Etymologie, Etimología, Étymologie, Etimologia, Etymology
UK Vereinigtes Königreich Großbritannien und Nordirland, Reino Unido de Gran Bretaña e Irlanda del Norte, Royaume-Uni de Grande-Bretagne et d'Irlande du Nord, Regno Unito di Gran Bretagna e Irlanda del Nord, United Kingdom of Great Britain and Northern Ireland
Logik, Lógica, Logique, Logica, Logic
mathematische Logik, Lógica matemática, Logique mathématique, Logica matematica, Mathematical logic

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Boolean algebra (W3)

The Mathematics of Boolean Algebra

First published Fri Jul 5, 2002; substantive revision Fri Feb 27, 2009

Boolean algebra is the algebra of two-valued logic with only sentential connectives, or equivalently of algebras of sets under union and complementation. The rigorous concept is that of a certain kind of algebra, analogous to the mathematical notion of a group. This concept has roots and applications in logic (Lindenbaum-Tarski algebras and model theory), set theory (fields of sets), topology (totally disconnected compact Hausdorff spaces), foundations of set theory (Boolean-valued models), measure theory (measure algebras), functional analysis (algebras of projections), and ring theory (Boolean rings). The study of Boolean algebras has several aspects: structure theory, model theory of Boolean algebras, decidability and undecidability questions for the class of Boolean algebras, and the indicated applications. In addition, although not explained here, there are connections to other logics, subsumption as a part of special kinds of algebraic logic, finite Boolean algebras and switching circuit theory, and Boolean matrices.

• •1. Definition and simple properties
• •2. The elementary algebraic theory
• •3. Special classes of Boolean algebras
• •4. Structure theory and cardinal functions on Boolean algebras
• •5. Decidability and undecidability questions
• •6. Lindenbaum-Tarski algebras
• •7. Boolean-valued models
• •Bibliography
• •Other Internet Resources
• •Related Entries

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disjunction (W3)

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Formal logic (W3)

Designation Standard Language

• "formal logic" is std English
• "mathematische Logik" is std German
• "formale Logik" is non-std German
• "lógica formal" is std Portuguese
• "lógica matemática" is non-std Portuguese

SEE: Symbolic Logic

The study of the meaning and relationships of statements used to represent precise mathematical ideas. Symbolic logic is also called formal logic.

SEE ALSO: Logic, Metamathematics

Mathematics and formal logic

• Ansatz (lit. "set down," roughly equivalent to "approach" or "where to begin", a starting assumption) - one of the most used German loan words in the English-speaking world of science.
• "Eigen-" in composita such as eigenfunction, eigenvector, eigenvalue, eigenform; in English "self-" or "own-". They are related concepts in the fields of linear algebra and functional analysis.
• Entscheidungsproblem
• Grossencharakter (German spelling: Größencharakter)
• Hauptmodul (the generator of a modular curve of genus 0)
• Hilbert's Nullstellensatz (without apostrophe in German)
• Ideal (originally "ideale Zahlen", defined by Ernst Kummer)
• Kernel (Ger.: Kern, translated as core)
• Krull's Hauptidealsatz (without apostrophe in German)
• Möbius band (Ger.: Möbiusband)
• quadratfrei
• Stützgerade
• Vierergruppe (also known as Klein four-group)
• "Neben-" in composita such as Nebentype
• "Z" from (ganze) Zahlen ((whole) numbers), the integers

Fuzzylogik (W3)

JBroFuzz is a stateless network protocol fuzzer that emerged from the needs of penetration testing. Written in Java, it allows for the identification of certain classess of security vulnerabilities, by means of creating malformed data and having the network protocol in question consume the data.

