The Art of Proof

In Example 1.2 we gave a proof of 1 + 1 = 2 in 17 (!) proof steps. At that point you may have asked yourself: If it takes that much effort to prove such a simple statement, how can one ever prove any non-trivial mathematical result using formal proofs? Th

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Gödel’s Theorems and Zermelo’s Axioms A Firm Foundation of Mathematics

Lorenz Halbeisen • Regula Krapf

Gödel’s Theorems and Zermelo’s Axioms A Firm Foundation of Mathematics

Lorenz Halbeisen Departement Mathematik ETH Zürich Zürich, Switzerland

Regula Krapf Institut für Mathematik Universität Koblenz-Landau Koblenz, Germany

ISBN 978-3-030-52278-0 ISBN 978-3-030-52279-7 (eBook) © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland


This book provides a self-contained introduction to the foundations of mathematics, where self-contained means that we assume as little prerequisites as possible. One such assumption is the notion of finiteness, which cannot be defined in mathematics. The firm foundation of mathematics we provide is based on logic and models. In particular, it is based on Hilbert’s axiomatisation of formal logic (including the notion of formal proofs), and on the notion of models of math¨ del’s Completeness ematical theories. On this basis, we first prove Go ¨ del’s Incompleteness Theorems, and then we introTheorem and Go duce Zermelo’s Axioms of Set Theory. On the one hand, G¨odel’s Theorems set the framework within which mathematics takes place. On the other hand, using the example of Analysis, we shall see how mathematics can be developed within a model of Set Theory. So, G¨odel’s Theorems and Zermelo’s Axioms are indeed a firm foundation of mathematics. The book consists of four parts. The first part is an introduction to FirstOrder Logic from scratch. Starting with a set of symbols, the basic concepts of formal proofs and models are developed,