What Do You Know About Infinity?

What do you think of the term “infinity”? In layman’s terms, infinity is something that has no end, something that is bigger than all the things we know. This is what most people think! Many people who have not seen advanced mathematics do not know about the fact that there are many different types of infinity, some of which are bigger than others. Now how is that possible? Isn’t there just one single infinity which quantifies the essence of uncountability?  

First of all, what exactly is infinity?

In most areas of mathematics, infinity is defined to be a quantity that is bigger than all the numbers we know. No matter what number you choose, infinity will always be greater than this number. If infinity is bigger than all the numbers, what about “infinity + 1”? Contrary to what some people believe, infinity itself is not a number. It is actually more of a concept! It is therefore improper for someone to talk about “infinity + 1”, because it is not valid. Addition only applies to numbers and infinity is not really a number.

Different sizes of infinity

There are many different types of infinity that mathematicians talk about. Georg Cantor, a German mathematician, started thinking about this and actually gave a proof. In fact, it gets so abstract after a certain point that it almost gets philosophical. When he first introduced this, it was met with a lot of opposition from the mathematics community at the time. It was just so counter-intuitive that people refused to believe it! What we will discuss here is only the tip of the iceberg. We will mostly discuss about two common sizes for sets: countably infinite and uncountably infinite.

Size me up!

Before we delve deeper, let’s see what cardinality means. The word cardinality refers to the size of a set. Cardinality of a set is a whole number. Simple enough right? A set is just a collection of objects. The objects can be numbers, alphabets, words, etc. For example, the cardinality of the set {12,43,1,65,29,54} is 6, since there are 6 elements in the set. When mathematicians talk about sets, knowing their cardinality can often tell many properties about the set. However, when it comes to sets that have infinitely many elements, finding the cardinality can prove to be somewhat difficult.

If the set has infinitely many elements, will the cardinality not be infinity?

This is where the haze begins. If you read the last paragraph closely, you will realize that cardinality has to be an actual number. Hence the cardinality of a set with infinitely many elements will not equal infinity. Let’s look at the cardinality of some sets. When we look at the natural numbers (1,2,3,…), it is clear that there is an infinite number of them. In other words, given any natural number n, there will always exist a natural number n+1, and this can continue on forever.

Now, any set that has a cardinality the same as that of the natural numbers, we call countably infinite. This is called countably infinite because there is no natural number between two consecutive natural numbers, hence the natural numbers are countable. Now the question would be ‘how can we tell the cardinality of the natural numbers if they are infinite?’. Well, the answer is, we don’t necessarily know the cardinality of the set of natural numbers. But on the other hand, it is not hard to tell when another set has the same cardinality as the set of natural numbers.

An example

We say that a set P has the same cardinality of the natural numbers if we can find some way to associate all of the elements in P with a unique natural number. For example, the set of prime numbers is countably infinite because it has the same cardinality as the set of natural numbers. We see this by associating each successive prime number with a successive natural number, basically counting the prime numbers. So now you have seen that the prime numbers are countably infinite, since the cardinality of the set of primes is the same as that of the natural numbers. In this way, both sets have the “same size” infinity. Same is true for the set of rational numbers as well. A rational number is obtained by taking the ratio of two integers (denominator being non-zero). Every rational number can be associated uniquely with a natural number.

What was the point of that example?

The point of that example was to get you to the next concept, uncountably infinite sets. An uncountably infinite set has no way to associate each of its elements to a unique natural number. A very easy example of an uncountable set is the set of real numbers. Real numbers are pretty much everything that you can think of, aside from ‘i’ (the imaginary number). Real numbers include 0, 1, -23/41, √3, pi, ‘e’ (natural log), and everything else you can think of. When you look at two real numbers, no matter what they are, there is always another real number hanging out in between them. This property explains why the real numbers are uncountable. Pick two real numbers a and b and start trying to associate natural numbers to every real number in between a and b. You will use an infinite number of natural numbers to count the real numbers in between a and b. After you are done with it, there is still the entire real number line outside of a and b left to count. It is in this way that the real numbers have more elements than the natural numbers, even though the sets are both infinite. This is exactly what mathematicians mean when they talk about “varying sizes of infinity”.

There is a whole theory about the sizes of infinite sets developed by Georg Cantor. It is a very deep theory, much of which depends on several things that are far beyond the scope of this blog post. If you are interested, you should consider reading this. If you go through it, you will learn that there are actually infinite types of infinities!

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