# Five Ways to Think About Infinity

Infinity is weird. All the more reason to explore it, says mathematician Eugenia Cheng.

Infinity is big. Really big. As mathematician Eugenia Cheng puts it, it’s the “biggest thing there is.”

In her new book, Beyond Infinity: An Expedition to the Outer Limits of Mathematics, Cheng tackles the enigmatic concept, and she recently unveiled some of its mysteries on Science Friday.

Infinity is definitely weird, said Cheng, but we shouldn’t let that scare us. “When you are faced with the fact that infinity causes strange things to happen, you can either run away—and unfortunately that’s what too many people do with mathematics,” or “you can delve into it and try and figure out what’s going on.”

Here are some cool infinity-related ideas that Cheng shared with us.

There isn’t one way to think about infinity.

“Infinity doesn’t have just one answer. It doesn’t just have one way of thinking about it,” said Cheng. “There are so many fascinatingly different ways of thinking about it that produce different weird scenarios, like can you eat infinite cookies?”

Infinity isn’t any ordinary number.

It’s a common tendency: Children like to tack on “times infinity” and “infinity plus one” after all kinds of statements, treating infinity as if it were any ordinary number. Since the idea was first proposed thousands of years ago, mathematicians have debated over how infinity should be applied. Equations have been manipulated so that infinity is used like other integers, Cheng writes in her book.

But infinity is not an ordinary number and can’t be treated as such, Cheng writes in her book. Adding, subtracting, or multiplying with infinity leads to a terribly confusing mess.

(Stephen Simpson, a mathematician at Pennsylvania State University, takes that argument further and says that we shouldn’t assume actual infinity exists in the mathematical universe, and that “potential infinity” should be used instead to prove theorems.)

If you “pretend that infinity is a number for a second, then you get to deduce logically that everything equals zero, which is not that great,” said Cheng. “If everything equaled zero, well, there wouldn’t really be anything, would there?”

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There are infinite infinities.

“There’s a hierarchy of infinities, because as soon as you have one infinity, then you can create a bigger one, Cheng said.

The idea of infinite infinites grew out of a concept put forth by the late 19th century, German mathematician Georg Cantor. He invented set theory, in which he proved that a collection of objects could go on infinitely. A list of real numbers in a set is never-ending because “it is always possible to come up with a real number that isn’t on the list,” Natalie Wolchover wrote in Quanta magazine.

The infinitely big and the infinitely small are infinitely related.

“When I was little, we grew carrots in the garden,” said Cheng. “So I knew perfectly well that if we pulled them out of the ground before they were big, then, well, they would be small.” Sure, it’s a logical observation, but it also relates to a theorem taught in high-level calculus courses—the intermediate value theorem.

This advanced theorem states that a continuous function growing from one point to another will take on every value in between. To better understand the theorem, take this example from Cheng’s book: Someone who is currently 6 feet tall was at some point in the past 5 feet tall and 4 feet tall. But that person was also 5-feet-6-inches, and 5-feet-6.5-inches, and 5-feet-6.55-inches and so on. An infinite number of values lie between the two points. The same concept applies to full-grown carrots, which must have been—at one point—baby carrots.

“It comes from understanding not just infinitely big things but infinitely small things,” said Cheng. “The infinitely big and the infinitely small are, well, infinitely related.”

Forever and a day is different than forever.

Let’s say that a guy named Douglas and his mother were born on the same day 24 years apart. If they both lived to infinity, they would both be infinity years-old, Cheng said. Since the difference in Douglas and his mother’s age came at the beginning rather than at the end, Cheng reasons they would be the same age. On the other hand, “infinity plus one, now that could be really something different,” Cheng said.

When Shakespeare writes “forever and a day” in the play The Taming of the Shrew, he must have known “that forever and a day is longer than forever. Otherwise he’d have just said forever.”

But Douglas and his mom would “also both be elephants,” said Cheng, because anything would be true in a world where you lived to infinity.

These quotes have been lightly edited for clarity. For a full transcript, click here and scroll down to “transcript.”