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Foolish question debunked. Move on, nothing to see.

Postby morriswalters » Sun Aug 06, 2017 3:41 pm UTC

Is the Banach–Tarski Paradox proof that you can't see the edge of the universe? To a non mathematician, that's what it looks like. Is there any reason I shouldn't think of it in that way? This thought seems to me, to lead inevitably to the idea that in our Universe, in as much as you can talk about this with words, shrinking away from the edges = expanding outward. Thus that relativistic limit on light, represents the indeterminate point at the edge of the of the expansion? So you can't get there from here. Just curious.
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Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby DavidSh » Sun Aug 06, 2017 4:01 pm UTC

Wikipedia describes the Banach-Tarski Paradox as follows:
The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them, without changing their shape. However, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points. The reconstruction can work with as few as five pieces.


I don't see any relationship with seeing the edge of the universe. Were you thinking of some other paradox?

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Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby doogly » Sun Aug 06, 2017 5:42 pm UTC

Yeah these are completely unrelated.
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Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby morriswalters » Sun Aug 06, 2017 10:54 pm UTC

DavidSh wrote:I don't see any relationship with seeing the edge of the universe. Were you thinking of some other paradox?
No I was thinking of spheres and points of reference. If you define a point as dimensionless and you sweep the surface with a vector from another dimensionless point at the origin, then you can point to an infinite number of places. Yet when you move to another point you can do exactly the same thing. At the new reference point. So place an observer in space so we can see him, somewhere close to the edge of the observable Universe. What can he see? Now if we place another observer where the third can see the second. What can he see? Each of these locations is a set with intersections with other sets. And it doesn't matter how many times you do this or in which direction. I believe this is representative of the Banach–Tarski Paradox.

I'm not representing this as anything other than a thought. I wondered if that were the case if you could transform the point of view so that the matter in the universe is shrinking with respect to the space.

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Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby Tub » Mon Aug 07, 2017 7:19 am UTC

I think you're assuming euclidean space somewhere. If the universe has an edge, you can bet that its surroundings are highly non-euclidean.

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Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby elliptic » Mon Aug 07, 2017 9:30 am UTC

So place an observer in space so we can see him, somewhere close to the edge of the observable Universe. What can he see? Now if we place another observer where the third can see the second. What can he see? Each of these locations is a set with intersections with other sets.


All you're saying is that different observers each have different sets of points in spacetime as their own observable universes.

Which is true (obviously) but there's no connection at all with Banach-Tarski - that's about partioning the points in *one* given set to generate two copies of it.

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Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby morriswalters » Mon Aug 07, 2017 2:12 pm UTC

I don't mind if you laugh, feel free. Having said that.
Tub wrote:I think you're assuming euclidean space somewhere. If the universe has an edge, you can bet that its surroundings are highly non-euclidean.
Well I know space is non-euclidean, it's what that means that I'm interested in. So framed by Banach-Tarski Paradox, I'm suggesting that any of those points of reference look out on different universes. What we see is the intersection of the two universes. So the observable Universe intersects all the other observable universes.

Each separate unto themselves except for the intersections. The degree of commonality at the intersection might somehow indicate how the structure might vary from our structure to some other universes structure???

This implies that you transit to a different universe simply by walking into the next room. Thus elliptic's comment. What it says to me is that wherever you go there is already something there, does not imply that there is a place where there is nothing. It merely implies there is a different universe already there, everywhere, separated only by a point of reference.

It's just an outgrowth of the nature of spheres. As elliptic noted most of these universes would be trivial, up close, in the same way as the surface of a sphere looks flatter the bigger the surface of the sphere with respect to the point of view.

As a purely science fiction thought you could consider the the variations away from expectations when looking at cosmological numbers, is jitter caused by the intersections. So you wouldn't need dark matter or energy. Hold time moving in one direction so nasty time travelers show up(sorry Dr. Whosil) and you're golden. Any way it's a thought.

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Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby arbiteroftruth » Mon Aug 07, 2017 2:27 pm UTC

Can you give a rough explanation in a few sentences of what you think the Banach-Tarski paradox has to do with any of that?

