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.