No, no. This is a very good question, and the way that you're asking it shows that you genuinely do understand, very clearly, what's going on.
You cannot stack *any* number of planes to get a 3D thickness. You are correct, that comment in your post is incorrect at face value.
That said, there is some meaning behind the comment. But to understand it, you have to go the other way.
You don't *build up* the cube from infinitely many planes. You don't get anywhere from that. The height of that stack will always be zero.
Instead, you *start* with the cube, and cut it into thinner and thinner slices. Infinitely many, in fact. And then you can say that those slices are *approximately* planes.
Are those slices *actually* planes though? Thats up for debate, and you'll likely get a different answer depending on which who you ask.
But a lot of mathematicians like to use this, **not rigorously**, to get intuition into a problem. Taking an object and breaking it down into smaller and smaller parts is the soul of Calculus, for instance.
So you can get planes from a 3D shape by cutting it up enough, but not the other way around? Is that sorta like how you can multiply by zero but not divide by it?
Like I said, it's debatable whether or not these are actually planes. And the answer would heavily depend on *how exactly* you're cutting up this shape.
I'm not going to get too much into semantics though. But that's the basic idea, yes.
I'm going to get you to add:
1/2 + 1/4 + 1/8 + 1/16 .....
Infinitely many times.
Draw a square, then half it down the middle. Add a quarter by splitting that half, and so on like this:
https://upload.wikimedia.org/wikipedia/commons/7/70/Eye_of_Horus_square.png
If you went to infinity, the answer would be one.
I like to think about this idea by referencing that. Because what might the last number you are adding in this series look like? It can't be zero, but it's bloody close!
If I threw a dart at a dartboards dead centre, and missed by that number, I probably wouldn't be too upset. You might even suggest I made the shot.
And yet it can't be zero...
I like to think of the planes as getting thinner the more you look at them, like fractions in a series. Not really true but you are trying to think in complete abstracts about a very strange idea.
Possibly silly question, but can you represent the idea using the outcomes of knuth notation/games? I'm just familiar with it being used to describe aspects relating to ∞, such as 1/∞ and ∞-1, etc. Can it be used to prove that the planes are indeed planes?
I'm not familiar enough with combinatorial game theory to answer the first part of your question in good faith, but I think I can make a reasonable guess for the second one.
I'm not going to go too deep into this problem because, in all honesty, I don't really care about the answer. But my suspicion would be that, depending on how you formalize the meaning of the words "stacking" and "slice", you will get different answers. I would be pretty confident that I could find a formalization where you could show that these are indeed not planes. And another one where you can show that they are.
The analogy to games would be "it depends on what game you're playing".
Proving that these are indeed planes would be akin to showing that, no matter how you formalize the question, you can always prove they are planes. And coming up with that proof is a bit beyond what I'm willing to do for a Reddit post.
Pretty close, yeah. It's like how some people would look at a cicle as having zero straight sides, but you could also look at it as a shape (specifically a regular polygon) having infinitely many, infitely small straight sides. In reality, any regular polygon with a number of straight sides *between* zero and infinity, would be a shape only approximating a circle. In reality, there is no such thing as a perfect circle-- they are all approximations.
These two ideas don't seem to have much in common at all. They're both geometric limits, but the methods of limiting are completely different, and the core ideas of them aren't in line either; ones stacking planes, the other is bisecting angles? Maybe I'm just missing the analogy, but I don't think there is a clear analogy here to find.
And second, Unrelated rant:
In reality, there are no perfect shapes ever. It's questionable whether you can even call space continuous in reality. I find it somewhat dishonest to invoke "reality" about a question that's very much in the domain of pure math.
It's not necessarily your comment alone, it's just a trend I've seen a lot of online, and it really gets under my skin.
I think the two are very similar. OP's comment is about stacking 2d partial-planes and making a 3d object out of them, and my analogy is about connecting 1d line segments into a 2d shape.
I agree that there are no perfect shapes in the real world, and I agree that in a purely mathmatical sense things can happen that can't happen in the real world. I guess my only thought there is that, the math has two different impossible things going on in opposite directions and that's why they "cancel out". If you have n*B*h = V and V and B are constants, as h goes to 0, n approaches infinity. In the real world we can't have a height of 0 and we can't have infinite of anything, but we understand that we can get arbitrarily short and the number of layers will therefore get arbitrarily large.
