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This has always been my struggle with all of this "infinity" maths. Infinity/Infinity is anything you like and many things you don't and these things always se

by harry8 4y ago

This has always been my struggle with all of this "infinity" maths. Infinity/Infinity is anything you like and many things you don't and these things always seem to boil down to dividing by infinity. Visit each room in the hotel in turn, each one for half the total time spent going to and visiting the last. Without explanation of why we can divide infinity by infinity when we want to and not get total garbage. Banach tarski, infinity/infinity = 2 because the numerator is exactly equaly to (infinity 2) then you cancel the infinity from top and bottom. Also (infinity 3) so clearly 2=3 and this is useful. infinity also equals infinity * infinity so 1 = infinity. As you say odd things happen when you divide by infinity and mostly they don't seem to be helpful things. They seem to be wholly invalid things. But not here? I'm sure that's not it and I'm totally missing the point, but that point is being skipped over and handwaved away with a sneer and a muttering of "mathematical maturity" rather often.

schoen 4y ago
What I got from this post is that the weirdness here comes in, not so much from the Hilbert's Hotel phenomenon about cardinalities and sets that can be put into correspondence with their own proper subsets, but from looking more deeply at something relatively familiar and something that we ordinarily use to tame infinities: volume. Even though there are infinitely many real numbers in [0,1], we have the idea that the unit cube [0,1]³ should have a finite volume of exactly 1, or the unit sphere { (x,y,z) | √(x²+y²+z²)≤1 } should have a finite volume of exactly 4π/3. Or indeed the unit line segment [0,1] should have a finite length of exactly 1, even though it contains infinitely many points. This stuff has felt totally normal and appropriate in mathematics ever since Euclid: Euclid would probably agree that you can't count how many points are in a line segment, but still endorses talking about lengths of line segments (or at least ratios of lengths of different line segments). While it feels like we know how to work with volumes, and that they're comfortably finite and well-behaved, things like Banach-Tarski suggest that if we want to have every set of points have a well-defined volume in any given number of dimensions, we're actually going to run into bad trouble. But the article suggests that there are several ways to avoid this trouble, including just saying that some weird geometric objects don't have a defined n-dimensional volume. Instead, maybe only some "nice" sets should have one?
- harry8 4y ago
  >...we have the idea that the unit cube [0,1]³ should have a finite volume of exactly 1... >...even though it contains infinitely many points. There is the division by infinity. Cut the volume in half. Still has infinite points. infinity/2 = infinity. 1 = 1/2, or 1=2 if you can cancel those. Any set of points is some number of points 1,2, ..infinity. Points are infinitely small (division by infinity) so you'd have to infinitely many of them to have a volume other than zero. And you're back to infinity points * (1/infinity) vol of a point. So yeah, no way I can see to have any set of points have a well defined volume because the volume of a point is a division by infinity, unless the set is finite in which case it is zero (or whatever you define a finite number divided infinitely to be - define it to be gerald if that helps? - Mathematical immaturity on display right there).
  - schoen 4y ago
    Yes, I see your point (no pun intended), but I would still maintain that this issue didn't necessarily bother mathematicians in the days before formalized set theory. The issue you mention is a reason that any definition of volume should not be based only on set cardinality, since there are those one-to-one correspondences with proper subsets (for infinite sets of points), yet volume should not be preserved by cutting something in half, or doubling something, in an ordinary way. That is, if X is the unit line segment and Y is a pair of unit line segments joined end-to-end, we want the measure function µ to obey µ(Y) = 2µ(X), even though |X|=|Y| in set theory. And even though that's weird, formalized mathematics didn't get existentially ambitious enough to make anything truly bizarre out of it, I suppose, from Euclid all the way up until Vitali!
    - harry8 4y ago
      >even though |X|=|Y| ... Yeah for me it simply doesn't. It isn't false, it just doesn't even make sense. Define X and Y as you did but make one of the segments joined to make Y /be/ X. Now if it is ever meaningful to have a one to one correspondence when infinity is lurking about surely the points on X correspond exactly, one to one, each with itself. Once all the points of X are accounted for, clearly half of Y remains, there is no other way. There is simply no point you can select in X that is free to correspond to the second half of Y if you select corresponding to itself in preference. You can never, ever select a point corresponding to the second half of Y if you admit preference to corresponding to itself in selection. But X has an inexhaustible number of points to chose from by definition and so does Y. So if you start picking points at random from Y and matching them up to a previously unused point from X you can do that forever and never exhaust either of the sets of points. Thus X and Y correspond one to one and no points remain of either X or Y. And there are the same number of integers as there are integers that are even. 1=2 QED The walls are different measured heights but that doesn't matter because we've proved it. Yeah ok, but maybe it does to the poor sap who needs a useful roof?
Smaug123 4y ago
If I asked you to divide an apple by a stone, what would your answer be? Infinity is simply not a thing that your carefully-honed real-world intuitions about division apply to. Hilbert's hotel is precisely a thought experiment to prove this: divide an infinitely large set among infinitely many people, and you can get many different results. Any true statement about infinities you construct with division is true only by coincidence, not for the usual maths reason that things are true (namely that they are the result of following some broadly-applicable rules). There is no number called "infinity", so there's no reason to expect the rules of the arithmetic of numbers to apply to it; you have to carefully go and prove that the rules you want to apply have some meaning when you extend the language in this way, and moreover that these rules are true. For example, there are multiple meaningful ways to add infinite quantities, e.g. depending on whether you are considering the cardinals or the ordinals. Those concepts are the same in finite-land, but are different in infinite-land. Cardinals do not admit the notion of "division"; ordinals at least have the division algorithm.
Karellen 4y ago
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?

