To Infinity and Beyond

Image: NASA/Paul E. Alers.

Image: NASA/Paul E. Alers.

In everyday life, infinity has multiple meanings. To most people, infinity means being bigger than any number, the entirety of everything, or something that has no ending. In mathematics, infinity is strictly defined, but for the purposes of this post we will use a more intuitive definition: bigger than any finite number or an unnamed number that is arbitrarily large.

Much like the number zero, infinity is a concept that had been controversial amongst ancient mathematicians and barely accepted by modern mathematicians before the twentieth century. The ancient Greeks are the first recorded society that used the concept of infinity, although they did not readily accept it. They believed (as did mathematicians and philosophers for two thousand years afterward) that there was a potential for something to be infinite but that it could not actually be infinite. Much like the set of integers could potentially be infinite by adding one to a get another larger number, but the set of all infinite integers did not exist.

It was not until the late 1800’s that mathematician Georg Cantor formalized and proved the concept of infinity. Not only did Cantor prove the existence of infinity but, much to everyone’s surprise, he also proved that there are different sizes of infinity.

Infinity. Image: Public domain, via Wikimedia Commons.

Infinity. Image: Public domain, via Wikimedia Commons.

How could something that is arbitrarily large be bigger than something else that is arbitrarily large? Take for example the set of all the even numbers, (2,4,6,…) and the set of all the natural numbers (1,2,3,…). The most obvious answer would be that the set of all even numbers is half as large as the set of all natural numbers, even though both of them are infinite. In fact, the correct answer is that the two sets of numbers are actually the same size. How is this so? Cantor showed this by assigning each element in each of the sets a number. The number 1 from the real’s and the number 2 from the evens was assigned the number 1, 2 and 4 were assigned the number 2 and so on. By doing this he showed that no matter how large either set becomes he can still assign them the same numbers, therefore proving that they are the same size. This is what is known as countable infinity.

So then how do we get infinities that are different sizes? Think of the integers on a number line, all the way from negative infinity (Yes, this is a real thing. It just means a negative number that is arbitrarily large) to positive infinity. It is quite large but as we talked about in the previous paragraph it is still countable. Now, take only the segment of line between one and two. In this segment you have the numbers 1.5, 1.2, 1.24, 1.247,… In fact you have so many numbers in this small segment that it would be impossible to list them all. To create a number that is different than any number you have chosen all I have to do is take the first digit of the first number and increase it by one, then take the second digit of the second number and increase that by one and so on and so forth. In doing this I have created a new number that you have not listed. We therefore have a set that is so large that it is uncountable.

The concept of infinity is very important in many areas of math. The field of calculus depends entirely on it and many other fields in mathematics use the concept in important theorems. In fact, the infinite arises much more in mathematics than does the finite.

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