How Big are All Infinities Combined? (Cantor’s Paradox) | Infinite Series

Infinities come in different sizes. There’s a whole tower of progressively larger “sizes of infinity”. So what’s the right way to describe the size of the whole tower?

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Previous Episode:
The Geometry of SET

Check out the solution to the Geometry of SET challenge problem right here:

Talking about the sizes of infinite things is tricky in part because the word “infinite” is often used in two distinct ways — to refer to the sets themselves, and also to refer to the *sizes* of those sets. In what follows, let’s try to keep as sharp a distinction as we can between infinite sets and infinite set *sizes*, because doing so will let me highlight an especially paradoxical feature about infinite sizing that I don’t think gets enough coverage. The technical term for a “size”, infinite or otherwise, is “cardinality”, and I should probably use a term like “numerousness” or “numerosity” rather than “size” because the idea it tries to generalize is the notion of “how many”. Still, I’m going to say “size” a lot in this episode just because it’s easier.

Written and Hosted by Tai-Denae Bradley
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow and Linda Huang
Made by Kornhaber Brown (

Thanks to Peleg Shilo, Anurag Bishnoi, and Yael Dillies for your comments on last week’s episode:

Special thanks to Roman Pinchuk for supporting us on our Converse level on Patreon.

Along with thanks to Matthew O’Connor, Yana Chernobilsky, and John Hoffman who are supporting us on Patreon at the Identity level!

And thanks to Mauricio Pacheco and Andrew Poelstra who are supporting us at the Lemma level!

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