On the Brink of Paradox

On the Brink of Paradox

Highlights from the Intersection of Philosophy and Mathematics

By Agustín Rayo

An introduction to awe-inspiring ideas at the brink of paradox: infinities of different sizes, time travel, probability and measure theory, and computability theory.
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An introduction to awe-inspiring ideas at the brink of paradox: infinities of different sizes, time travel, probability and measure theory, and computability theory.

This book introduces the reader to awe-inspiring issues at the intersection of philosophy and mathematics. It explores ideas at the brink of paradox: infinities of different sizes, time travel, probability and measure theory, computability theory, the Grandfather Paradox, Newcomb's Problem, the Principle of Countable Additivity. The goal is to present some exceptionally beautiful ideas in enough detail to enable readers to understand the ideas themselves (rather than watered-down approximations), but without supplying so much detail that they abandon the effort. The philosophical content requires a mind attuned to subtlety; the most demanding of the mathematical ideas require familiarity with college-level mathematics or mathematical proof.

The book covers Cantor's revolutionary thinking about infinity, which leads to the result that some infinities are bigger than others; time travel and free will, decision theory, probability, and the Banach-Tarski Theorem, which states that it is possible to decompose a ball into a finite number of pieces and reassemble the pieces so as to get two balls that are each the same size as the original. Its investigation of computability theory leads to a proof of Gödel's Incompleteness Theorem, which yields the amazing result that arithmetic is so complex that no computer could be programmed to output every arithmetical truth and no falsehood. Each chapter is followed by an appendix with answers to exercises. A list of recommended reading points readers to more advanced discussions. The book is based on a popular course (and MOOC) taught by the author at MIT.

Instructor Resources

Downloadable instructor resources available for this title: problem sets with answers, a file of figures in the book.


$45.00 X ISBN: 9780262039413 320 pp. | 7 in x 9 in 29 b&w illus.


  • On the Brink of Paradox is a surprisingly delightful read. The book contains an amazing amount of accesible material even to college mathematics majors, while offering value or review for mathematicians, but never in a stale way.

    Mathematical Association of America


  • “An intensely fun introduction to the fascinating boundary between mathematics and philosophy. A remarkable range of topics are treated at a high level of depth and mathematical precision (plus: exercises!). The book will be welcomed by curious and mathematically agile readers, who will appreciate Rayo's lucid prose, sense of humor, philosophical clarity, and, above all, enthusiasm for the subject.”

    Ted Sider

    Distinguished Professor, Andrew W. Mellon Chair, Department of Philosophy, Rutgers University

  • “This book would be excellent for a bright student who is not afraid of math, to point them in the direction of the paradoxical and philosophical ideas at the foundation of it all. There are many exercises in each chapter that mix mathematical calculations and philosophical considerations, with example answers at the end of each chapter. The paradoxes discussed include old classics as well as cutting-edge new ones. The book is written in an engaging style that asks the reader to work through exercises to see for themselves but doesn't require reading awkward academic citations. However, every example and paradox is credited, and there are pointers to academic research articles on each, as well as to various topics that readers can effectively learn more about through free internet resources or even straightforward Google searches.”

    Kenny Easwaran

    Associate Professor, Philosophy Department, Texas A&M University

  • "This engaging book packs a surprising amount of mathematics and philosophy into roughly 300 pages.  Rayo uses paradoxes and playfulness to guide the reader through tricky mathematical and philosophical topics, including the mathematics of the infinite, the nature of probability and its relationship to decision-making, the logical limits of our mathematical abilities, and some puzzles about free will and truth. Well-designed exercises (with solutions) will keep students curious and focused."

    Ray Briggs

    Department of Philosophy, Stanford University