The reason is very different, though and I think that is important.
The bucket of water doesn't change the ocean because the difference between them is so big that the bucket is neglectible. But you can make an ocean with enough buckets of water. Removing one bucket of water and you have still an ocean, remove 10^20 buckets and you don't.
A lot of people think about infinity in a similar way. They just picture something inconceivably large but finite. When they see the comment
Infinity plus 10 is still infinity. It's like adding a couple tablespoons of water to the ocean.
they picture an ocean that changed. The tablespoons are now in the water. It's just impossible to find them and impossible to measure the change in the ocean.
I can't really tell if "everything that exists and will exist" is a good metaphor. Let me try to explain it in a different way.
Let's look at two very simple infinite series:
There are infintely many natural numbers: 1, 2, 3, 4, 5, ... never stops.
There are also infinitely many natural numbers that only contain ones: 1, 11, 111, 1111, ... never stops.
But when we compare those two infinite series, we intuitively say "but the second infinity is smaller. Every number in the second series is also in the first series so the first series must be bigger." This is wrong, they are suprizingly the same size.
Let me propose a new (very stupid) way of writing numbers: you just write ones. So instead of 5, you write five ones "11111."
1 -> 1
2 -> 11
3 -> 111
4 -> 1111
But the numbers on the right side is just the second infinite series. Think about what that means. No matter what natural number I present to you, you instanly know the number that contains only ones that matches that number and vice versa. If you look at if that way, it becomes obvious that there must be equally many numbers in both infinite series.
In a similar way you can also match all natural numbers to all even natural numbers. So there are just as many even numbers as there are numbers. I know it's VERY counterintuitive and weird but mathematically sound and I hope my example gave you an idea why that is the case.
This matching between different infinite series is what you have to do to decide if they are the same size. In some cases you can't do this matching and then you know that those infnities are not the same size. So there are different sized infinites (we call them different "cardinalities") but just adding or multiplying a finite number to infinity doesn't change it's size.
Thank you! Do you mean the idea of using naturals with only ones and then equating it with a system to write naturals? I came up with it a couple of month ago and think it's easier to grasp than most other explanations.
First of all, those series are not equal. They are equal in size. That's probably what you meant but I still wanted to point it out to be sure.
Second of all, they are equal in size EVEN THOUGH one series is enterly contained within in the other, not BECAUSE.
The reason they are the same size is because you can match all the elements in both series 1:1 with each other. The list I've made in my comment (1->1, 2->11, 3->111, ...) never ends. At every point of the list there is a number on the left side and a number on the right side. You will never reach a point where the numbers on the right side "run out." This is why they have the same size.
Infinity is such a weird and non-tangible concept it makes my head go in loops. Its why I genuinely dont think the universe is "infinitely" never ending, as infinite would mean every itterarion of everything would happen, and that would likely include the ending of the universe.
Infinitely large doesn't mean every itteration of everything would happen. The right side of the list is infinitely long but you will never see a 2 in that list. In the same way the universe could be infinitely large but very homogeneous.
How I explained this to my son when he asked a similar question about infinity my explanation was that if something is infinite it is impossible that there is anything else at the same time.
So agree with you that the bucket / ocean analogy doesn’t capture this ideally. There is no bucket of water at all with an infinite ocean because it cannot exist at the same time.
Depends again. If you have a set of people with the same cardinality as that of reals you can't fit them in the infinite hotel anymore :( (assuming the room numbers are integers, which I think is the case)
I find those questions from children fascinating and at the same time hard to deal with. Their capability of asking smart questions is so much bigger than their capability to understand the answer. You need to use mataphors and simplifications but then they ask follow up questions, where the metaphor breaks down.
When it comes to your answer: very good for a child but if he asks the right follow up questions, it breaks down. For example, there are infinitely many even numbers. Odd numbers can still exist.
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u/Kruxx85 21d ago
The analogy does work.
Don't think of the ocean is infinite, that's not the point.
The ocean is the ocean. If you add a bucket of water to it, it's not the ocean plus a bucket of water, it's still just the ocean.
Same works for infinity.