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Infinity is an endlessly fascinating subject. You could talk about it forever.
Peter Alfeld, professor of mathematics at the University of Utah said that, “Infinity has a beauty and elegance that’s eternal.”
Prior to researching infinity, my concept of it was limited to the basics. I knew that space extended into infinity. I knew that time could extend into infinity, at least theoretically. I knew that numbers were infinite.
Infinity is a concept used in many fields, particularly mathematics, physics, philosophy and religion. And it quickly delves into realms of abstraction, which I needed help understanding.
I sat down with Alfeld, who explained to me how it works, its principles and patterns, as well as its “infinite beauty and elegance.”
Let’s explore infinity.
Infinity refers to anything that is without limit. In mathematics, infinity is often used as a number, but it’s used more like a special kind of number.
With infinity, things that may seem to be impossible are not. Take for instance the prime numbers (numbers that can only be divided by one and itself, such as three, five and 13). It seems reasonable that once you could find a large enough number such that every higher number could be divided by more than only one and itself. But, in fact, mathematicians tell us that not only are there a lot of prime numbers, but there are infinite prime numbers.
The field of infinity is filled with concepts that seem contradictory at first glance. It contains elements that can seem both logical and illogical at the same time, and many of its principles are counterintuitive. To explain infinity to non-mathematicians, the mathematician David Hilbert invented a concept he called Hotel Infinity. Imagine a hotel that is infinitely large, that has an infinite number of rooms, and all the rooms are filled. But if a new guest wants to move in, even though Hotel Infinity is already filled to capacity, there’s still room to add another guest. And if the next night an infinite number of new guests arrive and want to stay at Hotel Infinity, there’s room for them too, even though it’s already filled to capacity.
There is an infinity of numbers. We are all familiar with that idea, and that seems reasonable. You can just keep counting and never reach an end. But there are infinities in places you wouldn't expect, for instance, in between every number zero and one. Or between zero and two. or between zero and .0009. In between ever number.
A real number is a number that represents a quantity such as the integer -7, or the fraction ¾, and the irrational numbers, such as the familiar pi or tau. A natural number is a whole number such as one, seven and 15.
The really weird thing, perhaps the most difficult to understand thing about infinities, is that they come in different sizes, some really are bigger than others, and some are the same size that don't seem like they should be. For instance, the infinity of integers (1, 2, 3, 4, 5, and so on) is a actually smaller than the infinity of numbers between zero and one. Then integers are countably infinite, while the numbers between zero and one are uncountably infinite. How in the world can that be?
And that brings us to set theory. Set theory is one of the key elements of infinity. It was created by Georg Cantor, a German mathematician, in the late 19th century. Set theory is a way of quantifying different infinities, to determine if one set is larger, or equivalent, to another set.
A set is a collection of something. Anything at all. It could be a group of cars, elephants, molecules or numbers.
For example, all the natural numbers forms a set (of infinite size). And sets can be further broken down into subsets. For example, the natural numbers can be broken down into subsets of even numbers and odd numbers.
After the numbers have been broken down into different sets and subsets, one can determine if one set is larger than another using set theory.
“All sets are infinite,” Alfeld said, “so counting doesn’t work as a way of determining which set is bigger. Instead you have to pair them up.” If you are trying to determine if an infinity of even numbers is equivalent to an infinity of odd numbers, you begin pairing the numbers of one set to the numbers of the other set. You can pair zero to one, two to three, four to five. And through that we can see that the set of even numbers is equivalent to the set of odd numbers.
With infinity, things that may seem to be impossible are not.
Simple enough.
“But,” Alfeld said with a pause (and I can tell by the mad scientist twinkle in his eye that he’s about to say something that’s going to blow my mind), “the set of even numbers is also equivalent to the set of natural numbers.”
How can that be? The natural numbers include all even and odd numbers, so it seems like it must be twice as infinite.
“It all depends on how you pair the numbers,” Alfeld said. “You can pair zero to one, two to three, like we just did, but you can also pair 17 to 34, 25 to 50, and in this manner all the numbers from one set will be paired with all the numbers from the other set, and through that we can determine that they are equivalent sets.”
I think I can actually feel the neurons in my brain creating new neural pathways.He's talking about one-to-one correspondence and it is the centerpiece of comparing infinities. They are the same size (cardinality if you want to get technical) if for every number in one set, there is exactly one corresponding number in another set. But not all sets of numbers are the same.
“You cannot pair the real numbers with the natural numbers,” says Alfeld. “There will always be real numbers left.”
If we apply this to the Hotel Infinity analogy, we can say that Hotel Infinity is filled to capacity, and when the Real Number Family arrives and wants to move in, not only is there room for them throughout the hotel, but they can all fit into a single room. Another way of looking at it is, if real numbers and natural numbers each had their own Hotel Infinity, the real numbers couldn’t fit inside the natural numbers hotel, but the natural numbers could fit inside the real numbers hotel and have (infinite) room to spare.
“All right,” I tell Alfeld at the end of our discussion. “I think I’ve got it.” But as soon as I say that, the mad scientist twinkle returns to his eye.
“Think about this,” he says. “The set of prime numbers is equivalent to the set of natural numbers. They’re equally infinite.”
That neural pathway I was building just snapped.
If you have a science subject you'd like Steven Law to explore in a future article, send him your idea at curious_things@hotmail.com.