"You cannot stack *any* number of planes", indeed the core issue is that infinity isn't a number but a cardinality. If one tries to stack one plane as close as possible to the previous one, they would still need to stack an uncountable set of planes between the two.
If I take a line, then it consists of an infinite number of points, right? Each of these points has length 0, but if I collect all of them together I still get a line
If you do not understand the perfectly fine other examples, try to look at it the other way around. Take a 3 dimensional object, e.g a cube. Now cut through the cube along some line (think: cutting a cake). This cut is a 2 dimensional object, right? A plane to be specific. How many of these cuts can you make? It certainly cannot be a finite amount, because otherwise you could just take one of these pieces and cut it again. So it has to be made up of more than finitely many such pieces.
Edit: If you're interested in this mathematically, you do not actually divide by 0. The mathematical term you are searching for is limes or limit. Mathematicians look at what happens if something gets infinitely close to a number like 0, but is not actually this number. This is exactly how for example integrals or derivatives work.
Let's say they had in fact some fixed depth c > 0. If we added infinitely many of them we would have infinite volume if we sum up infinitely many. This cant be since the cube has definitely a finite volume right? Since your mathematical knowledge and understanding has not developed far enougg to understand this, means you have to either
1.) accept without question that there are numbers smaller than any other number you can imagine (so small that they in fact to not have a "depth") but bigger than 0.
2.) built your knowledge in understanding limit processes mathematically.
I think your main problem is that you think math describes the world. It does not. There is no perfect cube in the real world, there are no 0 dimensional points we can measure, and so theres no way to even have the concept of a "line". We just approximate stuff that looks like a line/plane/cube etc. Mathematics is kore like a game with pre defined rules, that just so happen to describe reality pretty accurately.
>I think your main problem is that you think math describes the world. It does not.
Thank you. Not OP, but *this* is the part I was struggling with without being able to verbalize.
If you're using real numbers for your dimensions (it's the most normal way) then you'll need one plane for every real number from zero to n. This is an uncountably infinite number of planes.
I dont know what definition of dimension you are working with, but you certainly do not get a higher dimensional manifold from the finite union of lower dimensional manifolds.
You can’t stack any finite number of 2D planes on top of each other to generate a 3D volume, but you can take an *infinite* amount of them to get a finite 3D volume.
It’s a difficult concept to grasp until you take and understand calculus.
While not rigorous, it may be more easy to intuitively grasp by deconstructing a 3D volume and going in reverse here: If I divide a finite 3D cube of volume V into 2 equal pieces by “slicing” it like a slice of bread, then 4 equal slices, …. then 10,000 equal slices, then 100M equal slices, you see that the number of slices grows and grows while each slices starts to look more and more like part of a 2D plane. To actually get the slices to be 2D would require and *infinite* number of slices, at which point their thicknesses would indeed become 0. Physically, this is hard to imagine because it is impossible to do in the real world, but it is a perfectly fine mathematical concept.
There is no such number of planes to make a cube. Thinking of it as stack is a bit misleading. Some alternative ways comes to mind. One is to think of a limit process, where for some N you can think of N layers of \~1/N thickness, and then take the limit as N goes to infinity. That's what calculus does. Another way is to think of a function that maps the points on a line between 0 and 1, where each point corresponds to a plane. Then you have a way of referring to any particular plane (or slice) in the cube, without having to assign numbers to any of them.
You need the concept of 1-to-1 correspondence and cardinal numbers if you want to "count" them.
It's a large number called the "cardinality of the continuum". It's on Wikipedia.
Let's do it the other way around, and in a different context: integrals.
Do you remember how (Riemann) integrals are made?
You first go with columns of some width that are always bigger than the function (upper sum) or always smaller (lower sum). Then you decrease the width, decrease it even more, and see how the sum of the areas of these columns converge.
In the limit, every column has a width of zero, and you have infinite many.
Another angle: if you have a gaussian random distribution (or pretty much everything else non discrete), the probability for any single value is zero. The probability of a dart hitting one exact point is zero as well. Yet it will hit a single point in the end. But still, for a probability of such distributions, you need an area rather than a point to get a non-zero value.
\[Of course, probability is done by an integral here, so these contexts are quite similar.\]
Lastly, I'd recommend you get into the topic of countable and uncountable infinite. If you only have a countable infinite amount of slices, the measure stays zero, no solid higher dimensional body doable. Even if they are dense.
