The Book of Infinity

Throughout Spring 2023 I shall be serializing here the chapters of my book, The Book of Infinity, a series of vignettes on infinity. I aim to cover a huge collection of topics—Zeno’s paradox, the coastline paradox, fractal dimension, supertasks, the paradox of the largest number contest, Galileo’s Salviati, Hilbert’s Grand Hotel, Cantor’s uncountable cardinals, Goodstein and the Hydra, counting in the ordinals, the infinitary Liar paradoxes, the continuum hypothesis, the axiom of choice, orders of infinity, infinitary utilitarianism, infinitary computability, indescribability, the sand reckoner, paradoxes of high dimension, the outer limits of reason via incompleteness, and much more.

I’ve decided to experiment with this serializing style of publishing my book, rather than publishing in the conventional manner, since I see several benefits. One of them is the stronger connection with all of you, my readers. Please participate in the comment sections on the posts. I try to read everything there and I shall often respond to posts.

Chapters will be released, about one or two a week, all through the Spring. I am using this book as the main text for a new undergraduate course I have designed on infinity, which I am teaching this semester at the University of Notre Dame.

Here are the currently-released chapters, in the book order. Please enjoy!

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The Book of Numbers
A curious order on the natural numbers Consider the natural numbers up to ten, but in the following curious order: 8 5 4 9 1 7 6 10 3 2 0 What order is this? Is it random? No, I have placed these numbers in a very definite and logical order, using an underlying order idea with which I am sure that you, gentle reader, are familiar. Can you discover th…
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Zeno's paradox
Zeno's paradoxes of motion The Greek philosopher Zeno of Elea (c. 490–430 BC) argued in antiquity that all motion is impossible. It is simply impossible to walk through town or even across the room, to go from here to there. What? We know, of course, that this…
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The infinite coastline paradox
A century ago Lewis Fry Richardson (1881–1953), noted polymath with prizes now named in his honor in metereology and scientific peace research, had a mathematical theory for predicting the likelihood of war between two countries based in part on the length and nature of their common border…
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Supertasks
Thomson's lamp You are sitting comfortably in your study reading in the twilight. It is beginning to become dark, and so you turn on the light. But after a little while, you realize the brightness of the lamp has ruined the twilight mood, and so you switch it off. But now it is becoming truly too dark to read, and so you turn it back on again. But of course it is still too bright, so you quickly turn it off, and then on again, and off, ever more rapidly in the vacillating panic of your indecision…
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