

In 1931 Kurt Godel published his fundamental paper, "On Formally Undecidable Propositions of "Principia Mathematica" and Related Systems." This revolutionary paper challenged certain basic assumptions underlying much research in mathematics and logic. Godel received public recognition of his work in 1951 when he was awarded the first Albert Einstein Award for achievement in the natural sciences--perhaps the highest award of its kind in the United States. The award committee described his work in mathematical logic as "one of the greatest contributions to the sciences in recent times." However, few mathematicians of the time were equipped to understand the young scholar's complex proof. Ernest Nagel and James Newman provide a readable and accessible explanation to both scholars and non-specialists of the main ideas and broad implications of Godel's discovery. It offers every educated person with a taste for logic and philosophy the chance to understand a previously difficult and inaccessible subject. With a new introduction by Douglas R. Hofstadter, this book will appeal students, scholars, and professionals in the fields of mathematics, computer science, logic and philosophy, and science.
A glimpse inside

The book demystifies Gödel’s 1931 discovery: in any sufficiently complex formal mathematical system, there exist true statements that cannot be proven within that system. Nagel and Newman break down the logic behind this, showing how Gödel ingeniously encoded statements about arithmetic into the language of mathematics itself, revealing inherent limitations in formal systems like those championed by Hilbert and Russell.
Nagel and Newman set the stage by describing the early 20th-century quest for absolute certainty in mathematics, led by thinkers like Hilbert. The authors explain the ambition to ground all mathematics on a finite set of axioms and rules, and how Gödel’s proof dashed hopes for such a complete and consistent foundation, fundamentally altering the philosophy of mathematics.
Ratings at a glance
- 1The Essence of Gödel’s Incompleteness Theorem
- 2The Dream and Limits of Formalism
- 3Self-Reference and Paradox
- 4Implications for Logic, Mathematics, and Beyond
- 5Making the Abstract Accessible
Popular quotes from Gödel's Proof
“The proof is not only unexpected but also deeply disturbing, for it shows that the dream of a complete and consistent set of axioms for all mathematics is unattainable.”