I have watched many videos on YouTube about it from people like vsauce, veratasium and others and even my math tutor a few years ago but still don’t understand.
Infinity is just infinity it doesn’t end so how can there be larger than that.
It’s like saying there are 4s greater than 4 which I don’t know what that means. If they both equal and are four how is one four larger.
In: Mathematics
Let’s say we have two bags of marbles, and we want to know if they both have the same number of marbles, BUT we do not know how to count. How can we check if they have the same amount without counting? We can take one marble out from bag A, one marble out from bag B, pair them up. Keep doing this. If every marble in bag A can be paired with a marble in bag B, with no leftovers, then we know both bags had the same number of marbles.
That’s how we can compare sizes of things without counting them, and how we can compare sizes of things that are “infinitely” large.
In math, we call those bags “sets”. Let’s start with two bags of infinite size that ARE the same size- consider a bag of all the positive whole numbers (1,2,3,..) and a bag of all the positive EVEN whole numbers (2,4,6,…). Since these “bags” contain an infinite number of objects, we cannot “count” how many there are in each to compare the size. So, we have to make pairs, like we did with the marbles. In this case, for every number in bag A, we can pair it with a number in bag B that is twice its value. 1 gets paired with 2, 2 gets paired with 4, 3 gets paired with 6, etc. You can see that for EVERY number in bag A, we can pair it with a number in bag B. So the “size”of all positive integers actually = the “size” of all positive EVEN integers.
Now, there are some “bags” of numbers where it is impossible to make these pairings. No matter how you can pair up numbers, there will always be some numbers leftover that can’t be paired up, meaning that one infinity contains more objects than another infinity, making it “larger”.
This is where it can get hard to explain an example, but we’ll give it a try anyways. Let’s look at these two bags of numbers: bag A will be all positive integers again (1, 2, 3,…) and bag B will be all the possible numbers between 0 and 1 (so for example 0.5, 0.51, 0.501, 0.837362773833333, 0.333, 0.33333333, 0.33333333333333333 repeating, you get the idea- basically all decimal numbers between 0 and 1).
For the sake of argument, let’s say we have come up with some pairing of the numbers in the two bags, and I will write out the first few pairings:
1 with 0.53827263727173000000010000…
2 with 0.8173637363839000000040000…
3 with 0.8387262222233474633000000…
4 with 0.3333333333333333333333333…
Imagine this list being infinitely long, exhausting all the marbles in bag A. However, I can ALWAYS come up with a number from bag B (the decimal number less than 1) that is guaranteed to NOT be on this infinitely long list. I will call that my magic number, and I will construct that number following this rule: I will start at row 1 and look at the digit in position 1 after the decimal point, (so 5 in this case) and for ease of illustration, +1 to that digit and append it to my magic number. At row 2, I will look at the digit in position 2 (1 in this case) and +1 to get 2. Continue down the list, and my magic number will start being 0.6294……
Remember that this list is infinitely long, so if we kept doing this, we would get some decimal number. HOWEVER, and this is the cool part, that magic number is GUARANTEED not to be in the original list. How? Well, let’s go down the list and compare it to all the numbers. Is it the same number as the decimal in row 1? Well it can’t be, since I altered the first digit. Is it the same number as the decimal number in row 2? Well it can’t be, since I altered the second digit. Is it the same number as……? You may start to see the point. So what we have shown is that is impossible to pair up positive integers with decimal numbers between 0 and 1, because no matter how you try to list all the decimals out you can always find a NEW decimal that was not on your original list. This means that the size of the bag containing all the decimals between 0 and 1 must be bigger than the size of the bag containing the positive integers, even though they are both infinitely large.
This is just one example of two different sizes of infinity, but there are many other cool examples that illustrate this. These concepts of infinite have always been one of my favorite things in math 🙂
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