With the release of the new Buzz Lightyear movie, the childhood memories came rushing back, and the catchphrase ”To infinity, and beyond” to this day gives me chills. It got me thinking about this encouraging quote, so I did a little digging.  Turns out that Mr. Lightyear is right!

Infinity. Just that simple word holds a tremendous amount of respect in my opinion, as it goes way beyond what we see in daily life. Of course, there are things that we can perceive as infinite. Looking out on the horizon is a good example, as it can definitely seem as infinitely long. However, somewhere we know that is not the case, and if we travel long enough towards the horizon, it will eventually cease to exist and be replaced by land. 

Understanding the theory of infinity is confusing to many people, as people in general often feel the urge to measure. Measure distance, quantities, anything to be able to compare and put the measurement in relation to something else to understand its dimensions. Nevertheless, infinity is not a number but is rather a concept, mostly pertaining to sets of objects, e.g stars in the sky or items in a bag that does not have a numbered ending. And believe it or not, despite the concept of not having an end, the theory of infinity allows for different sizes! During the late 19th century, German mathematician Georg Cantor demonstrated that some infinities are bigger than others.

Understanding the Difference in Size

How do we know if two sets are equal in size? To know this, we do not actually need to know the value of the size. Instead, all we need to verify is the relationship between the two sets. If the correspondence between the two sets is one-to-one, then we know for sure that they are equal in size, without measuring the actual size. A real simplistic example is to have two different bags, from which you simultaneously grab one item at a time. If the items run out at the same time, then you have just verified that the sizes are equal. If one bag still has items left in it, then the correspondence is not one-to-one, and the sizes are not equal. Now, let’s approach infinite sets the same way.

For this example, we are comparing natural numbers (positive integers, whole numbers, 1, 2, 3, etc.) with real numbers (positive and negative integers, basically any number that we can think of, for example, 4, 0, -3.5, 7/3, √13, etc.)

Cantor’s idea of understanding the different levels of infinity was to compare the infinite real numbers between zero and one, with the infinite natural numbers. His thesis was that there are more real numbers between zero and one, than there are natural numbers, even though they are both infinite. 

To apply Cantor’s idea when we compare the real numbers with the natural numbers, we initially assume that the two infinite sets of real and natural numbers are the same size, and can thus be put into a one-to-one correspondence as described before. Or more simply put, for every number within the scope of infinite real numbers between zero and one, there is a corresponding ”partner” within the scope of infinite natural numbers. For every natural number, N, there is a corresponding real number, RN. At first, this seems to follow a one-to-one correspondence. But oh, how wrong!

Diagonalization – Proof of Different Sized Infinities

In order to prove that infinity has different sizes, Cantor formulated a test. We already know that in order for any size to be equal, there has to be a one-to-one correspondence. As we are comparing natural numbers with real numbers, and in order for them to be equal, there has to be a corresponding real number, RN, for every natural number, N. As we are using natural numbers, we can actually sort the pairs in order as R1, R2, R3, and so on.

Now for the proof of different infinities.

Cantor creates a real number, referred to as RP, following this rule:

For the number P, replace the digit that is N places after the decimal point with something that is not the same as for RN.

To give some further explanation. The digit in the first decimal place of P can not be equal to the first decimal place of R1, the digit in the second decimal place of P can not be equal to the second decimal place of R2, and the same rule applies to every RN.

By following this mathematical method, known as diagonalization, we are generating a number P that will by construction, always differ from the other real numbers in at least one decimal place. Hence, the generated real number P does not have a partner among an infinite number of natural numbers! There is no longer a one-to-one correspondence between the real and the natural number; there are evidently more infinite real numbers than infinite natural ones.

In short, some infinities are bigger than others. How about that, huh?!