Intuition breaks down once we’re dealing with the endless. To begin with: Some infinities are bigger than others.

I nfinity invites resistance. Aristotle rejected the existence of the infinite entirely; to him, infinity was simply a limit that could never be reached, not a true mathematical entity. In the early 17th century, Galileo wrote that typical ways of thinking about sets and numbers held no meaning in the realm of the infinite, and that mathematicians would only find paradoxes if they tried to apply their usual tool kit to it. And when, 200 years later, Georg Cantor formalized the idea that infinity comes in many sizes, he was met with anger and fear. His colleagues dismissed his work as that of a madman.

But in time, Cantor’s work on sets and infinity would form the bedrock of modern mathematics. As David Hilbert, another mathematical great, later wrote: “No one shall expel us from the paradise that Cantor has created for us.”

So how can infinity have different sizes?

Welcome to Cantor’s paradise.

...read more at quantamagazine.org