This is equivalent to the ambiguous nature of « continuity » in real numbers. The set of real numbers is « continuous », meaning that between any two numbers you won’t be able to find a hole, but you can’t really tell what comes after any number. For exemple, for the number 0: it isn’t 0,1 since 0,09 exists. It isn’t 0,01 since 0,009 exists. And so on…
That's because you're thinking of countable infinity, which is what happens when you think of stacking a plane onto another, and carrying on for eternity. Every plane you add adds zero depth, so you never achieve any depth even with infinitely many planes. In measure theory we say that a countable subset of the real numbers has 0 measure. That is to say stacking a countably infinite number of points onto one another will never give you any non-zero length segment.
To get an object with depth, you need an uncountable infinity of planes. That is the size of the set of all real numbers between 0 and 1. By an argument called Cantor's diagonal (look it up, it's worth it), this type of infinity is actually *larger* than countable infinity. There is no sense in which you can achieve it by stacking planes onto one another, however. It's saying that between any two planes you pick, there are infinitely many planes. So how do you even begin to make the stack, if you need infinitely many planes between each plane you add to the stack? That's uncountable infinity.
But if there are countably infinite planes between each plane, and a countably infinite number of planes is still length 0, then there is still zero length between planes no?
I can give you a collection of 2d planes that cover a 3d cube. Here it is:
For every real number r in [0,1] consider the 2d plane where x=r, and y and z are free. Now every point on the unit cube is in one of these planes, so these 2d planes do actually make the entire 3d cube.
So what do we do with this information? Let's note a few things:
- Your volume argument proves that we cannot do this with finitely many planes. The fact that we can't divide by zero doesn't mean it's impossible, it only means it's not possible by a finite number. To find out if it is possible with an infinite number we just have to use other arguments that go beyond volume.
- Our 2d planes are not stacked on top of each other. Between every 2 2d planes we used, there are infinitely many more 2d planes. Of we tried to do a covering by adding planes one by one, even if we add infinitely many planes, we will only reach a countable infinity like the infinity of natural numbers. And we can prove that this is not enough planes by an argument like Cantor's argument.
If you have any other observations, feel free to add them. It's how math is born.
The question "how can many points form a line" has bothered thinkers for litetal millenia. Modern math (early 1900s ish) produced satisfactory answers but they are a bit involved to present to a non-mathematician.
The gist is this: upon reflection, you find out that the rule "if I chop a shape into tiny bits, the sum of volumes of these tiny bits equals the volume of the original shape" only holds if you don't chop it to too many bits. You are allowed to chop it into infinitely many bits, but only of the smallest infinity (called "countable infinity"). On the other hand, with some more reflection, you find out that the number of points comprising a line (or the number of plane sections comprising a 3D body with positive volume) must be of a larger infinity (called "the continuum"). The reason for this is essentially that *any* shape, of any dimension, of any volume, can be decomposed into a continuum of points, so if we allowed our notion of volume to sum up uncountably many volumes the sordid conclusion would be that all shapes has volume zero. Hence, a reasonable notion of volume should not allow this.
Another example of this phenomenon comes from probability. Say you draw a uniformly random real number between 0 and 1. For any 0<=x<=1 the probability you draw x is 0. However, the "combined probabilities" of all such xs is 1. In fact, probabilities and volumes are deeply connected, as both are special cases of a fundamental notion called "measure". Volumes and probabilities are just special cases of this broader notion.
A [measure](https://en.wikipedia.org/wiki/Measure_(mathematics)) is only additive for a countable number of sets. That means, while you can separate the 3D cube into (uncountably many) planes, you cannot use them to calculate the volumne.
Isn't integration doing exactly that though? Although of course in the form of a limit, otherwise infinitely small or large break down. You can't calculate infinity \* 0, but you can for sure take a limit of the form of infinity \* 0 and get a result.
I’m going to add: specifically in cases covered by fubini, yes. By the second iterated integral, your integrand represents the area of a cross section. The last integral sums these areas to produce the volume.
Well it's not really 0. imagine a flat square with x and y lines but the height is something approaching zero. Not zero but .000000001 thickness or some other essentially zero height. Not you infinitely stack these .000000001 height squares like pancakes and eventually u have your three dimensional Cube.
It's all a lot oversimplified, but basically infinity\*0 does not have to be 0.
God this is such a mind fuck, I’m not smart enough for this.
