmetic For Set Theory
    author = {Halbeisen, L. and Shelah, S. }, citeulikearticle-id = {71358}, keywords = {arithmetic, set, theory}, priority = {4}, title = {Consequences of
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      In this paper, we consider certain cardinals in ZF #set theory without AC, the Axiom of Choice#. In ZFC #set theory with AC#, given any cardinals C and D; either C # D or D # C: However, in ZF this is no longer so. For a given in#nite set A consider seq 1 1 #A#, the set of all sequences of A without repetition. We compare seq 1 1 #A# , the cardinality of this set, to P#A# , the cardinality of the power set of A. What is provable about these two cardinals in ZF? The main result of this...
      Note: You may cite this page as:

    32. EMail Msg <>
    To say these notions are definable_1 in set theory is to say that the relevant All nonstandard Models of firstorder arithmetic start with an initial
    Subject: Re: International STANDARD FOR LOGIC: CSMF/CG/KIF Message-id: Date: Fri, 9 Sep 1994 01:51:35 -0500 Sender: Precedence: bulk

    33. Homepage: Alex Wilkie
    On the theory of endextensions of Models of arithmetic,inSet theory and Hierarchy theory V,SLNM 619,Springer-Verlag,1997,305-310.
    The Homepage of Prof. Alex Wilkie FRS
    Personal Information
    Full name: Alex Wilkie Affiliation: Mathematical Institute University of Oxford Address: 24-29 St Giles Phone: Oxford, OX1 3LB Fax: United Kingdom email:
    1. On models of arithmetic-answers to two problems raised by H. Gaifman, J Symb Logic,40(1975)(1),41-47. 2. A note on products of finite structures with an application to graphs, J Lond Math Soc (2),14(1976),383-384. 3. On the theory of end-extensions of models of arithmetic,in:Set Theory and Hierarchy Theory V,SLNM 619,Springer-Verlag,1997,305-310. 4. On models of arithmetic having non-modular substructure lattices,Fund. Math.,XCV(1977),223-237. 5. Reconstruction theorems for families of sets (with R Rado),J Lond Math Soc (2),17(1978),5-9. 6. Applications of complexity theory to sigma-zero definability problems in arithmetic,in:Model Theory of Algebra and Arithmetic,SLNM 834, Springer-Verlag,1980,363-369. 7. Some results and problems on weak systems of arithmetic,in:Logic Colloquium '77,North-Holland,1980,285-296.

    34. "Reliable Computing" Special Issues: Calls For Papers
    The connection between interval mathematics and fuzzy set theory is evident in the extension principle, arithmetic, logic, and in the mathematics of
    Reliable Computing
    (formerly Interval Computations
    an International Journal
    Forthcoming Special Issues: Call for Papers
    Previous Special Issues: Call for Papers
    Call for Papers
    Special Issue on the Linkages Between Interval Mathematics and Fuzzy Set Theory
    Guest editor: Weldon A. Lodwick Reliable Computing will devote a special issue to papers that address the interrelationship between interval mathematics and fuzzy set theory. The connection between interval mathematics and fuzzy set theory is evident in the extension principle, arithmetic, logic, and in the mathematics of uncertainty. Much of the research to date has been in the use of interval mathematics in fuzzy set theory, in particular fuzzy arithmetic and fuzzy interval analysis. This may be because intervals can be considered as a particular type of fuzzy set. The impact of fuzzy set theory on interval mathematics is not quite as evident. For example, it is clear that fuzzy logic, fuzzy control, fuzzy neural networks, and fuzzy cluster analysis, are four important areas of fuzzy set theory. The impact of interval analysis on these four areas is not as apparent. Can the development in these areas of fuzzy set theory inform research in interval mathematics?