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Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby doogly » Mon Aug 07, 2017 2:28 pm UTC

Banach Tarski is about set theory. Questions about the edge of the universe are about geometry, and in particular how light would move through a curved space. Banach Tarski isn't really a geometric result. It is called a "paradox" because it defies geometric intuition, and it can defy geometric intuition because it isn't really about doing geometric processes. So I think it is still ultimately a mismatch to consider in this context.
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Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby MartianInvader » Mon Aug 07, 2017 9:07 pm UTC

Well, Banach-Tarski *is* about doing isometries, so it's about geometry at least a little bit.
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Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby morriswalters » Mon Aug 07, 2017 9:26 pm UTC

Wikipedia wrote:The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them, without changing their shape. However, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points. The reconstruction can work with as few as five pieces.[1]
I would point out that if this is true, than any of the copies are can be cut up to produce another two spheres. You end up with an infinite number of spherical balls, all from the first ball. Is this incorrect on my part? More explicitly the origin can be modeled as dimensionless sphere with exactly the same number of points on its surface as the sphere comprising the observable Universe, or any other sphere.
doogly wrote:it isn't really about doing geometric processes
I would have no idea in the sense you might be using it. I came to it from somewhere else. It comes from the nature of infinity. The Paradox just reflects something I was seeing in the geometry. It's the same reason there are no holes in the real number line. One set can contain all infinite sets.
doogly wrote:Questions about the edge of the universe are about geometry, and in particular how light would move through a curved space.
Why? I understand what you mean, but what has it to do with the things I asked? You can't see the edge. You always see more stars no matter where you look from. All that has changed is that you are looking from somewhere else. When you rotate his pieces all you do is transform the coordinates. In mine the observer can move in a way such that he always sees something new when he looks ahead. All I have suggested is that space can curve and anywhere along that curve is exactly like anywhere else. Curved space makes no difference to him. It isn't curved where he is at in any way he can see. Yet when he moves he follows the curve.

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Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby gmalivuk » Tue Aug 08, 2017 2:11 am UTC

You're supposing there is no edge, and then concluding that you can't see an edge. That has nothing to do with Banach-Tarski.
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Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby Eebster the Great » Tue Aug 08, 2017 8:11 am UTC

The way you are applying your misinterpretation of the Banach-Tarski paradox doesn't make any sense to me. It sounds like you could apply it to any volume and conclude that all finite volumes are infinite.

The Banach Tarski paradox relies on partitioning a ball into several pieces, mostly non-measurable sets, and translating and rotating them to form two balls. It isn't clear how that could have any physical application at all.

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Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby morriswalters » Tue Aug 08, 2017 2:51 pm UTC

When we're talking about this without mathematics other than its simplest forms, you are speaking to a brain damaged child. I am most probably wrong. I wish that I had a better notion of set theory and topology. This would be easier. Language has too much redundancy. I'm trying to define the defects in my understanding without the proper tools in my tool box. And I understand that makes me frustrating to deal with in these cases. I'd like to go back a slap that earlier me stupid.

Eebster the Great wrote:It sounds like you could apply it to any volume and conclude that all finite volumes are infinite.
I'm talking(and so apparently is he) about either volumes or surfaces of a mathematical sphere composed of dimensionless points. He says, in brief, that any sphere, formed that way, has that property. What I'm looking at is frames of reference. If a sphere is defined as dimensionless, then it is equivalent to a dimensionless point. Naively, this implies that it is true for any origin in space n, that is symmetric about all it axis. What follows under the spoiler may well be gibberish, almost certainly the form is amateurish. And it isn't a proof.
Spoiler:
Where ng is any coefficient which forms a set such that for each coefficient ng there is a unique corresponding variable.

So the Σ(ngcxgv)=d (for g found on the real number line, and for gc and gv independent of each other) represent an equation in n dimensional space differentiated by their coefficients.

Such that for 3 space 1x+n2y+n3z=d there is a plane of the coefficients on which there is an infinite number of axis where the equation is satisfied.

This equation defines the nature of what I am suggesting for three space. The general case in n dimensions where the coefficients define the space. Again from the Wikipedia article.
Unlike most theorems in geometry, the proof of this result depends in a critical way on the choice of axioms for set theory. It can be proven using the axiom of choice, which allows for the construction of nonmeasurable sets, i.e., collections of points that do not have a volume in the ordinary sense, and whose construction requires an uncountable number of choices.[2]

It was shown in 2005 that the pieces in the decomposition can be chosen in such a way that they can be moved continuously into place without running into one another.[3]
This is implicit from the geometry of spheres. He says in effect(simplifying to the real number line), I've created a set called the real number line. I can look at the real number line by picking my frame of reference, in such a way as to find a copy of the real number line, which is identical to the first real number line. And I can do this forever. I am quite prepared to accept that there is a defect in my understanding. I had never seen this proof for spheres.
gmalivuk wrote:You're supposing there is no edge, and then concluding that you can't see an edge. That has nothing to do with Banach-Tarski.
Not quite. I'm saying that there is an edge, and asking why I can't see it. Which simplifies to, there is no edge to see, there is always someplace else.

