I know the actual technical answer. I’m looking for a witty parallel that has a low chance of triggering an infinite “why?” procedure in a child.
In: 1335
Infinity isnt a number. It’s a concept. So you cant do math with infinity. Theres no such thing as 2 times infinity. Etc.
Infinity is not “everything”, it means , there is no end. Although you can double the numbers, for example, 1,2,3,4,5… And 2,4,6,8…, the counting just never ends. All numbers between 0 and 1 are also between 0 and 2, but not the other way around. But in the end, in both cases, there are infinitely many numbers.
So mathematically there’s this funky concept that some infinities are bigger than others. There are infinite numbers between 0 and 1, but there are more infinite numbers between 0 and 2.
Honestly, for your sanity I’d not think too hard about it
Take every real number between 0 and 1, and pair it up with a number between 0 and 2, according to the rule: numbers from [0,1] are paired with themselves-times-two.
See how every number in the set [0,1] has exactly one partner in [0,2]? And, though it takes a couple extra steps to think about, every number in [0,2] has exactly one partner, too?
Well, if there weren’t the same ~~number~~ quantity of numbers in the two sets, that wouldn’t be possible, would it? Whichever set was bigger would have to have elements who didn’t get paired up, right? Isn’t that *what it means* for one set to be bigger than the other?
Infinity has different sizes, and there’s no single way to think about it, so there’s no “right answer”, but one way to think about it is to use a type of math called “set theory”.
The smallest infinite size can be thought of as the natural numbers.
1, 2, 3, 4, and so on to infinite. We call this size of infinite “aleph null” (null is sometimes written as 0). Another term we use to describe them is that they are “countable”.
Interestingly, other sets of numbers have this same size. All of the even numbers are the same size.
But there should be twice as many natural numbers as even numbers, right? We’re missing all of the odd numbers!
But we can’t think of them like that, since there’s an infinite number of both natural numbers and even numbers. How do we understand what the heck this means?
We think about how they group up together.
The first natural number is 1. The first even number is 2.
The second natural number is 2. The second even number is 4.
The third natural number is 3. The third even number is 6.
Notice we are “counting” the even numbers.
We can “count” all of the odd numbers too. We can also “count” all of the integers (so negative numbers too – e.g. 0, 1, -1, 2, -2, 3, -3, …).
Another way to think of it is that we can “map” or match each natural number to each even number. We can do this forever and ever, but there will never be a point where we can say “this natural number has no corresponding even number”.
So what’s bigger than “countable infinite” or “aleph null”? When would it not be possible to map all the natural numbers to another set of numbers?
You’ve probably already guessed “real numbers!”.
Let’s try counting them:
The first natural number is 1. The first real number is… 0.1? Or 0.01? Hmm.
The second natural number is 2. The second real number is… 0.2? Or 0.01? We’ve got two directions – we could add up, or we could add a zero and make the number smaller each time.
The third number is… 0.3? Or 0.001?
There’s at least 2 obvious ways we can try and count them, but something feels wrong. It’s not clear we’re ever going to count them all.
As it happens, it doesn’t matter what method you use to try and map the natural numbers to the real numbers. There’s always going to be a way you can make up a real number that isn’t counted in that mapping. This size of infinity is “aleph 1”. The more complicated form of this argument is known as “Cantor’s diagonal argument”.
So where does this leave us with the original question – is the infinite size of real numbers between 0 and 2 “bigger than” the infinite size of real numbers between 0 and 1? The answer is “no”, it’s the same size of infinite. ~~It’s much more difficult to conceptualize, but~~ (see Jemdat_Nasr’s response for a great visual representation) it is possible to map the set of all the real numbers from 0 to 1 onto the set of all the real numbers from 0 to 2. The infinite size of each sets of numbers is “aleph one”.
How many of these aleph numbers representing infinity are there, anyway?
An infinite number, of course! 🙂
To start off with, let’s talk about how mathematicians count things.
Think about what you do when you count. You probably do something like looking at one object and saying “One”, then the next and saying “Two”, and so on. Maybe you take some short cuts and count by fives, but fundamentally what you are doing is pairing up objects with whole numbers.
The thing is, you don’t even have to use whole numbers, pairing objects up with other objects also works as a way to count. In ancient times, before we had very many numbers, shepherds would count sheep using stones instead. They would keep a bag of stones next to the gate to the sheep enclosure, and in the morning as each sheep went through the gate to pasture, the shepherd would take a stone from the bag and put it in their pocket, pairing each sheep with a stone. Then, in the evening when the sheep were returning, as each one went back through the gate, the shepherd would return a stone to the bag. If all the sheep had gone through but the shepherd still had stones in his pocket, he knew there were sheep missing.
Mathematicians have a special name for this pairing up process, *bijection*, and using it is pretty important for answering questions like this, because it turns out using whole numbers doesn’t always work.
Now, let’s get back to your question, but we’re going to rephrase it. Can we create a bijection and pair up each number between 0 and 1 to a number between 0 and 2, without any left over?
We can, it turns out. One way is to just take a number between 0 and 1 and multiply it by two, giving you a number between 0 and 2 (or do things the other way around and divide by 2). If you’re a more visual person, [here’s another way to do this](https://i.imgur.com/exWVWkU.mp4). The top line has a length of one and the bottom line a length of two. The vertical line touches a point on each line, pairing them up, and notice that as it sweeps from one end to the other it touches every point on both lines, meaning there aren’t any unpaired numbers.
If there is one infinity between 0 and 1, then 0 to 2 would be twice that. While these are technically the same, the second is slightly larger because of the two infinity and beyond.
Get all the numbers from 0-1.
For each number, double it’s (e.g. 0.25 becomes 0.5, and pi/4 becomes pi/2, and .99 becomes 1.98, etc).
You now have all the numbers between 0-2.
Doubling the value of a number still results in a single number.
So this process has not change how many numbers we had.
Therefore, 0-1 and 0-2 have the same amount of numbers.
Pretty much everyone else in this thread is wrong (as of the time of me posting this).
The correct answer is: it depends what you mean by “amount”.
If by “amount” you mean cardinality, then they have the same.
If by “amount” you mean Lebesgue measure, then there are twice as many between 0 and 2.
If you’re talking to a child, or any adult who has not yet learned Set theory, then they don’t know what either of those words mean, or even that there can be different competing definitions that could match the English word “amount”. But when they use that word they probably are thinking of something closer to the Lebesgue measure than cardinality (which is weird and unintuitive and less useful in simpler problems related to the real world that non-mathematicians face), in which case the correct answer would be that there are twice as many between 0 and 2.
If you’re talking to someone who has learned Set theory but not measure theory (usually undergrads/bachelors and/or math-adjacent majors, since measure theory is usually taught much later), they will confidently assert that Cardinality and “amount” are synonyms, or just bake the assumption into all their explanations without even thinking about it.
What is 2 * infinity? 2 infinities? no, it is still infinity.
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