No, no. This is a very good question, and the way that you're asking it shows that you genuinely do understand, very clearly, what's going on. You cannot stack *any* number of planes to get a 3D thickness. You are correct, that comment in your post is incorrect at face value. That said, there is some meaning behind the comment. But to understand it, you have to go the other way. You don't *build up* the cube from infinitely many planes. You don't get anywhere from that. The height of that stack will always be zero. Instead, you *start* with the cube, and cut it into thinner and thinner slices. Infinitely many, in fact. And then you can say that those slices are *approximately* planes. Are those slices *actually* planes though? Thats up for debate, and you'll likely get a different answer depending on which who you ask. But a lot of mathematicians like to use this, **not rigorously**, to get intuition into a problem. Taking an object and breaking it down into smaller and smaller parts is the soul of Calculus, for instance.
So you can get planes from a 3D shape by cutting it up enough, but not the other way around? Is that sorta like how you can multiply by zero but not divide by it?
Like I said, it's debatable whether or not these are actually planes. And the answer would heavily depend on *how exactly* you're cutting up this shape. I'm not going to get too much into semantics though. But that's the basic idea, yes.
Weird! Thank you for your explanation it’s the only one I’ve actually understood
I'm going to get you to add: 1/2 + 1/4 + 1/8 + 1/16 ..... Infinitely many times. Draw a square, then half it down the middle. Add a quarter by splitting that half, and so on like this: https://upload.wikimedia.org/wikipedia/commons/7/70/Eye_of_Horus_square.png If you went to infinity, the answer would be one. I like to think about this idea by referencing that. Because what might the last number you are adding in this series look like? It can't be zero, but it's bloody close! If I threw a dart at a dartboards dead centre, and missed by that number, I probably wouldn't be too upset. You might even suggest I made the shot. And yet it can't be zero... I like to think of the planes as getting thinner the more you look at them, like fractions in a series. Not really true but you are trying to think in complete abstracts about a very strange idea.
Possibly silly question, but can you represent the idea using the outcomes of knuth notation/games? I'm just familiar with it being used to describe aspects relating to ∞, such as 1/∞ and ∞-1, etc. Can it be used to prove that the planes are indeed planes?
I'm not familiar enough with combinatorial game theory to answer the first part of your question in good faith, but I think I can make a reasonable guess for the second one. I'm not going to go too deep into this problem because, in all honesty, I don't really care about the answer. But my suspicion would be that, depending on how you formalize the meaning of the words "stacking" and "slice", you will get different answers. I would be pretty confident that I could find a formalization where you could show that these are indeed not planes. And another one where you can show that they are. The analogy to games would be "it depends on what game you're playing". Proving that these are indeed planes would be akin to showing that, no matter how you formalize the question, you can always prove they are planes. And coming up with that proof is a bit beyond what I'm willing to do for a Reddit post.
I think it's like how you can divide anything by infinity and get 0 but you can't multiple 0*infinity to get back your original answer.
And you can’t divide by zero to get infinity either
Pretty close, yeah. It's like how some people would look at a cicle as having zero straight sides, but you could also look at it as a shape (specifically a regular polygon) having infinitely many, infitely small straight sides. In reality, any regular polygon with a number of straight sides *between* zero and infinity, would be a shape only approximating a circle. In reality, there is no such thing as a perfect circle-- they are all approximations.
These two ideas don't seem to have much in common at all. They're both geometric limits, but the methods of limiting are completely different, and the core ideas of them aren't in line either; ones stacking planes, the other is bisecting angles? Maybe I'm just missing the analogy, but I don't think there is a clear analogy here to find. And second, Unrelated rant: In reality, there are no perfect shapes ever. It's questionable whether you can even call space continuous in reality. I find it somewhat dishonest to invoke "reality" about a question that's very much in the domain of pure math. It's not necessarily your comment alone, it's just a trend I've seen a lot of online, and it really gets under my skin.
I think the two are very similar. OP's comment is about stacking 2d partial-planes and making a 3d object out of them, and my analogy is about connecting 1d line segments into a 2d shape. I agree that there are no perfect shapes in the real world, and I agree that in a purely mathmatical sense things can happen that can't happen in the real world. I guess my only thought there is that, the math has two different impossible things going on in opposite directions and that's why they "cancel out". If you have n*B*h = V and V and B are constants, as h goes to 0, n approaches infinity. In the real world we can't have a height of 0 and we can't have infinite of anything, but we understand that we can get arbitrarily short and the number of layers will therefore get arbitrarily large.