    35. FOM: Urbana Thoughts; Model Theory; Spirit Of Generosity?
    Among the unfashionable topics are infinitary logics, generalized quantifiers, twocardinal theorems, Models of set theory, Models of arithmetic
    FOM: Urbana thoughts; model theory; spirit of generosity?
    Stephen G Simpson simpson at
    Wed Jun 14 20:25:33 EDT 2000 More information about the FOM mailing list

    36. Arbeitsgruppe Mathematische Logik | Main / Set Theory Browse
    The research areas of modern set theory are. The theory of ZFC; Large Cardinals; Inner Models and Fine Structure; Descriptive set theory; Forcing
    Arbeitsgruppe Mathematische Logik
    Set Theory at the University of Munich
    Research Interests
    • Inner Models and Fine Structure Infinite Combinatorics Equiconsistency Results Large Cardinals
    Set Theory was invented by Georg Cantor (1845 - 1918), and revolutionized mathematics. It's main theme is infinity. The research areas of modern set theory are:
  • The theory of ZFC Large Cardinals Inner Models and Fine Structure Descriptive Set Theory Forcing Infinite Combinatorics
  • Set Theory provides an universal framework in which all of mathematics can be interpreted. There is no competing theory in that respect. A well-known formulation of the basic set theoretic principles is given by the axiomatic system ZFC of Ernst Zermelo and Abraham Fraenkel, formalized in first order logic (the C denotes the axiom of choice).
    ZFC, however, does not decide the size of the reals: Cantor's Continuum Hypothesis (CH) is independent of ZFC (Goedel 1938, Cohen 1963). This result marks the beginning of modern Set Theory.
    Not only (CH) but many other propositions were shown to be independent. The methods used to prove independence are inner models - natural realizations of the axioms, the most prominent being Goedel's constructible universe L - and forcing - Cohen's method to enlarge given models of set theory keeping control of the new sets.

    37. Learning Restricted Models Of Arithmetic Circuits: Theory Of Computing: An Open
    theory of Computing, An Open Access Journal (2000) can be used to learn depth3 set-multilinear arithmetic circuits. Previously only versions of depth-2 ISSN 1557-2862
    Volume 2 (2006) Article 10 pp. 185-206
    Learning Restricted Models of Arithmetic Circuits
    by Adam Klivans, Amir Shpilka Received: August 31, 2005
    Published: September 28, 2006 Download article from ToC site:
    [PS (400K)]
    [PS.GZ (104K)] [PS.ZIP (105K)]
    [PDF (229K)]
    ... [TeX (57K)] Misc.:
    [HTML Bibliography]
    [Bibliography Source] [BibTeX]
    [About the Author(s)]
    ... [Comments, updates] Keywords: learning, exact learning, arithmetic circuit, partial derivative, multiplicity automata
    ACM Classification: I.2.6, F.2.2
    AMS Classification:
    [Plain Text Version] We present a polynomial time algorithm for learning a large class of algebraic models of computation. We show that any arithmetic circuit whose partial derivatives induce a low-dimensional vector space is exactly learnable from membership and equivalence queries. As a consequence, we obtain polynomial-time algorithms for learning restricted algebraic branching programs as well as noncommutative set-multilinear arithmetic formulae. In addition, we observe that the algorithms of Bergadano et al. (1996) and Beimel et al. (2000) can be used to learn depth-3 set-multilinear arithmetic circuits. Previously only versions of depth-2 arithmetic circuits were known to be learnable in polynomial time. Our learning algorithms can be viewed as solving a generalization of the well known polynomial interpolation problem where the unknown polynomial has a succinct representation. We can learn representations of polynomials encoding

    38. MPLA :: Graduate Program In Logic, Algorithms And Computation
    E. Koutsoupias; 6. set theory Y. Moschovakis; 6. Databases F. Afrati; 12. Model theory C. Dimitracopoulos; 02 . arithmetic Complexity
    Graduate Program in Logic, Algorithms and Computation. Old website
    Full courses archive
    By Semester
    • C. Dimitracopoulos Y. Moschovakis E. Koutsoupias Ì11. Recursion Theory Y. Moschovakis

