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
To see the other Super Facts click here
Super Facts 
I enjoyed the video!
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That is cool. Thank you Lynette.
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My mind is blown 😊 I had never thought much of infinity before, but found this bizarre and fascinating. Maggie
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Thank you Maggie. Yes before I found this out, I just thought that infinite is one concept, one thing that you cannot grasp, and I would not have believed infinities can come in different sizes.
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Another fascinating Super Fact. I remember encountering Cantor’s Theorem in college and finding it very mind-blowing!
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Thank you so much David. I don’t remember when I first came across it, it might have been college too but it could have been college. I also think it is mind-blowing.
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Truly mind-blowing, Thomas, both your post and the video! 🤯 Then again, I’m more of a “right-brain” person and didn’t get past grade 11 in math. Still, I enjoyed learning more about infinity and had no idea there were so many different types.
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Thank you Debbie. I think it is very mind-blowing too. I don’t remember when I first came across it. It was a long time ago but I think it is one of those strange and mind-blowing facts.
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Java Bean: “Ayyy, so they really do contain multitudes!”
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Yes Java Bean, there are small infinities and infinite infinities and infinite infinite infinities, and to infinity and beyond.
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What an elegant reminder that even infinity — the very idea of the unbounded — can have hierarchy and structure. Cantor’s vision turned chaos into order, showing that logic can measure even what the mind cannot contain. It’s both humbling and thrilling to realize that “forever” comes in sizes.
Your post captures that wonder perfectly — translating abstract mathematics into something almost poetic. The thought that there are infinities beyond infinity makes one feel small, yes, but also beautifully connected to the endless curiosity that drives science itself. ♾️✨
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Thank you so much Livora. You put that so wonderfully clear and poetic.
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Thomas, one might say that contemplating infinities over a cup of coffee offers its own quiet paradox: the finite warmth of the cup against the boundless expanse of thought it inspires. Like Cantor’s sets, each sip seems small, yet within it lies an uncountable spectrum of ideas, proof, perhaps, that even in the most ordinary rituals, the infinite finds a way to be present. I enjoyed this article.
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That is certainly a very interesting and profound thought. I will think about that for a while. I guess infinity is more a concept than a number and it is everywhere and always with us. Thank you so much for your philosophical commentary.
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Another fascinating topic Thomas. Some infinities are larger than others is an interesting concept. It makes me think of an infinite amount of bricks as opposed to an infinite number of feathers. No matter how infinite, the bricks will weigh more. 😊
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Thank you Debby. It is fascinating and I agree that your thought about feathers, bricks and infinity is an interesting thought.
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Thanks. 😊
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