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Gödel's incompleteness theorem (W3)

• incompleteness theorem (logic)
• incompleteness theorem, Gödel’s first (logic)
• incompleteness theorem, Gödel’s second (logic)

• Gödel's incompleteness theorem
• Gödel's theorem

On Gödel's Philosophy of Mathematics
by Harold Ravitch, Ph.D.
Chairman, Department of Philosophy
Los Angeles Valley College

Table of Contents

• Title and Signature Page [1.1K]
• Abstract [1.2K]
• On Gödel's Philosophy of Mathematics
• Chapter I: Gödel's Methodology of Mathematics [41.2K]
• 1.) Gödel's Defense of Classical Mathematics
• 2.) The Vicious Circle Principle
• 3.) Gödel's Research in Intuitionistic Mathematics
• 4.) Gödel's Dilemma of Higher Axioms
• 5.) Truth Criteria for Higher Axioms
• 6.) Some Concluding Remarks on Gödel's Methodology
• Appendix A
• Appendix B
• Chapter II: Gödel on the Existence of Mathematical Objects [33.5K]
• 1.) Gödel's Realism
• 2.) Gödel's Interpretation of 'exist'
• 3.) Some Criticisms of Gödel's Realism
• 4.) Gödel's Realism vis à vis Gödel's Methodology of Mathematics
• Appendix A
• Notes and References [33.8K]
• Bibiography [7.5K]

Palle Yourgrau,
A World Without Time,
The Forgotten Legacy of Gödel and Einstein
Basic Books, 2005
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The Kurt Gödel Society was founded in 1987 and is chartered in Vienna. It is an international organization for the promotion of research in the areas of Logic, Philosophy, History of Mathematics, above all in connection with the biography of Kurt Gödel, and in other areas to which Gödel made contributions, especially mathematics, physics, theology, philosophy and Leibniz studies.

Short biography of Kurt Gödel (collected by Rosalie Iemhoff)
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Title: THE MODERN DEVELOPMENT OF THE FOUNDATIONS OF MATHEMATICS IN THE LIGHT OF PHILOSOPHY
Author: GÖDEL KURT

• Logic and Foundations: Gödel Numbers [185]
• History of Mathematics: Gödel, Kurt [184]

Gödel, Kurt (1906-1978)
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Gödel, Kurt (Juliette Kennedy)

Kurt Gödel

First published Tue Feb 13, 2007; substantive revision Tue Jul 5, 2011

Kurt Friedrich Gödel (b. 1906, d. 1978), “established, beyond comparison, as the most important logician of our times,” in the words of Solomon Feferman (Feferman 1986), founded the modern, metamathematical era in mathematical logic. His Incompleteness Theorems, among the most significant achievements in logic since, perhaps, those of Aristotle, are among the handful of landmark theorems in twentieth century mathematics. His work touched every field of mathematical logic, if it was not in most cases their original stimulus. In his philosophical work Gödel formulated and defended mathematical Platonism, involving the view that mathematics is a descriptive science, and that the concept of mathematical truth is an objective one. On the basis of that viewpoint he laid the foundation for the program of conceptual analysis within set theory (see below). He adhered to Hilbert's “original rationalistic conception” in mathematics (as he called it); he was prophetic in anticipating and emphasizing the importance of large cardinals in set theory before their importance became clear.