Consider this analogy. A man walks up to a cube of 100 foot on its side. On the side of the cube as he faces it is a number 1. He draws a smiley face on the cube and walks around the cube so that his left hand touches the wall. He then walks around the cube drawing smiley faces on each side as he does. When he gets to where he started, the number reads 2, and there is no smiley face. Obviously I have transformed his frame of reference. The only stipulation I place on it is, that time always points in the same direction in his frame of reference. He can walk back to his starting point, but in his frame, his watch never runs backward. I don't care what weird path in my hypothetical space he had to walk to get there.

What I've suggested is that space time does something like this at the edges. I have also suggested by symmetry, that when you use this type of transform, it allows you to look at some things in different fashions. If the topology of space time isn't closed than the curve needn't curve back on itself. In my analogy, space time is either a descending spiral or a ascending one. Or some other weird plot in space time. But the topology isn't closed unless I choose to close it by not walking about the cube.

I'm using closed in this sense specifically. I could be misunderstanding it.
Consider a sphere in 3 dimensions. Implicitly there are two regions of interest created by this sphere; the sphere itself and its interior (which is called an open 3-ball). It is useful to be able to distinguish between the interior of 3-ball and the surface, so we distinguish between the open 3-ball, and the closed 3-ball - the closure of the 3-ball. The closure of the open 3-ball is the open 3-ball plus the surface.
Connect the observers in such a fashion that if one has a line of sight then they all have a line of sight which passes through all, such that the line of sight appears to be a straight line. It doesn't matter that it really can't be. It's a way of defining the frame of reference of any observer. You could travel in that fashion forever and never reach a position where you can't see any more. There is no edge. If the topology isn't closed than there isn't any reason that I'm aware of, that I couldn't say, you would never see the same space twice, in a line of sight defined that way. So in a Universe defined by the Banach–Tarski Paradox there infinite other universes exactly like like the original, differing only by the frame of reference you're looking from. Cosmologists bounded our space when they set their initial conditions. These identical spheres, are identical only in their geometry.

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Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby Tub » Tue Aug 08, 2017 4:23 pm UTC

I cannot claim to fully understand your proof, but I get the impression that you're only using a simple geometric translation of a sphere or of a point of reference. The interesting part of Banach-Tarski is that you can disassemble the sphere and re-assemble it into *two* equal spheres, effectively doubling the volume. But you don't seem to rely on that property in your argument.

There are several possibilities for the topology of our universe. It could be infinite and borderless. It could have an observable edge, but the laws of physics would get weird in the vicinity. It can be finite and closed, thus without an edge. It can be finite and open, but expanding, having an edge that's never observable. The last case requires space to be curved in such a way that the edge is an event horizon, i.e. moving away at the speed of light from every observer. (I wish I could find the article.. it was a good read.)

Considering that we can only observe the observable part of the universe, and that part is pretty flat, we'll never find out. We can only hope for a strong theoretical argument.

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Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby morriswalters » Tue Aug 08, 2017 10:58 pm UTC

It isn't a proof, merely an observation about spheres. It is certainly fantasy.
Tub wrote:The last case requires space to be curved in such a way that the edge is an event horizon, i.e. moving away at the speed of light from every observer. (I wish I could find the article.. it was a good read.)
I might be able to read it, maybe. But the math would require more time than I have to learn. I thought about some of those but, it's just beyond my skills. I considered one case, where rather than expanding, matter was collapsing in upon itself.
Tub wrote:The interesting part of Banach-Tarski is that you can disassemble the sphere and re-assemble it into *two* equal spheres, effectively doubling the volume. But you don't seem to rely on that property in your argument.
I would be deceptive if I said I knew for certain but this seems to fit that bill. All I did was superimpose these balls. It occurs to me to ask if this is why you see a sphere when you look at the sun from any location.
Using the Banach–Tarski paradox, it is possible to obtain k copies of a ball in the Euclidean n-space from one, for any integers n ≥ 3 and k ≥ 1, i.e. a ball can be cut into k pieces so that each of them is equidecomposable to a ball of the same size as the original. Using the fact that the free group F2 of rank 2 admits a free subgroup of countably infinite rank, a similar proof yields that the unit sphere Sn−1 can be partitioned into countably infinitely many pieces, each of which is equidecomposable (with two pieces) to the Sn−1 using rotations. By using analytic properties of the rotation group SO(n), which is a connected analytic Lie group, one can further prove that the sphere Sn−1 can be partitioned into as many pieces as there are real numbers (that is, 20 pieces), so that each piece is equidecomposable with two pieces to Sn−1 using rotations. These results then extend to the unit ball deprived of the origin. A 2010 article by Valeriy Churkin gives a new proof of the continuous version of the Banach–Tarski paradox.[11]