    39. School Of Mathematics
    Peano arithmetic and Goedel numbering. Goedel s first incompleteness theorem. In Gödel s method one would take a model of set theory (much harder to
    School of Mathematics
    Course 371 - Computability, logic, and set theory
    Duration: 21 weeks (54 lectures + tutorials)
    Number of lectures per week:
    Regular homeworks and final exam
    End-of-year Examination: One 3-hour examination - end of year Description: Peano Arithmetic - axioms for N . Resolution principle for propositional logic. Complete axiom system for propositional logic. Predicate logic, models, and completeness. Axioms for equality. Turing machines and partial recursive functions. Peano arithmetic and Goedel numbering. Goedel's first incompleteness theorem. Goedel-Rosser theorem Hilbert-Bernays derivability conditions. Goedel's second incompleteness theorem. Further analysis of Goedel's second theorem. Goedel's First theorem and partial recursive functions. ZF set theory. Ordinals. Foundation axiom and its relative consistency. Cardinals, the Axiom of choice, and the General Continuum Hypothesis. The constructible universe. Relative consistency of V=L. V=L implies AC. V=L implies GCH. Additional notes. A considerable advance in nineteenth-century mathematics was the introduction of rigour to suspect areas of analysis. The notion of `real number,' for example, can now be defined in terms of Cauchy sequence or Dedekind cut. Both of these are generally acceptable reductions of the intuitive continuum of real numbers to sets of sets or sequences of rational numbers.

    40. Model Theory. Goedel's Completeness Theorem. Skolem's Paradox. Ramsey's Theorem.
    Model theory is using the full power of set theory. Its results and proofs can be .. This model is called traditionally the standard model of arithmetic.
    model theory, Skolem paradox, Ramsey theorem, Loewenheim, categorical, Ramsey, Skolem, Gödel, completeness theorem, categoricity, Goedel, theorem, completeness, Godel Back to title page Left Adjust your browser window Right
    Appendix 1. About Model Theory
    Some widespread Platonist superstitions were derived from other important results of mathematical logic (omitted in the main text of this book): Goedel's completeness theorem for predicate calculus, Loewenheim-Skolem theorem, the categoricity theorem of second order Peano axioms. In this short Appendix I will discuss these results and their methodological consequences (or lack of them). All these results have been obtained by means of the so-called model theory . This is a very specific approach to investigation of formal theories as mathematical objects. Model theory is using the full power of set theory. Its results and proofs can be formalized in the set theory ZFC Model theory is investigation of formal theories in the metatheory ZFC. The main structures of model theory are interpretations . Let L be the language of some (first order) formal theory containing constant letters c , ..., c

    41. Skolem (print-only)
    He made refinements to Zermelo s axiomatic set theory, publishing work in 1922 and work in metalogic and constructed a nonstandard model of arithmetic.
    Thoralf Albert Skolem
    Born: 23 May 1887 in Sandsvaer, Buskerud, Norway
    Died: 23 March 1963 in Oslo, Norway
    Thoralf Skolem 's parents were Helene Olette Vaal and Even Skolem, who was primary school teacher. Although his father was a teacher, Thoralf came from a farming family with most of his relations being farmers. He attended secondary school taking the final examination, the Examen artium, in Kristiania (later renamed Oslo) in 1905. He then entered Kristiania University to study mathematics, but he also took courses on physics, chemistry, zoology and botany. In 1909 Skolem took a job as assistant to the physicist Kristian Birkeland, who was famed for his experiments with the aurora-like effect obtained by bombarding a magnetized sphere with electrons, and Skolem's first publications were physics papers written jointly with Birkeland. Skolem took his state examination in 1913, passing with distinction. His dissertation Undersokelser innenfor logikkens algebra ... Viggo Brun and Skolem agreed that neither of them would bother to obtain the degree of Doctor, probably feeling that, in Norway, it served no useful function in the education of a young scientist. Skolem became a Docent in Mathematics in Kristiania in 1918, and in the same year he was elected to the Norwegian Academy of Science and Letters. Despite his earlier agreement with Viggo Brun, he decided to submit a thesis for a doctorate in 1926 [5]:-

