One question has preoccupied humankind for thousands of years: Do infinities exist? More than 2,300 years ago Aristotle distinguished between two types of infinity: potential and actual. The former deals with abstract scenarios that would result from repeated processes. For example, if you were asked to imagine counting forever, adding 1 to the previous number, over and over again, this situation, in Aristotle’s view, would involve potential infinity. But actual infinities, the scholar argued, could not exist.

Most mathematicians gave infinities a wide berth until the end of the 19th century. They were unsure of how to deal with these strange quantities. What results in infinity plus 1—or infinity times infinity? Then the German mathematician Georg Cantor put an end to these doubts. With

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