Proven Impossible

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.

• Presents rigorous yet elementary proofs for readers with limited mathematical background, starting from first principles and definitions, and using only arithmetic and simple logic
• Covers proofs slowly, with plentiful examples and explanation, and less jargon and notation than is typical
• Provides the historical and philosophical context for many of the theorems to help readers understand the larger context and their impact