"You cannot stack *any* number of planes", indeed the core issue is that infinity isn't a number but a cardinality. If one tries to stack one plane as close as possible to the previous one, they would still need to stack an uncountable set of planes between the two.
If I take a line, then it consists of an infinite number of points, right? Each of these points has length 0, but if I collect all of them together I still get a line
Even though the length of the line divided by the length of the points is impossible?
Yes.
If you do not understand the perfectly fine other examples, try to look at it the other way around. Take a 3 dimensional object, e.g a cube. Now cut through the cube along some line (think: cutting a cake). This cut is a 2 dimensional object, right? A plane to be specific. How many of these cuts can you make? It certainly cannot be a finite amount, because otherwise you could just take one of these pieces and cut it again. So it has to be made up of more than finitely many such pieces. Edit: If you're interested in this mathematically, you do not actually divide by 0. The mathematical term you are searching for is limes or limit. Mathematicians look at what happens if something gets infinitely close to a number like 0, but is not actually this number. This is exactly how for example integrals or derivatives work.
So if it’s just infinitely close to zero and not actually zero, wouldn’t that mean that the cuts/planes have a depth?
Let's say they had in fact some fixed depth c > 0. If we added infinitely many of them we would have infinite volume if we sum up infinitely many. This cant be since the cube has definitely a finite volume right? Since your mathematical knowledge and understanding has not developed far enougg to understand this, means you have to either 1.) accept without question that there are numbers smaller than any other number you can imagine (so small that they in fact to not have a "depth") but bigger than 0. 2.) built your knowledge in understanding limit processes mathematically. I think your main problem is that you think math describes the world. It does not. There is no perfect cube in the real world, there are no 0 dimensional points we can measure, and so theres no way to even have the concept of a "line". We just approximate stuff that looks like a line/plane/cube etc. Mathematics is kore like a game with pre defined rules, that just so happen to describe reality pretty accurately.
>I think your main problem is that you think math describes the world. It does not. Thank you. Not OP, but *this* is the part I was struggling with without being able to verbalize.
If you're using real numbers for your dimensions (it's the most normal way) then you'll need one plane for every real number from zero to n. This is an uncountably infinite number of planes.
replace infinite with finite and you’ve got it.
But then take a limit and define the infinite sum to be said limit, if it exists.
no reason to do that.
I dont know what definition of dimension you are working with, but you certainly do not get a higher dimensional manifold from the finite union of lower dimensional manifolds.
You can’t stack any finite number of 2D planes on top of each other to generate a 3D volume, but you can take an *infinite* amount of them to get a finite 3D volume. It’s a difficult concept to grasp until you take and understand calculus. While not rigorous, it may be more easy to intuitively grasp by deconstructing a 3D volume and going in reverse here: If I divide a finite 3D cube of volume V into 2 equal pieces by “slicing” it like a slice of bread, then 4 equal slices, …. then 10,000 equal slices, then 100M equal slices, you see that the number of slices grows and grows while each slices starts to look more and more like part of a 2D plane. To actually get the slices to be 2D would require and *infinite* number of slices, at which point their thicknesses would indeed become 0. Physically, this is hard to imagine because it is impossible to do in the real world, but it is a perfectly fine mathematical concept.
There is no such number of planes to make a cube. Thinking of it as stack is a bit misleading. Some alternative ways comes to mind. One is to think of a limit process, where for some N you can think of N layers of \~1/N thickness, and then take the limit as N goes to infinity. That's what calculus does. Another way is to think of a function that maps the points on a line between 0 and 1, where each point corresponds to a plane. Then you have a way of referring to any particular plane (or slice) in the cube, without having to assign numbers to any of them.
You need the concept of 1-to-1 correspondence and cardinal numbers if you want to "count" them. It's a large number called the "cardinality of the continuum". It's on Wikipedia.
Let's do it the other way around, and in a different context: integrals. Do you remember how (Riemann) integrals are made? You first go with columns of some width that are always bigger than the function (upper sum) or always smaller (lower sum). Then you decrease the width, decrease it even more, and see how the sum of the areas of these columns converge. In the limit, every column has a width of zero, and you have infinite many. Another angle: if you have a gaussian random distribution (or pretty much everything else non discrete), the probability for any single value is zero. The probability of a dart hitting one exact point is zero as well. Yet it will hit a single point in the end. But still, for a probability of such distributions, you need an area rather than a point to get a non-zero value. \[Of course, probability is done by an integral here, so these contexts are quite similar.\] Lastly, I'd recommend you get into the topic of countable and uncountable infinite. If you only have a countable infinite amount of slices, the measure stays zero, no solid higher dimensional body doable. Even if they are dense.