by harry8 4y ago

schoen 4y ago

What I got from this post is that the weirdness here comes in, not so much from the Hilbert's Hotel phenomenon about cardinalities and sets that can be put into correspondence with their own proper subsets, but from looking more deeply at something relatively familiar and something that we ordinarily use to tame infinities: volume. Even though there are infinitely many real numbers in [0,1], we have the idea that the unit cube [0,1]³ should have a finite volume of exactly 1, or the unit sphere { (x,y,z) | √(x²+y²+z²)≤1 } should have a finite volume of exactly 4π/3. Or indeed the unit line segment [0,1] should have a finite length of exactly 1, even though it contains infinitely many points. This stuff has felt totally normal and appropriate in mathematics ever since Euclid: Euclid would probably agree that you can't count how many points are in a line segment, but still endorses talking about lengths of line segments (or at least ratios of lengths of different line segments). While it feels like we know how to work with volumes, and that they're comfortably finite and well-behaved, things like Banach-Tarski suggest that if we want to have every set of points have a well-defined volume in any given number of dimensions, we're actually going to run into bad trouble. But the article suggests that there are several ways to avoid this trouble, including just saying that some weird geometric objects don't have a defined n-dimensional volume. Instead, maybe only some "nice" sets should have one?

harry8 4y ago
>...we have the idea that the unit cube [0,1]³ should have a finite volume of exactly 1... >...even though it contains infinitely many points. There is the division by infinity. Cut the volume in half. Still has infinite points. infinity/2 = infinity. 1 = 1/2, or 1=2 if you can cancel those. Any set of points is some number of points 1,2, ..infinity. Points are infinitely small (division by infinity) so you'd have to infinitely many of them to have a volume other than zero. And you're back to infinity points * (1/infinity) vol of a point. So yeah, no way I can see to have any set of points have a well defined volume because the volume of a point is a division by infinity, unless the set is finite in which case it is zero (or whatever you define a finite number divided infinitely to be - define it to be gerald if that helps? - Mathematical immaturity on display right there).
- schoen 4y ago
  Yes, I see your point (no pun intended), but I would still maintain that this issue didn't necessarily bother mathematicians in the days before formalized set theory. The issue you mention is a reason that any definition of volume should not be based only on set cardinality, since there are those one-to-one correspondences with proper subsets (for infinite sets of points), yet volume should not be preserved by cutting something in half, or doubling something, in an ordinary way. That is, if X is the unit line segment and Y is a pair of unit line segments joined end-to-end, we want the measure function µ to obey µ(Y) = 2µ(X), even though |X|=|Y| in set theory. And even though that's weird, formalized mathematics didn't get existentially ambitious enough to make anything truly bizarre out of it, I suppose, from Euclid all the way up until Vitali!
  - harry8 4y ago
    >even though |X|=|Y| ... Yeah for me it simply doesn't. It isn't false, it just doesn't even make sense. Define X and Y as you did but make one of the segments joined to make Y /be/ X. Now if it is ever meaningful to have a one to one correspondence when infinity is lurking about surely the points on X correspond exactly, one to one, each with itself. Once all the points of X are accounted for, clearly half of Y remains, there is no other way. There is simply no point you can select in X that is free to correspond to the second half of Y if you select corresponding to itself in preference. You can never, ever select a point corresponding to the second half of Y if you admit preference to corresponding to itself in selection. But X has an inexhaustible number of points to chose from by definition and so does Y. So if you start picking points at random from Y and matching them up to a previously unused point from X you can do that forever and never exhaust either of the sets of points. Thus X and Y correspond one to one and no points remain of either X or Y. And there are the same number of integers as there are integers that are even. 1=2 QED The walls are different measured heights but that doesn't matter because we've proved it. Yeah ok, but maybe it does to the poor sap who needs a useful roof?

Smaug123 4y ago

If I asked you to divide an apple by a stone, what would your answer be? Infinity is simply not a thing that your carefully-honed real-world intuitions about division apply to. Hilbert's hotel is precisely a thought experiment to prove this: divide an infinitely large set among infinitely many people, and you can get many different results. Any true statement about infinities you construct with division is true only by coincidence, not for the usual maths reason that things are true (namely that they are the result of following some broadly-applicable rules). There is no number called "infinity", so there's no reason to expect the rules of the arithmetic of numbers to apply to it; you have to carefully go and prove that the rules you want to apply have some meaning when you extend the language in this way, and moreover that these rules are true. For example, there are multiple meaningful ways to add infinite quantities, e.g. depending on whether you are considering the cardinals or the ordinals. Those concepts are the same in finite-land, but are different in infinite-land. Cardinals do not admit the notion of "division"; ordinals at least have the division algorithm.

Karellen 4y ago

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?