    42. Ralf Schindler's Home Page
    set theory meeting at Oberwolfach, Dec. 99, The core model for almost linear Review of Cardinal arithmetic by M. Holz, K. Steffens, and E. Weitz for
    My son's online diary
    LIST OF OPEN PROBLEMS IN INNER MODEL THEORY , edited jointly with John Steel , which comes with a Bibliography of inner model theory
    Ralf Schindler
    Professor (C4)
    Fachbereich Mathematik und Informatik
    Einsteinstr. 62
    e-mail: rds at math dot uni-muenster dot de (with the "at" replaced by "@," and each occurence of "dot" replaced by a dot)
    phone: +49-251-83-33790 or +49-251-83-33761
    fax: +49-251-83-33078
    : 2; Research interest: set theory.
    Sprechstunde: Di 13-14 (Zi. 804)
    Contents: Publications Notes and preprints Editorial and other activities Ph.D. students ... scripts
    See also here or here Books:
  • (with John Steel) The core model induction PS PDF
    Research papers:
  • Erkenntnis
  • A Dilemma in the Philosophy of Set Theory, Notre Dame Journal of Formal Logic
  • Weak Covering at Large Cardinals, Mathematical Logic Quarterly
  • On a Chang Conjecture, Israel Journal of Mathematics
  • The core model up to one strong cardinal, Bonner Mathematische Schriften Nr. 295 (1997).
  • On a Chang Conjecture II, Archive for Mathematical Logic
  • Successive weakly compact or singular cardinals
  • 43. Set Theory And Topology
    allows one to solve settheoretic problems with natural number arithmetic. One rarely uses this model to solve natural number problems using set theory.

    44. LoFrm For Relations
    Since arithmetic and the other branches of mathematics can be derived from set theory, it is possible to explain ontologically how mathematics is true by

    45. Department Of Mathematics, University Of Illinois At Urbana-Champaign
    (3) arithmetic complexity simple ideas from model theory and diophantine Rosendal works on descriptive set theory and its interactions with Banach
    Logic: Research
    Some Specific Projects Research Support, Collaborations and Exchange Programs Research done by members of our group tends to be centered in model theory and its applications and in descriptive set theory. There are also research projects involving other members of the Mathematics Department, especially in nonstandard analysis and in aspects of group theory that have significant connections to logic. Much of this work emphasizes the interactions between logic and other parts of mathematics. Recent successes based on these connections demonstrate that mathematical logic can provide powerful new methods for many areas of mathematics and can be the basis for breakthroughs on critical problems. Logic typically comes into the picture in two ways: (a) it provides a new set of tools; (b) it provides a new language within which to formulate results and problems. Some Specific Projects These brief descriptions give representative examples of the research projects that members of our group are involved in. Van den Dries has in recent years pursued research on three fronts:
    (1) O-minimal structures on the real field, with connections to real analytic geometry and geometric measure theory.

    46. Logic Seminar - Archive
    Mechanical proofs, combinatorial logic, and Quine s set theory November 27, 1998 at 10.oo Abstract dimension theory in model theory and in arithmetic
    Logic seminar: archive 1995 - 2006
  • S. Riis (Aarhus)
    Symmetrical equations -> symmetrical solutions?
  • A. Woods (Australia)
    Random finite functions standard and nonstandard

    January 4, 1996
  • J. Hruska (Charles University)
    Open-point a Open-open hry na regularnich topologickych prostorech a jejich ekvivalenty na Booleovskych algebrach
    March 25, 1996
  • Dan Willard (SUNY, Albany)
    Introspective Semantics for Turing Machine Languages

    March 4, 1996
  • Yuri Gurevich (Ann Arbor) Finite model theory June 17, 1996
  • Stevo Todorcevic (Toronto) Borel chromatic numbers July 10, 1996
  • L. van den Dries (Urbana) O-minimal extensions of R and quantifier elimination with Exp July 15, 1996
  • A. Sochor (MU) Nestandartnost nadmodelu September 16, 1996
  • A. Sochor (MU) Nestandartnost nadmodelu II. September 30, 1996
  • A. Sochor (MU) Nestandartnost nadmodelu III. October 7, 1996
  • P. Pudlak (MU) O Dawkingskove knize "Selfish gene" October 21, 1996
  • J. Krajicek (MU) Forcing v jazyce teorie miry November 11, 1996
  • J. Krajicek (MU) Spodni odhad pro polynomialni kalkulus podle A.A.Razborova December 2, 1996
  • 47. Mathematics Archives - Topics In Mathematics - Logic & Set Theory
    Axiomatic set theory, First order arithmetic, Hilbert s Tenth problem, Incompleteness theorems, Around Goedel s theorem, About model theory
    Topics in Mathematics

