Set Theory Logic and Their Limitations

Author: Moshe Machover
Publisher: Cambridge University Press
ISBN: 0521479983
Release Date: 1996-05-23
Genre: Mathematics

Rigorous coverage of logic and set theory for students of mathematics and philosophy.

Set Theory Logic and their Limitations

Author: Moshe Machover
Publisher: Cambridge University Press
ISBN: 0521474930
Release Date: 1996-05-23
Genre: Mathematics

In this introduction to set theory and logic, the author discusses first order logic, and gives a rigorous axiomatic presentation of Zermelo-Fraenkel set theory. He includes many methodological remarks and explanations, and demonstrates how the basic concepts of mathematics can be reduced to set theory. He explains concepts and results of recursion theory in intuitive terms, and reaches the limitative results of Skolem, Tarski, Church and Gödel (the celebrated incompleteness theorems). For students of mathematics and philosophy, this book provides an excellent introduction to logic and set theory.

Cantorian Set Theory and Limitation of Size

Author: Michael Hallett
Publisher: Oxford University Press
ISBN: 0198532830
Release Date: 1986
Genre: Mathematics

Cantor's ideas formed the basis for set theory and also for the mathematical treatment of the concept of infinity. The philosophical and heuristic framework he developed had a lasting effect on modern mathematics, and is the recurrent theme of this volume. Hallett explores Cantor's ideas and, in particular, their ramifications for Zermelo-Frankel set theory.

Logik

Author: Gerhard Schurz
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 9783110590630
Release Date: 2018-09-10
Genre: Philosophy

Diese Einführung in die Logik umfaßt einen Grundkurs und einen Aufbaukurs. Der Grundkurs ist voraussetzungsfrei geschrieben und führt in die Semantik und Beweistheorie der Aussagenlogik und elementaren Prädikatenlogik ein, eingebettet in die allgemeine Theorie des rationalen Schließens. Logische Zusammenhänge werden in Verbindung mit sorgfältig ausgewählten Übungsbeispielen – inklusive Lösungen – einsichtig gemacht. Auf die philosophische Anwendung der Logik in der logischen Rekonstruktion natursprachlicher Texte und Argumente liegt besonderes Augenmerk. Zusammenhänge zwischen alternativen logischen Notationen und Techniken, die anfangs oft Schwierigkeiten bereiten, werden sorgfältig erklärt. Der anschließende Aufbaukurs schlägt die Brücke zwischen einer philosophischen Logikeinführung und dem fortgeschrittenen Niveau moderner formaler Logik. Nach einer gründlichen Einführung in die volle Prädikatenlogik und ihrer mengentheoretischen Semantik wendet sich der Band metalogischen Methoden zu. Prominente Resultate zur Korrektheit und Vollständigkeit der Prädikatenlogik, zur Entscheidbarkeit der monadischen und Unentscheidbarkeit der vollen Prädikatenlogik sowie zur Unvollständigkeit der Arithmetik 1. Stufe werden Schritt um Schritt erklärt. Abgerundet wird der Band durch zahlreiche Exkurse zur philosophischen Vertiefung logischer Grundlagenfragen. Zahlreiche Übungsbeispiele mit Lösungen zum Download vertiefen den Stoff. Die Lösungen werden ab Oktober 2018 verfügbar sein.

Introduction to Model Theory

Author: Philipp Rothmaler
Publisher: CRC Press
ISBN: 9056993135
Release Date: 2000-10-31
Genre: Mathematics

Model theory investigates mathematical structures by means of formal languages. So-called first-order languages have proved particularly useful in this respect. This text introduces the model theory of first-order logic, avoiding syntactical issues not too relevant to model theory. In this spirit, the compactness theorem is proved via the algebraically useful ultrsproduct technique (rather than via the completeness theorem of first-order logic). This leads fairly quickly to algebraic applications, like Malcev's local theorems of group theory and, after a little more preparation, to Hilbert's Nullstellensatz of field theory. Steinitz dimension theory for field extensions is obtained as a special case of a much more general model-theoretic treatment of strongly minimal theories. There is a final chapter on the models of the first-order theory of the integers as an abelian group. Both these topics appear here for the first time in a textbook at the introductory level, and are used to give hints to further reading and to recent developments in the field, such as stability (or classification) theory.

