Why do some infinities seem bigger than others, and how do mathematicians compare their sizes?
In: Mathematics
When scientists talk about different infinites, they are talking about the size of infinite sets. A set is a collection of items where each item in the set is unique. For finite sets it’s easy to compare their sizes, you just count how many items are in each set and compare the numbers. For infinite sets we have to get creative.
For infinite sets mathematicians use the buddy system to figure out if two sets are the same size. As an example, let’s use the set of all whole numbers (1, 2, 3…) and the set of all even numbers (2, 4, 6…). At first you might think there are more whole numbers than even numbers, but that is incorrect. If we create a rule where we multiply each whole number by two to find it’s buddy, we can see that each whole number has a “buddy” in the even number set, and each even number has a “buddy” in the whole number set.
It turns out that a lot of infinite sets are actually the same size, but there are some that are bigger. In 1891 a mathematician by the name of Georg Cantor came up with a clever proof to show that the set of Real Numbers (numbers with an infinitely long decimal part) is actually bigger than the set of whole numbers. He showed that no matter how you do it you can never set up a ‘”buddy system” between these two sets. There will always be real numbers that aren’t included in whatever buddy system you set up.
This proof is known as Cantor’s diagonal argument, and it’s actually pretty easy to understand. You can read about it here: https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
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