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
The different sizes come from countability and uncountability. The counting numbers are the natural numbers 0,1,2,… We can prove that all integers, all coordinates, all rational numbers and all rational complex numbers are the same size as the naturals. We call that countable.
However if we consider infinite decimal expansions, we get an issue where with the Cantor diagonalization argument you can’t list them all. So you can’t match a natural number with them, they’re not the same size. There are more size differences between uncountable sets too because you cannot create an injective or surjective function between the power set of a set and the set.
Infinity is not a number with a value, it is a concept.
Between 1 and 2 there are infinite numbers 1.00-1, 1.00-2 all the way until you get to 1.99-9 with an indeterminable amount of numbers in the dash.
Guess what has twice as many as that indeterminable amount?
The numbers between 1 and 3!
So you really gotta separate your view of numbers from the concept of the infinite. They just don’t gel well together
Is 1997 your birth year? If you have watched all those videos and still don’t understand, then maybe abstract reasoning is not your strength. The concepts are not easy, and you can function just fine without them. Nobody here is going to explain better than the Veritassium guy.
Lets say you will walk on forever.
If you take 5 steps for every meter you walk you will take more steps than if you take 1 step for every meter you walk.
The timeframe is the same, forever, but one of those walks has more steps than the other even though you won’t ever stop in both cases.
How many water molecules are there in a lake? God knows. Infinite, right? Okay cool, how many water molecules are there in two lakes? It’s still infinite but it’s more. How about the ocean?
Imagine Superman is challenged by Batman reach a goal post 10 meters away. Batman says that Superman can only cross this distance by jumping and that Superman can only jump half the remaining distance at a time. So Super jumps once for 5 meters and now there are 5 meters left between Superman and the goal post. Superman jumps again, but this time he can only go 2.5 meters because that is half of the distance, so now Superman has gone 7.5 meters. Another jump and Superman is at 8.75 meters, one more and he is at 9.375 meters. Another way to think about it is that the distance between Superman and the goal is halved every time he jumps. If the distance between Superman and the goal post is 0, he made it.
Now ask yourself the question: If Superman keeps going like this, will he ever reach the goal post? No matter how many times he jumps, there will be some distance left. Think of the biggest number you can and if Superman jumps that many times, there would still be some distance from the goal post.
Eventually, Superman realizes this, and starts using his super speed to jump at an incredibly fast rate. Each jump is a shorter and shorter distance, so Superman can jump faster and faster. Eventually, because he’s Superman, he finally reaches the goal post. But it took him an infinite amount of jumps. Notice how I didn’t say he made infinity jumps? That’s what infinity is: not a number, but a concept. You can say someone made 4 or 5 or 6 things (jumps), but not infinity things (usually described as an infinite amount).
Batman asks to repeat the challenge, but this time Superman can only jump one third of the distance instead of one half. It’s definitely slower than before, but Superman again speeds up and finally reaches the goal post.
Now Batman asks the question: “Which goal post took more jumps to reach?” Both took an infinite amount of jumps (Specifically not “infinity jumps”), but the one-third challenge definitely took more. Thus, one set of an infinite amount of something was more than another.
∞> π. Make sense now?
You are getting such bad answers here that I feel compelled to write something.
Lets imagine you have a big pile of marbles and so does another guy. You want to see who has more marbles. Obviously you can just count yours, and he counts his, and you compare the results. But here is the catch: The other guy only speaks only French (and you don’t). So if you try this then neither of you will understand what the number was that the other person reached.
Here is a better idea. Instead of counting marbles, you iteratively roll a marble out of your pile. He does the same. You continue until one of you runs out of marbles. Whoever has marbles left at the end is the one who has more marbles.
The second method of comparing sizes still makes sense with infinite sets so this is how mathematicians talk about the “size” of a set (we use the word cardinality). Of course you might simply guess that when comparing infinite sets using the second method, both piles will always run out of marbles at the same time. It turns out that this isn’t the case. The most famous example is the set of reals and the set of naturals. In this example, the naturals run out of marbles before the reals. Hence, we say that the reals are a “larger” infinite set than the naturals.
You’re reading too much into it. Infinities don’t physically exist, they are just a mathematical toy. It’s just saying “This is the way we generate numbers and in this way we can generate an unlimited amount of numbers”.
“Larger” in this case is also a mathematical trick. It’s not your normal everyday “larger”. We can’t compare infinities like we can compare numbers. So we say okay if we can somehow define a “bijection” between two infinite sets then we call them “equal”. Bijection means each element of these two sets has exactly one pair in the other set. If we can’t do this then the one that always has leftover elements is “larger”.
So you follow this train of thought and the conclusion is under this specific definition of “larger”, some infinities are larger. It’s not something you can visualize.
Imagine you are standing on the edge of an infinite forest. This forest has an infinite number of trees, assume they are 1 m apart for simplicity. Now, the floor of the forest is covered with infinite grass. And grass is so much finer than trees, that between any pair of trees, there is an infinite amount of grass between them.
The analogy is that trees are integers, grass is real numbers. Both are infinite, but the latter a lot more so.
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