    48. Set Theory And Its Neighbours, Seventh Meeting
    The nineth oneday conference in the in the series set theory and its neighbours, Abstract Finite model theory has strong connections with a number of
    Set theory and its neighbours , nineth meeting:
    The nineth one-day conference in the in the series Set theory and its neighbours , took place on Wednesday, 25th April 2001 at the London Mathematical Society building, De Morgan House, 57-58 Russell Square, London WC1. The speakers at the meeting were:
    • Russell Barker (Oxford), Robinson-type relations and the relationship between the k-size and cardinality of finite structures
        In this talk I will introduce the notions of L^k, the restriction of first order logic to k-variables, the k-size of a model and, two conjectures proposed by Anuj Dawar. Then I shall define define a special kind of relation which I shall call a Robinson-type relation and prove some results about these relations. I shall go on to give a translation between these relations and the set of L^3 theories and then use the earlier results to disprove Dawar's second conjecture.
    • (UEA), Combinatorial principles that follow from GCH-like cardinal arithmetic assumptions
        Abstract: We discuss various results showing that at certain cardinals diamond-like principles follow just from local GCH-like assumptions on cardinal arithmetic.
    • Peter Koepke (Bonn)

    49. Logic Colloquium 2007: Contributed Talk Schedule
    Topic, set theory, Proof theory, Computable Model theory, Modal Logic Provably Recursive Functions in Extensions of a Predicative arithmetic

    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):
    -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:

    50. Pete L. Clark's Papers
    These notes, mostly written after I attended the 2003 Arizona Winter School on model theory and arithmetic, give a sort of introduction to the model theory
    Pete L. Clark
    Return to Pete's home page
    Shimura Curves
      Notes from my 2005 ISM course:
      These are all the notes I typed up for my 2005 ISM course on Shimura varieties. In the lectures, I presented more material on Hilbert and Siegel modular varieties, adelic double coset constructions, and strong approximation than has survived in the lecture notes. Most of the omitted material is of a rather standard sort it appears in many places which is not to say that it shouldn't appear here as well. The reader will notice that the notes are significantly more polished at the beginning and the end than in the middle. I am quite pleased with the very last lecture, which seems to put some of the pieces of the theory together in a new way. I would like to see more detail on arithmetic groups and lots more detail on quaternion orders and trace formulas. Inevitably for notes of this length, the most important results like the existence of rational and integral canonical models get stated and kicked around a bit but not proved. To remedy this will require significantly more work.
    • Lecture 0: Modular curves. (

    51. Thoralf Albert Skolem (Norwegian Logician) -- Britannica Online Encyclopedia
    infinitesimals, logic, recursive function theory, set theory. gave an explicit construction of what is now called a nonstandard model of arithmetic,
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    Thoralf Albert Skolem (Norwegian logician)
    A selection of articles discussing this topic.
    The infinitesimal i
    recursive function theory
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    52. The Journal Of Symbolic Logic, Volume 50
    468475 BibTeX Ali Enayat Weakly Compact Cardinals in Models of set theory. 476-486 BibTeX John K. Slaney 3088 Varieties A Solution to the Ackermann
    The Journal of Symbolic Logic , Volume 50
    Volume 50, Number 1, March 1985

    53. Gödel’s Theorems (PRIME)
    If this contradiction stands then arithmetic is inconsistent. . Like Gödel, he built a model of set theory, but unlike in the previous case, this model
    Basic Math
    Biography Calculus Comp Sci Discrete Economics Foundations Geometry Graph Thry History Number Thry Physics Statistics Topology Trigonometry logic and beyond, this result is only the middle movement, so to speak, of a metamathematical symphony of results stretching from 1929 through 1937. These results are: (1) the Completeness Theorem; (2) the First and Second Incompleteness Theorems; and (3) the consistency of the Generalized Continuum Hypothesis (GCH) and the Axiom of Choice (AC) with the other axioms of Zermelo-Fraenkel set theory . These results are discussed in detail below. THE COMPLETENESS THEOREM (1929) In 1928, David Hilbert and Wilhelm Ackermann published , a slender but potent text on the foundations of logic. In this text they posed the question of whether a certain system of axioms for the first-order predicate calculus is complete, i.e., whether every logically valid sentence in first-order logic can be derived from the

    54. Book's Contents
    5.2 Cardinal arithmetic, 68. 5.3 Cofinality, 74. III, The Power of recursive definitions, 77 9.5 Model for MA+nonCH, 196. A, Axioms of set theory, 211
    Set Theory for the Working Mathematician
    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

