Proven Impossible
Elementary Proofs of Profound Impossibility from Arrow, Bell, Chaitin, Gödel, Turing and More

A highly readable presentation of elementary yet rigorous proofs of profound impossibility theorems for a broad, lay audience.

Dan Gusfield (Author)

9781009349499, Cambridge University Press

Paperback / softback, published 18 January 2024

270 pages
23 x 15 x 1.8 cm, 0.448 kg

'… a great book both for the layperson and for people who know some of the material.' William Gasarch, SIGACT News

In mathematics, it simply is not true that 'you can't prove a negative'. Many revolutionary impossibility theorems reveal profound properties of logic, computation, fairness and the universe, and form the mathematical background of new technologies and Nobel prizes. But to fully appreciate these theorems and their impact on mathematics and beyond, you must understand their proofs. This book is the first to present these proofs for a broad, lay audience. It fully develops the simplest rigorous proofs found in the literature, reworked to contain less jargon and notation, and more background, intuition, examples, explanations, and exercises. Amazingly, all of the proofs in this book involve only arithmetic and basic logic – and are elementary, starting only from first principles and definitions. Very little background knowledge is required, and no specialized mathematical training – all you need is the discipline to follow logical arguments and a pen in your hand.

Preface
1. Yes you can prove a negative!
2. Bell's impossibility theorem(s)
3. Enjoying Bell magic
4. Arrow's (and friends') impossibility theorems
5. Clustering and impossibility
6. Gödel-ish impossibility
7. Turing undecidability and incompleteness
8. Chaitin's theorem: More devastating
9. Gödel (for real, this time).

Subject Areas: Mathematics [PB]