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Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby Eebster the Great » Wed Aug 09, 2017 3:45 am UTC

If the universe has an edge, then by definition we don't live in a Euclidean space. In order to actually decompose one ball into two, you need to translate some of the pieces. If there is no room for you to move them to, that is simply not possible. The B-T paradox is just not relevant to what you're trying to demonstrate.

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Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby morriswalters » Wed Aug 09, 2017 3:29 pm UTC

Eebster the Great wrote:If the universe has an edge, then by definition we don't live in a Euclidean space.
Yeah, but then comes the thought, if I go that way long enough where will I end up? So I was answering my question, if I can't go where I want where will I go if I try anyway? I showed me a way not to need an edge.

His proof doesn't work if "room to do something? is meaningful, as he doesn't seem to be talking about volumetric solids. He seems to be discussing an shape composed of a point cloud. He bounds it so you can see that the new shape is distinct from the original shape even having an infinite number of points. The general proof of infinite spheres takes care of the origin.

Because you have infinite points you now have two spheres with infinite points distinguished by their axis of rotation. He then makes a new set containing both those spheres. Given this I can stop. Whatever he was trying to prove generally, he proved what geometry tells me is true when he did this, and reminded me why my intuition says his construction is nuts.

A Rubik's Cube is an example of this type of thing. The faces change but not the rest shape. The way he seems to talk about n(1,2), I believe, implies a condition where both the axis of rotation and translation must act through the geometric centers of the pieces. Whole spheres are non distinct, and hemispheres move out of the boundaries, at those value of n, thus out of the set as he defined it. But that isn't clear to me.

The long and the short of it is that he chose dimensionless points. They have no volume. They have shape. If the shape is the same, it matters not one wit if the volumes are different.

When I walked the character around the sphere he took a path in such a way that as he moved through space through a series of discrete axis as he traveled on the path. In the case of the cube he was traveling around the vertical axis on a flat spiral staircase. With n continuous and increasing as t increased, and his velocity was non zero. It's a very crude model but it felt a little creepy. I like it, I would write a story, but Heinlein did it already using doors. I would like to take blame for this mess, but instead I must instead cede to numerous others. First and foremost Bertrand Russell, anonymous poster, Terry Pratchett and Steven Barnes, and some others. Anyway thank you to all who responded.

I have now been thinking about spheres longer than I believe is healthy.

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Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby Soupspoon » Wed Aug 09, 2017 4:22 pm UTC

morriswalters wrote:Yeah, but then comes the thought, if I go that way long enough where will I end up? So I was answering my question, if I can't go where I want where will I go if I try anyway? I showed me a way not to need an edge.

I'm a big fan of the finite-but-unbounded (i.e. wraparound) model, ultimately. But reaching the (nominal) wraparound point is likely beyond the light-cone edge of any such expedition, and indicators as to whether it wraps or stops or fazes out or the edge stays just ahead of you must be inferred from past lightcone of information, rather than reaching out and touching the future edge. Which means cosmopaleology and a lot, of mathematics is your only tool to try to verify what our future selves may never manage to discover first-hand, for many of the possible limitations that it may actually be..

Any which way, heading in any direction for long enough isn't long enough to go anywhere interestingly "beyond the boundary" or "coming back from the other direction", may only lead us to places we can already see (plus a significant amount of time, which renders them no longer anything much like what we see). And we also will not see any thing new for om this new vantage point that we did not previously see from this one (plus the rigours of time passed).


(And greater minds than mine are working on which subset of models accurately portray our universe, or potentially still so. I'm not melded to the wraparound model, it just seems simplest. Allied to non-Euclidean warping of space and inclusive of expansion too, probably, rather than anything as simple as an Asteroids setup.)