    55. Is Mathematics Consistent?
    For example, Peano arithmetic has a model (e.g. in set theory), so we call it consistent. The problem is that set theory is too comprehensive to be

    56. Faculty At The Mathematics Ph.D. Program At CUNY
    Kossak, Roman mathematical logic, model theory, nonstandard Models of arithmetic Room 4432 / 212817-8142 rkossak (at) Bronx Community College




    Listed by Research
    DOCTORAL FACULTY A B C D ... Z A Anshel, Michael: combinatorial group theory, algebraic cryptography
    Room 4307 / 212-817-8554 mikeat1140 (at) City College Apter, Arthur: mathematical logic and set theory
    Room 4432 / 212-817-8143 awapter (at) Baruch College Artemov, Sergei logic, artificial intelligence, automated deduction and verification, optimal control
    Room 4319/(212) 817-8661 sartemov (at) [Graduate Center] B Baider, Alberto: analysis, dynamical systems, and Hamiltonian systems
    Room 4307 / 212-817-8553 abaider (at) Hunter College Basmajian, Ara: hyperbolic geometry; riemann surfaces; geometric structures on manifolds
    Rm. 4307 / 212-817-8553

    57. 2.3 Approaches For Representation Of Uncertainty
    Using fuzzy arithmetic, based on the grade of membership of a parameter of interest in a set, the grade of membership of a model output in another set can
    Next: 2.4 Sensitivity and Sensitivity/Uncertainty Up: 2. BACKGROUND Previous: 2.2 Reducible and Irreducible
    2.3 Approaches for Representation of Uncertainty
    Various approaches for representing uncertainty in the context of different domains of applicability are presented by Klir [ ], and are briefly summarized in the following:
    • Classical set theory: Uncertainty is expressed by sets of mutually exclusive alternatives in situations where one alternative is desired. This includes diagnostic, predictive and retrodictive uncertainties. Here, the uncertainty arises from the nonspecificity inherent in each set. Large sets result in less specific predictions, retrodictions, etc., than smaller sets. Full specificity is obtained only when one alternative is possible.
    • Probability theory: Uncertainty is expressed in terms of a measure on subsets of a universal set of alternatives (events). The uncertainty measure is a function that, according to the situation, assigns a number between and 1 to each subset of the universal set. This number, called probability of the subset, expresses the

    58. Model Theory: An Introduction
    Model theory is a branch of mathematical logic where we study . More sophisticated ideas from combinatorial set theory are needed in Chapter 5 but are
    Model Theory: an Introduction
    David Marker
    Springer Graduate Texts in Mathematics 217
    Model theory is a branch of mathematical logic where we study mathematical structures by considering the first-order sentences true in those structures and the sets definable by first-order formulas. Traditionally there have been two principal themes in the subject:
    • starting with a concrete mathematical structure, such as the field of real numbers, and using model-theoretic techniques to obtain new information about the structure and the sets definable in the structure;
    • looking at theories that have some interesting property and proving general structure theorems about their models.
    A good example of the first theme is Tarski's work on the field of real numbers. Tarski showed that the theory of the real field is decidable. This is a sharp contrast to Godel's Incompleteness Theorem, which showed that the theory of the seemingly simpler ring of integers is undecidable. For his proof, Tarski developed the method of quantifier elimination which can be used to show that all subsets of R^n definable in the real field are geometrically well-behaved. More recently, Wilkie extended these ideas to prove that sets definable in the real exponential field are also well-behaved.
    For some time, these two themes seemed like opposing directions in the subject, but over the last decade or so we have come to realize that there are fascinating connections between these two lines. Classical mathematical structures, such as groups and fields, arise in surprising ways when we study general classification problems, and ideas developed in abstract settings have surprising applications to concrete mathematical structures. The most striking example of this synthesis is Hrushovski's application of very general model-theoretic methods to prove the MordellLang Conjecture for function fields.

    59. Program Annoucement For ASL Annual Meeting
    General conference announcements should go to the theoryA list send . Papers By Title GunWon Lee HOP Model Zhou Xunwei set theory in Geometrical Logic
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    Program Annoucement for ASL Annual Meeting

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