Godel's Incompleteness Theorems

A Guided Tour Through Kurt Godel's Historic Proof

By: Dirk W. Hoffmann

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eText | 30 August 2024

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In 1931, the mysterious-sounding article "On Formally Undecidable Propositions of Principia Mathematica and Related Systems I" shook the mathematical world. In this article, Kurt Godel proved two incompleteness theorems that have fundamentally changed our view of mathematics. Godel's theorems manifest that the concept of truth and the concept of provability cannot coincide. Since their discovery, the incompleteness theorems have attracted much attention, and a flood of articles and books have been devoted to their striking consequences. For good reasons, however, hardly any work deals with Godel's article in its original form: His complex lines of thought described with meticulous precision, the many definitions and theorems, and the now largely outdated notation turn Godel's historical masterpiece into a difficult read. This book explores Godel's original proof in detail. All individual steps are carefully explained and illustrated with numerous examples. However, this book is more than just an annotated version of the historical article, as the proper understanding of Godel's work requires a solid grasp of history. Thus, numerous excursions take the reader back to the beginning of the twentieth century. It was the time when mathematics experienced one of its greatest crises, when type theory and axiomatic set theory were taking shape, and Hilbert's formalistic logic and Brouwer's intuitionistic mathematics were openly confronting each other. This book is the revised translation of the second edition of the author's German language book "Die Godel'schen Unvollstandigkeitssatze".