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"There are many textbooks available for a so-called transition course from calculus to abstract mathematics. I have taught this course several times and always find it problematic. The Foundations of Mathematics (Stewart and Tall) is a horse of a different color. The writing is excellent and there is actually some useful mathematics. I definitely like this book."--The Bulletin of Mathematics Books
Mathematical logic grew out of philosophical questions regarding the foundations of mathematics, but logic has now outgrown its philosophical roots, and has become an integral part of mathematics in general. This book is designed for students who plan to specialize in logic, as well as for those who are interested in the applications of logic to other areas of mathematics. Used as a text, it could form the basis of a beginning graduate-level course. There are three main chapters: Set Theory, Model Theory, and Recursion Theory. The Set Theory chapter describes the set-theoretic foundations of all of mathematics, based on the ZFC axioms. It also covers technical results about the Axiom of Choice, well-orderings, and the theory of uncountable cardinals. The Model Theory chapter discusses predicate logic and formal proofs, and covers the Completeness, Compactness, and Lowenheim-Skolem Theorems, elementary submodels, model completeness, and applications to algebra. This chapter also continues the foundational issues begun in the set theory chapter. Mathematics can now be viewed as formal proofs from ZFC. Also, model theory leads to models of set theory. This includes a discussion of absoluteness, and an analysis of models such as H( ) and R( ). The Recursion Theory chapter develops some basic facts about computable functions, and uses them to prove a number of results of foundational importance; in particular, Church's theorem on the undecidability of logical consequence, the incompleteness theorems of Godel, and Tarski's theorem on the non-definability of truth.
Was war noch mal die Catalan’sche Vermutung? Und woher kommt eigentlich das Wurzelsymbol? Was hat die Zahl Pi mit dem Sternenhimmel zu tun? Wer erfand das Gleichheitszeichen? Der britische Matheguru Ian Stewart breitet in diesem Band Schätze aus, die er in Jahrzehnten gesammelt hat: über 180 interessante Matherätsel, Lösungen, Spiele, Tricks, Geschichten, Anekdoten und Logeleien. Zudem ist Stewarts Schatztruhe mit interessanten historischen Exkursen angereichert, zum Beispiel einer kurzen Einführung in das Rechnen der Maya und der alten Ägypter und auch in die Vergangenheit unseres eigenen Rechnens: Wer erfand das Gleichheitszeichen – und warum? Ein Buch zum Blättern und Stöbern, zum Spaßhaben und Dazulernen, für Laien und für Fortgeschrittene.
Classic undergraduate text acquaints students with fundamental concepts and methods of mathematics. Topics include axiomatic method, set theory, infinite sets, groups, intuitionism, formal systems, mathematical logic, and much more. 1965 second edition.
This volume discusses various aspects of Harvey Friedman's research in the foundations of mathematics over the past fifteen years. It should appeal to a wide audience of mathematicians, computer scientists, and mathematically oriented philosophers.
First published in 2000. Routledge is an imprint of Taylor & Francis, an informa company.

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