ELI5: There are infinite numbers between 0 and 1. There are also infinite numbers between 0 and 2. There would more numbers between 0 and 2. How can a set of infinite numbers be bigger than another infinite set?

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I think the problem is you are thinking of “infinite” to be “a very big number”. It is not a very big number, it’s a different kind of thing. A similar problem exists with zero, in that it’s not just “a really small number”, it’s actually zero. For example if I take a really small number like 0.0000001 and double it, I get 0.0000002. If I take 0 and double it, I still get zero. 2×0 is not bigger than 1×0. If I have an infinite number of numbers between 0 and 1, then they are separated by 0. If I double all of those numbers, then they are separated by 2×0, so they are still separated by 0. Edit: thanks for the kind words and shiny tokens of appreciation. This is now my second highest voted post after a well timed Hot Fuzz quote, I guess that’s what reddit is like.


Here’s a way to see that there are the same “size”. We’re going to show that for each number between 0 and 1, there exists a number between 0 and 2, and vice versa. 1. Pick any number between 0 and 1. 2. Multiply it by 2. 3. You now have a number between 0 and 2. 4. Vice versa, pick any number between 0 and 2 5. Divide it by 2. 6. You now ave a number between 0 and 1. This works both for the case of rational and real numbers. We just constructed a so-called bijection between the intervals [0,1] and [0,2].


Your intuition for size comes from the structure of intervals, rather than the amount of elements they have. The intervals [0, 1] and [0, 2] have the same quantity of points, because you can pair them up. However, the interval [0, 2] is twice as long as the interval [0, 1]. The particular elements within [0, 2] and their relation to each other is what gives it that length, not the amount of elements.


Well, two things are happening here. There are different kinds of infinities, some of which are larger than others. However, the number of real numbers between 0 and 1 is the same as the number of real numbers between 0 and 2. You can prove this second one by creating what’s called a bijection – showing that for every member of group A there is exactly one member of group B. This is easier to show with another set but it does carry over into this situation. Let’s say we’re comparing every even number with every even AND odd number. It seems like the second one should be larger, right? But if we take every even number and divide it by two, we go from 0, 2, 4, 6… to 0, 1, 2, 3… That second set sure looks like the set of all even and odd numbers. The same thing applies here. If you take every real number between 0 and 2, and divide them all by 2, you get every real number between 0 and 1. There is also a way to show that some infinities are larger than others. This one is a bit harder to picture, but imagine a list of every real number between 0 and 1. This is every rational number, but also every irrational, every transcendental, every number that is between all of those forever. It’s not obvious how you could sort such a list but let’s say you just write down the numbers randomly. Well, this is a list that you can order 1, 2, 3 etc. Sure, it’s infinite, but so is the list of counting numbers. Right now there’s no obvious problem; if they’re both infinite, you’re good to say that they’re the same size. However, we can do something that breaks this. Let’s create a new number; the rule is that it’s different from the first number in the first decimal place, different from the second number in the second decimal place, and so on forever. This is definitely a real number, meaning it should be on the list, but it’s definitely not on the list, since it’s different from every number on the list in at least one place. Even if you added this new number to the list, you could just do this again. What we’ve done is shown that, even if we use all the counting numbers, all infinity of them, we can still create numbers that are not on that list and for which there is no matching number. There are numbers left over after we’ve used all the counting numbers. Even though they’re both infinite, there are *more* real numbers than there are counting numbers. I hope this makes sense.


The concept of size that’s used for infinite sets is basically this: Two sets are the same size if you can pair the members from one up with the members of the other with no leftovers. You can do that with the two sets OP asked about, so they’re actually the same size. But you can’t do that with the set of all integers and the set of all numbers between 0 and 1.