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The breakthrough for me was when someone described what it means for two sets of things to be the same size. This is super basic, because innumerate shepherds

The breakthrough for me was when someone described what it means for two sets of things to be the same size. This is super basic, because innumerate shepherds can use it to count sheep. As your sheep go out in the morning, for each sheep, take a small rock from a pile and put it in a bag. When the sheep come back in, take a small rock from your bag and put it back on the pile. Any rocks left in your bag at the end of this? If yes, you've lost a sheep and have to go find it. If a sheep comes back and you don't have any rocks left, oops, you've got someone else's sheep, so you should probably see if any other local shepherds are missing any. Or, how can you tell if you've got as many students in your auditorium as there are seats? You look for students without seats, or seats without students. If there are no spare students or spare seats, you have the same number of both. In both cases, you don't actually need to know how many sheep, rocks, students or seats there are. You just know that the sets of "sheep that went out" and "rocks in a bag" and "sheep that came back" are the same size, and "students" and "seats" are the same size, because you can establish a 1-to-1 mapping between every element of each set. This is the important result - if you can establish a 1-to-1 mapping between every element of two sets, those sets are the same size Therefore, if you can show a 1-to-1 mapping between every element of "all integers" and every element of "all even integers", then those sets are the same size. And you can. For each x in the set of integers, you map it to 2x in the set of even integers. Every member of either set has a single equivalent member in the other. And that's how you can make an infinite amount of space in your infinite-but-full hotel, by moving every guest from room x to room 2x. Yes, infinity is weird.

harry8 4y ago
> "And that's how you can make an infinite amount of space in your infinite-but-full hotel." If you re-frame that as an inexhaustible resource than you can make any amount of space in it because by definition it is inexhaustible. > "infinite-but-full..." "If it were already full then..." It can't be full. By definition it is inexhaustible. "but we have an infinite supply of hotel guests." So by definition we can't house them all because the supply cannot be exhausted. "We fill this inexhaustible space with an inexhaustible quantity of stack-able objects..." It seems as nuts as saying "We apply this irresistible force to this immovable object.." The existence of one implies the other _cannot_ exist. Pick one, reason with it. Include a second in your reasoning and... No. A one to one mapping between two resources that we define to be inexhaustible boils down to take infinity and divide it by infinity. How can it be anything else? So yes you can get 2, or 10, or pi, or literally anything at all. Infinity - infinity is no better, division is repeated subtraction so you still get "Gerald" as the answer. Sure two infinite things are, by some view of it, "The same size" where that size makes them not comparable. The set of points contained in a sphere in my hand has the "same size" as the set of points in the sphere we call the earth which is the same as the set of points in the universe. But so what? How does this tell us anything? We divided by infinity to define what a point is and there you have to stop or you get nonsense, even if it is convenient nonsense I would hesitate to build my house there. Maybe it doesn't make any sense to talk about size when it isn't finite. Maybe we can only really reason about it using limits for the same reason we note dividing by zero makes no sense in any way that is really useful?
- Karellen 4y ago
  If it helps, don't think about operations on infinite sets as "things that are true (or false)". Rather, think about it as "an abstract model with certain rules, that can produce interesting results if applied in careful ways." Like √-1. Does that exist? Well, not really. But if we make up a pretend number i that we say is equal to √-1, we can perform interesting mathematical operations on it that can still produce useful results in the real world. But only if we follow a given set of rules carefully and consistently. "All models are wrong, but some models are useful." Infinity isn't real - at least, not in our universe. It certainly isn't a number. But it's a concept we can use to produce useful results sometimes, if we handle it carefully. But it's also a concept that can produce some weird results, even if we follow all the rules.

harry8 4y ago

> "And that's how you can make an infinite amount of space in your infinite-but-full hotel." If you re-frame that as an inexhaustible resource than you can make any amount of space in it because by definition it is inexhaustible. > "infinite-but-full..." "If it were already full then..." It can't be full. By definition it is inexhaustible. "but we have an infinite supply of hotel guests." So by definition we can't house them all because the supply cannot be exhausted. "We fill this inexhaustible space with an inexhaustible quantity of stack-able objects..." It seems as nuts as saying "We apply this irresistible force to this immovable object.." The existence of one implies the other _cannot_ exist. Pick one, reason with it. Include a second in your reasoning and... No. A one to one mapping between two resources that we define to be inexhaustible boils down to take infinity and divide it by infinity. How can it be anything else? So yes you can get 2, or 10, or pi, or literally anything at all. Infinity - infinity is no better, division is repeated subtraction so you still get "Gerald" as the answer. Sure two infinite things are, by some view of it, "The same size" where that size makes them not comparable. The set of points contained in a sphere in my hand has the "same size" as the set of points in the sphere we call the earth which is the same as the set of points in the universe. But so what? How does this tell us anything? We divided by infinity to define what a point is and there you have to stop or you get nonsense, even if it is convenient nonsense I would hesitate to build my house there. Maybe it doesn't make any sense to talk about size when it isn't finite. Maybe we can only really reason about it using limits for the same reason we note dividing by zero makes no sense in any way that is really useful?

Karellen 4y ago
If it helps, don't think about operations on infinite sets as "things that are true (or false)". Rather, think about it as "an abstract model with certain rules, that can produce interesting results if applied in careful ways." Like √-1. Does that exist? Well, not really. But if we make up a pretend number i that we say is equal to √-1, we can perform interesting mathematical operations on it that can still produce useful results in the real world. But only if we follow a given set of rules carefully and consistently. "All models are wrong, but some models are useful." Infinity isn't real - at least, not in our universe. It certainly isn't a number. But it's a concept we can use to produce useful results sometimes, if we handle it carefully. But it's also a concept that can produce some weird results, even if we follow all the rules.