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 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! 🙂
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