Infinity, Explained: Why Some Infinities Are Bigger Than Others

by Jerry Kaplan

Some infinities are bigger than others. This book will explain why.

You have probably heard that infinity is just infinity, one single idea meaning "it goes on forever." But what if someone told you there are different sizes of infinity, and that most numbers can never be written down? It sounds impossible. It sounds like a trick. It is neither. It is one of the most stunning discoveries in the history of mathematics, and you do not need anything beyond basic arithmetic to follow the proof yourself.

Infinity, Why Some Infinities Are Bigger Than Others starts where you already counting, sets, and simple patterns. Step by step, author and veteran math teacher Jerry Kaplan walks you through one-to-one correspondence, rational and irrational numbers, and the real number line. Each concept builds on the last, with no leaps of faith required. By the final chapters, you will see for yourself why Georg Cantor's revolutionary discovery holds not all infinities are the same size. And you will glimpse the astonishing possibility that there are more than two infinities.

Inside you will

- Why there are just as many even numbers as whole numbers, even though that seems wrong

- How mathematicians compare infinite sets.

- Why the real numbers are a bigger infinity than the fractions, and what that means for the nature of numbers themselves

No memorization. No advanced formulas. Just clear thinking, surprising questions, and answers that will change how you see mathematics.

134 pages, Paperback

Expected publication October 20, 2026

About the author

Kaplan is widely known as a serial entrepreneur, technical innovator, bestselling author, and futurist. He co-founded four Silicon Valley startups, two of which became publicly traded companies. His best-selling non-fiction novel "Startup: A Silicon Valley Adventure" was selected by Business Week as one of the top ten business books of the year, was optioned to Sony Pictures, and is available in Japanese, Chinese, and Portuguese. Kaplan has been profiled in The New York Times, The Wall Street Journal, Forbes, Business Week, Red Herring, and Upside, and is a frequent public speaker.

Kaplan is currently a Fellow at The Stanford Center for Legal Informatics. He also teaches Philosophy, Ethics, and Impact of Artificial Intelligence in the Computer Science Department, Stanford University. He holds a BA (1972) from the University of Chicago in History and Philosophy of Science, and an MSE (1975) and PhD (1979) in Computer and Information Science, specializing in Artificial Intelligence and Computational Linguistics, from the University of Pennsylvania.