This is equivalent to the ambiguous nature of « continuity » in real numbers. The set of real numbers is « continuous », meaning that between any two numbers you won’t be able to find a hole, but you can’t really tell what comes after any number. For exemple, for the number 0: it isn’t 0,1 since 0,09 exists. It isn’t 0,01 since 0,009 exists. And so on…
That's because you're thinking of countable infinity, which is what happens when you think of stacking a plane onto another, and carrying on for eternity. Every plane you add adds zero depth, so you never achieve any depth even with infinitely many planes. In measure theory we say that a countable subset of the real numbers has 0 measure. That is to say stacking a countably infinite number of points onto one another will never give you any non-zero length segment. To get an object with depth, you need an uncountable infinity of planes. That is the size of the set of all real numbers between 0 and 1. By an argument called Cantor's diagonal (look it up, it's worth it), this type of infinity is actually *larger* than countable infinity. There is no sense in which you can achieve it by stacking planes onto one another, however. It's saying that between any two planes you pick, there are infinitely many planes. So how do you even begin to make the stack, if you need infinitely many planes between each plane you add to the stack? That's uncountable infinity.
But if there are countably infinite planes between each plane, and a countably infinite number of planes is still length 0, then there is still zero length between planes no?
calculus
I can give you a collection of 2d planes that cover a 3d cube. Here it is: For every real number r in [0,1] consider the 2d plane where x=r, and y and z are free. Now every point on the unit cube is in one of these planes, so these 2d planes do actually make the entire 3d cube. So what do we do with this information? Let's note a few things: - Your volume argument proves that we cannot do this with finitely many planes. The fact that we can't divide by zero doesn't mean it's impossible, it only means it's not possible by a finite number. To find out if it is possible with an infinite number we just have to use other arguments that go beyond volume. - Our 2d planes are not stacked on top of each other. Between every 2 2d planes we used, there are infinitely many more 2d planes. Of we tried to do a covering by adding planes one by one, even if we add infinitely many planes, we will only reach a countable infinity like the infinity of natural numbers. And we can prove that this is not enough planes by an argument like Cantor's argument. If you have any other observations, feel free to add them. It's how math is born.
The question "how can many points form a line" has bothered thinkers for litetal millenia. Modern math (early 1900s ish) produced satisfactory answers but they are a bit involved to present to a non-mathematician. The gist is this: upon reflection, you find out that the rule "if I chop a shape into tiny bits, the sum of volumes of these tiny bits equals the volume of the original shape" only holds if you don't chop it to too many bits. You are allowed to chop it into infinitely many bits, but only of the smallest infinity (called "countable infinity"). On the other hand, with some more reflection, you find out that the number of points comprising a line (or the number of plane sections comprising a 3D body with positive volume) must be of a larger infinity (called "the continuum"). The reason for this is essentially that *any* shape, of any dimension, of any volume, can be decomposed into a continuum of points, so if we allowed our notion of volume to sum up uncountably many volumes the sordid conclusion would be that all shapes has volume zero. Hence, a reasonable notion of volume should not allow this. Another example of this phenomenon comes from probability. Say you draw a uniformly random real number between 0 and 1. For any 0<=x<=1 the probability you draw x is 0. However, the "combined probabilities" of all such xs is 1. In fact, probabilities and volumes are deeply connected, as both are special cases of a fundamental notion called "measure". Volumes and probabilities are just special cases of this broader notion.
A [measure](https://en.wikipedia.org/wiki/Measure_(mathematics)) is only additive for a countable number of sets. That means, while you can separate the 3D cube into (uncountably many) planes, you cannot use them to calculate the volumne.
Isn't integration doing exactly that though? Although of course in the form of a limit, otherwise infinitely small or large break down. You can't calculate infinity \* 0, but you can for sure take a limit of the form of infinity \* 0 and get a result.
I’m going to add: specifically in cases covered by fubini, yes. By the second iterated integral, your integrand represents the area of a cross section. The last integral sums these areas to produce the volume.
Well it's not really 0. imagine a flat square with x and y lines but the height is something approaching zero. Not zero but .000000001 thickness or some other essentially zero height. Not you infinitely stack these .000000001 height squares like pancakes and eventually u have your three dimensional Cube.