• 1. Biographical Sketch
• 2. Gödel's Mathematical Work?2.1 The Completeness Theorem¦2.1.1 Introduction
• 2.1.2 Proof of the Completeness Theorem
• 2.1.3 An Important Consequence of the Completeness Theorem
• 2.2 The Incompleteness Theorems¦2.2.1 The First Incompleteness Theorem
• 2.2.2 The proof of the First Incompleteness Theorem
• 2.2.3 The Second Incompleteness Theorem
• 2.2.4 Did the Incompleteness Theorems Refute Hilbert's Program?
• 2.3 Speed-up Theorems
• 2.4 Gödel's Work in Set theory¦2.4.1 The consistency of the Continuum Hypothesis and the Axiom of Choice
• 2.4.2 Gödel's Proof of the Consistency of the Continuum Hypothesis and the Axiom of Choice with the Axioms of Zermelo-Fraenkel Set Theory
• 2.4.3 Consequences of Consistency
• 2.4.4 Gödel's view of the Axiom of Constructibility
• 2.5 Gödel's Work in Intuitionistic Logic and Arithmetic¦2.5.1 Intuitionistic Propositional Logic is not Finitely-Valued
• 2.5.2 Classical Arithmetic is Interpretable in Heyting Arithmetic
• 2.5.3 Intuitionistic Propositional Logic is Interpretable in S4
• 2.5.4 Heyting Arithmetic is Interpretable into Computable Functionals of Finite Type.
• 3. Gödel's philosophical work?3.1 Documents¦3.1.1 “My Notes, 1940–1970”
• 3.2 Gödel's Philosophical Views¦3.2.1 Gödel's Rationalism
• 3.2.2 Gödel's Realism
• 3.2.3 Gödel's Turn to Phenomenology
• 3.2.4 A Philosophical Argument
• Bibliography?Primary Sources ¦Gödel's Writings
• The Collected Papers of Kurt Gödel
• Selected Works of Kurt Gödel
• Secondary Sources
• Other Internet Resources
• Related Entries

Bernays-Gödel Set Theory
SEE: von Neumann-Bernays-Gödel Set Theory

Gödel's Completeness Theorem

Gödel's First Incompleteness Theorem

Gödel's Incompleteness Theorems

Gödel Number

Gödel's Second Incompleteness Theorem

von Neumann-Bernays-Gödel Set Theory

Gödel, Kurt (1906-1978)

Kurt Gödel (1906 - 1978)
Gödel proved fundamental results about axiomatic systems showing in any axiomatic mathematical system there are propositions that cannot be proved or disproved within the axioms of the system.

Kurt Gödel
Born: 28 April 1906 in Brünn, Austria-Hungary (now Brno, Czech Republic)
Died: 14 Jan 1978 in Princeton, New Jersey, USA
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Kurt Gödel - Gödel's incompleteness theorem

Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible, giving a negative answer to Hilbert's second problem.

The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system.

The second incompleteness theorem, a corollary of the first, shows that such a system cannot demonstrate its own consistency.
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Kurt Gödel - Gödel's ontological proof

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Informal logic (W3)

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logic (W3)

Earliest Uses of Symbols of Set Theory and Logic
Last updated: Sept. 1, 2010

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." 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 19th and early 20th 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.
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History Topics

• Bolzano's manuscripts references
• Bolzano publications.html
• Set theory references
• Jaina mathematics references
• Mathematical games references
• U of St Andrews History
• Mathematical games references
• Ledermann interview
• Jaina mathematics references
• Christianity and Mathematics
• Word problems
• Set theory
• Measurement
• function concept
• Mathematical games
• Amusements.html
• Infinity
• Squaring the circle
• Calculus history
• Greek astronomy
• 20th century time references
• Topology history references
• Classical time references
• Infinity references
• Real numbers 3 references
• Real numbers 2 references
• 20th century time references
• Topology history references
• Classical time references
• Infinity references
• Real numbers 3 references
• Real numbers 2 references
• Bolzano's manuscripts
• Harriot's manuscripts

• •1. Sources of our information on the Stoics
• •2. Philosophy and life
• •3. Physical Theory
• •4. Logic
• •5. Ethics
• •6. Influence
• •Bibliography
• •Other Internet Resources
• •Related Entries

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Mathematical logic (W3)

Peano was the founder of "symbolic logic" and his interests centred on the foundations of mathematics and on the development of a formal logical language.
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In 1888 Peano published the book Geometrical Calculus which begins with a chapter on "mathematical logic". This was his first work on the topic that would play a major role in his research over the next few years and it was based on the work of Schröder, Boole and Charles Peirce.
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Symbolic logic (W3)

