Diagonalization
A collection of objects or things is called a set. For example, we can put all of the cats in your neighborhood in a set, using their name to represent each one as an element of the set. This kind of set would be called a finite set, meaning that we can count every element (or cat!) in the set and eventually run out of cats to count. However, we can also have a set that doesn’t end, also called an infinite set. If we made our set of cats into an infinite set and tried to count it, we would be counting cats forever.
We can also say that a set is countable or uncountable. A set is countable if there is a 1-to-1 mapping from the set S to the Natural Numbers. We can demonstrate countability by representing elements from the set S with the natural numbers, which is also an infinite set. We can call both finite and infinite sets countable, but all uncountable sets are infinite. Infinite sets are countable when we can assign an element from the (infinite) natural number set to each element in the infinite set. But how do we show that an infinite set is uncountable?
Diagonalization is used to show that there are infinite sets that are uncountable, which means that the set does not have 1-1 matching with the set of natural numbers, which are infinite. This technique is famously known as Cantor’s Proof.
Look at every line and row in this 8x8 grid above. If we take each “on” light to represent a 1 and each “off” light to represent a 0, we can see how this grid could be translated into binary. We can also imagine each row and column extends infinitely, like they would if we had a set of infinite bit-strings in an infinite set. All the strings shown in the grid of rows and columns are countable, even if they are infinite, since we can assign an element from the natural numbers to each of them.
When looking at our representation, we can see that there are bits that make up a diagonal through the center of the grid. These highlighted bits create a new bit-string that is different from the counted bit-strings, meaning it’s not represented in the rows or columns. (This is the same string that we see in the 1x8 grid next to the large grid.) Since we can see that we can find an uncounted string from an already counted infinite set, this means that the superset of this set is uncountable.
In this way, we can see that it’s possible for an infinite set to be uncountable!