Sizes of Infinity

by Agustín Rayo

Explore the structure of infinity with Georg Cantor and David Hilbert, two of the great minds of twentieth-century math.


Agustín Rayo

Agustín Rayo has served as Associate Dean of the School since February 2016, with a portfolio that includes oversight of the School’s mission in undergraduate education and diversity efforts. A Professor of Philosophy in the Department of Linguistics and Philosophy, Rayo’s research lies at the intersection of the philosophy of logic and the philosophy of language.

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Hilbert's Hotel

In this Wireless Philosophy video, Agustín Rayo teaches us about some weird properties of infinity, using an example due to mathematician David Hilbert called ‘Hilbert’s Hotel’. He shows us a result proved by another mathematician, Georg Cantor: that many infinite collections of things are the same size. Things that are the same size include: the natural numbers, the natural numbers plus one, the natural numbers plus the natural numbers, and as many copies of the natural numbers as there are natural numbers! Amazing!


Getting Real

After part one, you might have thought that all different infinite collections of things are the same size. Not so! In this video, Agustin Rayo shows us another of Georg Cantor’s results: that for every size of infinity, there is a bigger one! An example: there are way more real numbers than there are natural numbers.