Links to Other Sites

Organizations

• American Association for the Advancement of Science (AAAS)
• American Mathematical Society (AMS)
• American Philosophical Association (APA)
• Association for Computing Machinery (ACM)
• Conference Board of the Mathematical Sciences (CBMS)
• Cognitive Science Society
• European Association for Computer Science Logic (EACSL)
• European Association for Logic, Language and Information (FoLLI)
• Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• International Mathematical Olympiad (IMO)
• Linquistic Society of America
• Logic in Computer Science (LICS)
• London Mathematical Society
• Mathematical Association of America (MAA)
• The Danish Network for Philosophical Logic and Its Applications Newsletter (PHINEWS)
• Society for Industrial and Applied Mathematics (SIAM)
• The Philosophy of Science Association

Research Institutes

• Fields Institute for Research in Mathematical Sciences
• Fuzziness and Uncertainty Modelling Research Group
• Institute for Logic, Language, and Computation (ILLC)
• Institute for Mathematics and its Applications (IMA)
• Mathematical Institutes and Centers (Site organized by AMS)
• Mathematical Sciences Research Institute (MSRI)

Scholarly Resources

• http://www.nd.edu/~cholak/computability/computability.html - Bibiliographic Site for Computability Theory
• http://wwwagr.informatik.uni-kl.de/~akademie/contents.html - Bibliography of Mathematical Logic and Related Fields
• http://www.math.ufl.edu/~jal/set_theory.html - Bibliographic Site for Set Theory
• http://dblp.uni-trier.de/db/index.html - Computer Science Bibliography
• http://directory.google.com/Top/Science/Math/Logic_and_Foundations/ - Google Web Directory for Logic and Foundations
• http://www.jstor.org/ - JSTOR - Scholarly Journal Archive
• http://www.cirs-tm.org - International Center for Scientific Research (CIRS)
• http://xxx.lanl.gov/archive/math - Los Alamos National Laboratory Mathematics Archive
• http://www-groups.dcs.st-and.ac.uk/~history/index.html - The MacTutor History of Mathematics Archive
• http://dmoz.org/Science/Math/Logic_and_Foundations/ - Open Directory Project
• http://www.math.uu.se/logik/logic-server/ - Research Groups in Logic and Theoretical Computer Science (Maintained by the Upsala Group for Mathematical Logic)

Peano was the founder of "symbolic logic" and his interests centred on the foundations of mathematics and on the development of a formal logical language.
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The study of the meaning and relationships of statements used to represent precise mathematical ideas. "Symbolic logic" is also called "formal logic".

SEE ALSO: Logic, Metamathematics

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Etymologie, Etimología, Étymologie, Etimologia, Etymology
UK Vereinigtes Königreich Großbritannien und Nordirland, Reino Unido de Gran Bretaña e Irlanda del Norte, Royaume-Uni de Grande-Bretagne et d'Irlande du Nord, Regno Unito di Gran Bretagna e Irlanda del Nord, United Kingdom of Great Britain and Northern Ireland
Logik, Lógica, Logique, Logica, Logic

amazon - Logik, Lógica, Logique, Logica, Logic

      Logica (IT)    

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Smith, Peter (Author)
An Introduction to Formal Logic

Product Description
Formal logic provides us with a powerful set of techniques for criticizing some arguments and showing others to be valid. These techniques are relevant to all of us with an interest in being skilful and accurate reasoners. In this highly accessible book, Peter Smith presents a guide to the fundamental aims and basic elements of formal logic. He introduces the reader to the languages of propositional and predicate logic, and then develops formal systems for evaluating arguments translated into these languages, concentrating on the easily comprehensible 'tree' method. His discussion is richly illustrated with worked examples and exercises. A distinctive feature is that, alongside the formal work, there is illuminating philosophical commentary. This book will make an ideal text for a first logic course, and will provide a firm basis for further work in formal and philosophical logic.

Book Description
This book introduces the reader to the languages of propositional and predicate logic, and then develops formal systems for evaluating arguments translated into these languages. It will make an ideal text for a first logic course, and will provide a firm basis for further work in formal and philosophical logic.

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