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Alasdair Wilkins — The notion of infinity is fundamentally beyond the human ability to comprehend, but that hasn't stopped mathematicians from trying. So just what is infinity, and why is there more than one of them? And just what is infinity plus one?

Last week, we searched for the largest meaningful number in the universe, but all of these must of course pale in comparison to infinity. Mathematicians define "infinity" very strictly. But we'll stick with a broader, everyday definition: Infinity covers any number that isn't finite. Now, without further ado, let's expand our minds and tiptoe towards infinity.

The Beginning of Infinity

In order to talk about infinity, we first have to find a way to define it mathematically. That isn't an easy task - while the concept of infinity was known to the ancient Greeks, and it features prominently in the calculus of Isaac Newton and Gottfried Liebniz, infinity wouldn't be rigorously defined until the late 1800s. Before that, it was just some vast, amorphous concept, more an artifact of certain mathematical operations than something worth understanding in its own right.

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Indeed, many 19th century mathematicians found infinity to be vaguely distasteful, and they felt it had no place in serious mathematical discussion. At best, infinity was something for philosophers to discuss, and you can imagine the sort of disdain with which such pronouncements were made. It was in that context that Georg Cantor published the first proof of the existence of infinity in 1874.

Born in Russia but raised in Germany, Cantor provided a stunning and instantly controversial proof that not only defined the nature of infinity, but it also revealed that multiple infinities existed, and some were larger than others. What made his achievement all the more remarkable was that he had built the entire thing out of an ancient and seemingly useless branch of mathematics known as set theory. Basically, it was the mathematical equivalent of building an interstellar drive out of a wheelbarrow.

Set Theory

Set theory really does seem laughably simple, but it's proven to be among the most powerful tools in modern mathematics. The basic idea can be found as far back as Aristotle, and it's simply this: numbers can be grouped into sets. That's it. Hell, even that can be simplified: things can be grouped into sets. You can take the numbers 1, 2, 3, and 4 and put them in the set {1,2,3,4}, which we'll call Set A. You could also take the letter D, a tuna sandwich, a Thomas Hardy novel, and the planet Neptune and put them in the set {D, tuna sandwich, Thomas Hardy novel, Neptune}, which we'll call Set B.

A brief introduction to infinityNot exactly what you'd call impressive, right? But amazingly, we're only a couple of steps away from the big insight that reveals infinity. Let's say you took those two sets I just described and compared them. Which one is bigger, Set A or Set B? If you think about it in individual terms, that might seem like a nonsense assignment - how could you compare a Thomas Hardy novel to the number 3, for instance? The key here isn't to look at the specific terms, but to look at how many terms there are. Since there are four terms in both sets, they're of equal size.

Let's take nothing for granted though. How did we deduce there were four terms in both sets? I'm guessing most of you would have simply counted how many were in each set and then compared them...again, this is basic, basic stuff.

Last week, we searched for the largest meaningful number in the universe, but all of these must of course pale in comparison to infinity. Mathematicians define "infinity" very strictly. But we'll stick with a broader, everyday definition: Infinity covers any number that isn't finite. Now, without further ado, let's expand our minds and tiptoe towards infinity.

The Beginning of Infinity

In order to talk about infinity, we first have to find a way to define it mathematically. That isn't an easy task - while the concept of infinity was known to the ancient Greeks, and it features prominently in the calculus of Isaac Newton and Gottfried Liebniz, infinity wouldn't be rigorously defined until the late 1800s. Before that, it was just some vast, amorphous concept, more an artifact of certain mathematical operations than something worth understanding in its own right.

Full size

Indeed, many 19th century mathematicians found infinity to be vaguely distasteful, and they felt it had no place in serious mathematical discussion. At best, infinity was something for philosophers to discuss, and you can imagine the sort of disdain with which such pronouncements were made. It was in that context that Georg Cantor published the first proof of the existence of infinity in 1874.

Born in Russia but raised in Germany, Cantor provided a stunning and instantly controversial proof that not only defined the nature of infinity, but it also revealed that multiple infinities existed, and some were larger than others. What made his achievement all the more remarkable was that he had built the entire thing out of an ancient and seemingly useless branch of mathematics known as set theory. Basically, it was the mathematical equivalent of building an interstellar drive out of a wheelbarrow.

Set Theory

Set theory really does seem laughably simple, but it's proven to be among the most powerful tools in modern mathematics. The basic idea can be found as far back as Aristotle, and it's simply this: numbers can be grouped into sets. That's it. Hell, even that can be simplified: things can be grouped into sets. You can take the numbers 1, 2, 3, and 4 and put them in the set {1,2,3,4}, which we'll call Set A. You could also take the letter D, a tuna sandwich, a Thomas Hardy novel, and the planet Neptune and put them in the set {D, tuna sandwich, Thomas Hardy novel, Neptune}, which we'll call Set B.

A brief introduction to infinityNot exactly what you'd call impressive, right? But amazingly, we're only a couple of steps away from the big insight that reveals infinity. Let's say you took those two sets I just described and compared them. Which one is bigger, Set A or Set B? If you think about it in individual terms, that might seem like a nonsense assignment - how could you compare a Thomas Hardy novel to the number 3, for instance? The key here isn't to look at the specific terms, but to look at how many terms there are. Since there are four terms in both sets, they're of equal size.

Let's take nothing for granted though. How did we deduce there were four terms in both sets? I'm guessing most of you would have simply counted how many were in each set and then compared them...again, this is basic, basic stuff.

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