Super fact 68 : Infinities come in different sizes. Some infinities are larger than others. In fact, it is possible to create an infinite number of different-sized infinities. Some infinities are countable and others are uncountable.
How can infinity be of different sizes? Infinity is infinity after all. Infinity is an absolute, an absolute we cannot imagine. However, at the end of the 19th century a German mathematician by the name Georg Cantor demonstrated that there exists a variety of infinities, and some are larger than others. In fact, his theorem implies the existence of an infinity of infinities. For those who have not heard of this concept before it is surprising and yet it is true, and it is also kind of an important thing to know about mathematics. This is why I consider this a super fact.
Let me explain. Think about all natural numbers, 1, 2, 3, 4, 5, …, a billion, a quintillion, … You can keep counting forever. There are infinitely many natural numbers. Now take two natural numbers, the numbers 3 and 4. How many numbers are in between 3 and 4? Well, you have 3.5, 3.55555, 3.016893, and you have the number pi, 3.14159265358979323846, and infinitely many other numbers with infinitely many decimals. 3 and 4 are only two numbers but you have infinitely many decimal numbers, or so called real numbers, between 3 and 4. That is true for the two numbers 4 and 5 as well, and 5 and 6, and all natural numbers. Even though there are infinitely many natural numbers there are infinitely many more real numbers.
For the explanation of the mathematical proof see this article in Scientific American. This article explains why there are an infinite number of different-sized infinities.
I should say that today Saturday November 8, 2025, is National STEM Day, or STEAM Day. Stem stands for Science, Technology, Engineering, and Mathematics. The extra ‘A’ in STEAM stands for Art. Therefore, I thought that posting something related was appropriate for today, in this case it is Mathematics. I have to admit I am posting it a little bit late in the day though.
To infinity … and beyond!
As mentioned, infinites can have different sizes. How about, for example, the set of all natural numbers and the set of even numbers. After all, it seems like the set of all even numbers should be half as big. As it turns out they are the same size. That’s because for all natural numbers 1, 2, 3, 4, 5 you can always find a buddy, 2, 4, 6, 8, 10 among the even numbers. This is called a bijection. However, you cannot do this with natural numbers versus real numbers. You cannot pair up a natural number with the real numbers because there are infinitely many for each single natural number.
The set of all real numbers is really a larger infinity than natural numbers, which by the way is the smallest infinity. The sets of natural numbers and, for example, even numbers are called countable infinities, and the set of real numbers are called uncountable infinities. In addition, there are infinitely many different sizes of infinities that are larger than both natural numbers and real numbers.
Another interesting, related issue is that Cantor tried to prove that there are no infinities with sizes that lie in between the natural numbers and the real numbers. This was called the continuum hypothesis. As it turned out this hypothesis is not possible to prove (you can prove that it cannot be proved) at the same time as it is not possible to disprove. It is related to Gödels incompleteness theorem but is not the same. However, it is another absolute limit to human knowledge, things we know that we can never know.
If you want to have your mind blown, watch this 8 minute YouTube video about infinities of different sizes and the continuum hypothesis.
Other Mathematics Related Super Facts
- The Surprising Monty Hall Problem
- Every Symmetry is Associated with a Conservation Law
- The Euler Number Math Magic
- Eulers Polyhedra Math Magic
Super Facts 