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This book provides a concise and self-contained introduction to the
foundations of mathematics. The first part covers the fundamental
notions of mathematical logic, including logical axioms, formal
proofs and the basics of model theory. Building on this, in the
second and third part of the book the authors present detailed
proofs of Goedel's classical completeness and incompleteness
theorems. In particular, the book includes a full proof of Goedel's
second incompleteness theorem which states that it is impossible to
prove the consistency of arithmetic within its axioms. The final
part is dedicated to an introduction into modern axiomatic set
theory based on the Zermelo's axioms, containing a presentation of
Goedel's constructible universe of sets. A recurring theme in the
whole book consists of standard and non-standard models of several
theories, such as Peano arithmetic, Presburger arithmetic and the
real numbers. The book addresses undergraduate mathematics students
and is suitable for a one or two semester introductory course into
logic and set theory. Each chapter concludes with a list of
exercises.
This book provides a concise and self-contained introduction to the
foundations of mathematics. The first part covers the fundamental
notions of mathematical logic, including logical axioms, formal
proofs and the basics of model theory. Building on this, in the
second and third part of the book the authors present detailed
proofs of Goedel's classical completeness and incompleteness
theorems. In particular, the book includes a full proof of Goedel's
second incompleteness theorem which states that it is impossible to
prove the consistency of arithmetic within its axioms. The final
part is dedicated to an introduction into modern axiomatic set
theory based on the Zermelo's axioms, containing a presentation of
Goedel's constructible universe of sets. A recurring theme in the
whole book consists of standard and non-standard models of several
theories, such as Peano arithmetic, Presburger arithmetic and the
real numbers. The book addresses undergraduate mathematics students
and is suitable for a one or two semester introductory course into
logic and set theory. Each chapter concludes with a list of
exercises.
Dieses Lehrbuch nimmt Sie mit auf eine Entdeckungsreise durch die
Welt der klassischen Geometrie: Beginnend beim Satz von Thales und
den Apolloniuskreisen fuhrt die Reise uber Steiner'sche Kreisketten
bis in die Welt der Kegelschnitte. Dabei werden verborgene
Zusammenhange aufgedeckt und Perlen der Elementargeometrie
prasentiert. Hierbei werden Sie durch harmonische Verhaltnisse
geleitet, welche eine zentrale Rolle spielen und sich wie ein roter
Faden durch das ganze Buch ziehen. Einerseits ist dieses Buch fur
alle Liebhaberinnen und Liebhaber der Geometrie geschrieben,
andererseits ist es durch die leicht zugangliche Theorie und die
kurzen Beweise besonders auch fur Schulerinnen und Schuler der
Sekundarstufe sowie Lehramtsstudierende geeignet. Die zweite
Auflage des Buches enthalt neu Loesungen und Hinweise zu allen
Aufgaben. Zusatzlich wurde ein Abschnitt uber die Zyklographie
eingefugt. Diese heute fast in Vergessenheit geratene Abbildung
erlaubt eine verbluffend einfache und elementare Loesung des
Apollonischen Beruhrungsproblems. Die ebenfalls neuen Abschnitte
uber trilineare Koordinaten und das Ceva- und das Anti-Ceva-Dreieck
ermoeglichen einen Ausblick in die moderne Dreiecksgeometrie.
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