Perceiving the impossible is an intellectual achievement that sets humans apart. For instance, Einstein’s theory of relativity and Heisenberg’s uncertainty principle impose physical boundaries we will never surpass. As disturbing as they are, these discoveries possess an irresistible allure; throughout human history, the impossible has captivated more people than the possible.
However, such phenomena are not limited to physics. Twentieth-century mathematical logic has shown that similar negative results impact nearly all branches of mathematics. Today, we know that the notion of truth and the notion of provability do not coincide, even for seemingly straightforward theories such as number theory. It is impossible to encapsulate mathematics within a formal system in which all true statements are provable, and all false statements are not.
I invite you on a journey through the realms of mathematical logic. My goal when writing this book was to present the concepts, methods, and findings of this fascinating branch of mathematics as clearly as possible without sacrificing depth. Wherever appropriate, I have motivated definitions and theorems with examples and placed them in their factual and historical context through numerous side notes. Additionally, I have chosen to present theorems that play only a marginal role in a sketchy manner or to indicate where an elaborated proof can be found. In this sense, the book cannot replace the formally precise literature in mathematical logic in all aspects -- and it does not intend to. Above all, I aim to convey the fascination that this branch of mathematics undoubtedly exudes. It is up to you, dear readers, to judge how much I have succeeded in this endeavor.
I want to take the opportunity to thank my publisher, as this text would not exist without permitting me to prepare English translations of two of my logic books initially published in German. One of these books is the one you are reading right now. It is the revised translation of the third edition of my German language book [Grenzen der Mathematik](http://www.dirkwhoffmann.de/GM). The other one is [Gödel's Incompleteness Theorems -- A Guided Tour Through Kurt Gödel's Historic Proof](http://www.dirkwhoffmann.de/GDE), the revised translation of the second edition of my German book [Die Gödel'schen Unvollständigkeitssätze](http://www.dirkwhoffmann.de/GD). It details Gödel's paramount article from 1931, which sent shockwaves through large parts of mathematics that still reverberate today. Gödel's two incompleteness theorems are also central to this book, as their negative propositions permeate almost all areas of mathematical logic. However, I kept their presentation concise due to the wide range of covered topics. If you become as mesmerized by Gödel’s incompleteness theorems as I became many years ago, you might find my other book an enjoyable follow-up read.
For me, as a native German speaker, the translation was a considerable linguistic challenge with an open outcome. At the beginning of the project, I was unsure whether I could complete it to a reasonable standard. After several months of hard work, however, I finished the manuscripts with a quality I am personally satisfied with, and I am pleased to present the result to an international readership. Undoubtedly, this book is not perfect. I am therefore grateful to any reader pointing out errors or suggesting improvement.
Karlsruhe, 1 January 2025
Dirk W. Hoffmann