Enzyklop die Philosophie und Wissenschaftstheorie

Author: Jürgen Mittelstraß
Publisher: Springer-Verlag
ISBN: 9783476001405
Release Date: 2016-08-16
Genre: Philosophy

Das gesamte Wissen der Philosophie und Wissenschaftstheorie. Die Sach- und Personenartikel des Nachschlagewerks erfassen nicht nur den klassischen Bestand des philosophischen Wissens, sondern werden auch den neueren Entwicklungen in der Philosophie gerecht. Insbesondere in den Bereichen Logik, Erkenntnis- und Wissenschaftstheorie sowie Sprachphilosophie. Jetzt erscheint der fünfte Band der Neuauflage mit über 70 neuen Artikeln u. a. zu diesen Begriffen: antike Logik, Lüge, Macht, Medizin, Nano und Neurowissenschaften. Mit neuen Personenartikeln, darunter Luhmann, Lyotard, Maturana.

Numbers Sets and Axioms

Author: A. G. Hamilton
Publisher: Cambridge University Press
ISBN: 0521287618
Release Date: 1982
Genre: Mathematics

Following the success of Logic for Mathematicians, Dr Hamilton has written a text for mathematicians and students of mathematics that contains a description and discussion of the fundamental conceptual and formal apparatus upon which modern pure mathematics relies. The author's intention is to remove some of the mystery that surrounds the foundations of mathematics. He emphasises the intuitive basis of mathematics; the basic notions are numbers and sets and they are considered both informally and formally. The role of axiom systems is part of the discussion but their limitations are pointed out. Formal set theory has its place in the book but Dr Hamilton recognises that this is a part of mathematics and not the basis on which it rests. Throughout, the abstract ideas are liberally illustrated by examples so this account should be well-suited, both specifically as a course text and, more broadly, as background reading. The reader is presumed to have some mathematical experience but no knowledge of mathematical logic is required.

Principia Mathematica

Author: Alfred North Whitehead
Publisher:
ISBN: STANFORD:36105039675058
Release Date: 1984-01
Genre: Logic, Symbolic and mathematical


Logic Induction and Sets

Author: Thomas Forster
Publisher: Cambridge University Press
ISBN: 0521533619
Release Date: 2003-07-21
Genre: Mathematics

This is an introduction to logic and the axiomatization of set theory from a unique standpoint. Philosophical considerations, which are often ignored or treated casually, are here given careful consideration, and furthermore the author places the notion of inductively defined sets (recursive datatypes) at the center of his exposition resulting in a treatment of well established topics that is fresh and insightful. The presentation is engaging, but always great care is taken to illustrate difficult points. Understanding is also aided by the inclusion of many exercises. Little previous knowledge of logic is required of the reader, and only a background of standard undergraduate mathematics is assumed.

Mathematical Logic

Author: H.-D. Ebbinghaus
Publisher: Springer Science & Business Media
ISBN: 9781475723557
Release Date: 2013-03-14
Genre: Mathematics

This introduction to first-order logic clearly works out the role of first-order logic in the foundations of mathematics, particularly the two basic questions of the range of the axiomatic method and of theorem-proving by machines. It covers several advanced topics not commonly treated in introductory texts, such as Fraïssé's characterization of elementary equivalence, Lindström's theorem on the maximality of first-order logic, and the fundamentals of logic programming.

Lectures in Logic and Set Theory Volume 1 Mathematical Logic

Author: George Tourlakis
Publisher: Cambridge University Press
ISBN: 1139439421
Release Date: 2003-01-09
Genre: Mathematics

This two-volume work bridges the gap between introductory expositions of logic or set theory on one hand, and the research literature on the other. It can be used as a text in an advanced undergraduate or beginning graduate course in mathematics, computer science, or philosophy. The volumes are written in a user-friendly conversational lecture style that makes them equally effective for self-study or class use. Volume 1 includes formal proof techniques, a section on applications of compactness (including nonstandard analysis), a generous dose of computability and its relation to the incompleteness phenomenon, and the first presentation of a complete proof of Godel's 2nd incompleteness since Hilbert and Bernay's Grundlagen theorem.

Grundz ge der Mengenlehre

Author: Felix Hausdorff
Publisher: American Mathematical Soc.
ISBN: 082840061X
Release Date: 1949-01-01
Genre: Mathematics

This reprint of the original 1914 edition of this famous work contains many topics that had to be omitted from later editions, notably, Symmetric Sets, Principle of Duality, most of the ``Algebra'' of Sets, Partially Ordered Sets, Arbitrary Sets of Complexes, Normal Types, Initial and Final Ordering, Complexes of Real Numbers, General Topological Spaces, Euclidean Spaces, the Special Methods Applicable in the Euclidean Plane, Jordan's Separation Theorem, the Theory of Content and Measure, the Theory of the Lebesgue Integral. The text is in German.

Conceptus

Author:
Publisher:
ISBN: STANFORD:36105121666940
Release Date: 2001
Genre: Philosophy