Limits of Mathematics - A Journey Through the Key Areas of Mathematical Logic

Is mathematics free of contradictions? Are there truths beyond what can be proven? Is it possible to encode our mathematical knowledge into a single number?
Modern mathematical logic of the twentieth century provides astonishing answers to these questions.
This book takes you on a journey through the core areas of mathematical logic, leading to the limits of mathematics. The covered topics include the history of mathematical logic, formal systems, axiomatic number theory and set theory, proof theory, Gödel's incompleteness theorems, computability theory, algorithmic information theory, and model theory.
The book contains numerous two-color illustrations and more than 70 exercises (with solutions available on the author's website). This translation is based on the third edition of the original German book.

List of contents

1 Historic Notes.- 2 Formal Systems.- 3 Foundations of Mathematics.- 4 Peoof Thory.- 5 Computability Theory.- 6 Algorithmic Information Theory.- Model Theory.

About the author

Prof. Dr. Dirk W. Hoffmann is a professor at the Department of Computer Science and Business Information Systems at Karlsruhe University of Applied Sciences in Germany. He has also authored the book Gödel’s Incompleteness Theorems - A Guided Tour Through Kurt Gödel’s Historic Proof.

Summary

Is mathematics free of contradictions? Are there truths beyond what can be proven? Is it possible to encode our mathematical knowledge into a single number?
Modern mathematical logic of the twentieth century provides astonishing answers to these questions.
This book takes you on a journey through the core areas of mathematical logic, leading to the limits of mathematics. The covered topics include the history of mathematical logic, formal systems, axiomatic number theory and set theory, proof theory, Gödel's incompleteness theorems, computability theory, algorithmic information theory, and model theory.
The book contains numerous two-color illustrations and more than 70 exercises (with solutions available on the author's website). This translation is based on the third edition of the original German book.

Product details

• Authors: Dirk Hoffmann, Dirk W Hoffmann, Dirk W. Hoffmann
• Publisher: Springer, Berlin
• Original title: Grenzen der Mathematik. Eine Reise durch die Kerngebiete der mathematischen Logik
• Languages: English
• Product format: Paperback / Softback
• Release: 31.07.2025
• EAN: 9783662709986
• ISBN: 978-3-662-70998-6
• No. of pages: 418
• Illustrations: X, 418 p. 262 illus. in color.
• Series: Mathematics Study Resources
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Basic principles; mathematische Grundlagen, Theoretische Informatik, Hilbert, Gödel, Mathematical Logic and Foundations, Formal Languages and Automata Theory, Set theory, Model Theory, forcing, Chow, Turing machines, Undecidability, Goedel's incompleteness theorems, algorithmic information theory, provability, continuum hypothesis, fundaments of mathematics