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Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby Xanthir » Wed Aug 09, 2017 4:50 pm UTC

Yeah, either there's an edge, or it's closed (wraparound), or it's infinite. As people repeatedly say, none of those involve B-T.
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Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby morriswalters » Wed Aug 09, 2017 8:14 pm UTC

Soupspoon, I never said you could go there, I just said there was a place to go, if you thought about it. I would talk about this with my neighbor, I'm pretty sure he doesn't care though. My options are limited outside that. Since I was merely translating the frame of reference, space itself, isn't all that important. I live in 3 space. I don't care what space does because I can't see space time. I'm little and it's big. :D But my mind sneers at space time, laughs at the cone of visibility, tickles time in the ribs, and goes where it pleases. And is pleased in fact because I'm little.)

Xanthir wrote:As people repeatedly say, none of those involve B-T.
You are probably right. I'm primarily interested in infinite spherical sets around a common axis. I've had two other arguments about those sets in the last two weeks. Both of those involved frames of reference in terms of lines of sight. What I thought the Banach–Tarski Paradox said was that if I had an infinitely small sphere at the center of the sun made with infinite viewpoints is exactly the same as a sphere centered on the sun at the orbit of Mars made of infinite viewpoints if you consider only the viewpoint cloud in both. How is this view defective?


Edit
Just in case you missed it the pre edit post also says I can rotate to any axis. Since they are all the same when I'm in them.

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Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby Xanthir » Wed Aug 09, 2017 9:58 pm UTC

B-T has nothing to do with viewpoints, or translations of viewpoints. It's just a funny example of a non-intuitive result of the Axiom of Choice.
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Big No. Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby morriswalters » Thu Aug 10, 2017 12:44 am UTC

I actually asked you to tell me "How is this view defective?". And your answer is basically because I said so. But that's okay, I wandered far afield. I wrote the answer to my question in the title text.

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Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby ConMan » Thu Aug 10, 2017 2:21 am UTC

B-T is about breaking a thing up into an infinite number of pieces and moving them around to get a strange result. It doesn't prove anything about seeing the edge of the universe any more than Zeno's paradoxes prove that the existence of space and time are impossible and you can't actually finish a 100m race against a tortoise (with the biggest difference being that Zeno was actually trying to prove something like that).
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Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby Eebster the Great » Thu Aug 10, 2017 2:27 am UTC

morriswalters wrote:
Eebster the Great wrote:If the universe has an edge, then by definition we don't live in a Euclidean space.
Yeah, but then comes the thought, if I go that way long enough where will I end up? So I was answering my question, if I can't go where I want where will I go if I try anyway?

The edge. That's what "edge" means. It's where you end up when you go towards it as long as possible.

Again, can you explain how your proof (which I still do not understand at all) would not apply equally well to anything smaller than the universe? For instance, why can't I use the same logic to prove that a pizza box has no edge?

His proof doesn't work if "room to do something? is meaningful, as he doesn't seem to be talking about volumetric solids. He seems to be discussing an shape composed of a point cloud. He bounds it so you can see that the new shape is distinct from the original shape even having an infinite number of points. The general proof of infinite spheres takes care of the origin.

Because you have infinite points you now have two spheres with infinite points distinguished by their axis of rotation. He then makes a new set containing both those spheres. Given this I can stop. Whatever he was trying to prove generally, he proved what geometry tells me is true when he did this, and reminded me why my intuition says his construction is nuts.

The theorem does not rely necessarily on any particular decomposition; it merely states that a sphere is equidecomposible with two copies of the sphere. The sets chosen for the constructive proof were merely convenient, not unique. I don't know what a point cloud is, but I have never seen that term in connection with the Banach-Tarski paradox. The theorem is about actual 3-balls in continuous Euclidean 3-space, not some ersatz facsimile of them.

The long and the short of it is that he chose dimensionless points. They have no volume. They have shape. If the shape is the same, it matters not one wit if the volumes are different.

Well, points don't have "shape." Points have no properties except location. The 3-ball has a shape, but it turns out that the shape actually doesn't matter. The theorem can be extended to all bounded solids with nonempty interior (so, for instance, it doesn't apply to spheres, but it does apply to spherical shells with finite thickness, or to cubes, or to busts of Pallas). But it is true that any decomposition of any solid demonstrating this theorem must involve nonmeasurable sets, i.e. sets that don't have a defined volume. (It's not that they have zero volume; they do not have a volume at all.)

When I walked the character around the sphere he took a path in such a way that as he moved through space through a series of discrete axis as he traveled on the path.

It's sentences like this where I get lost. what does "he moved through space through a series of discrete axis" mean?

ConMan wrote:B-T is about breaking a thing up into an infinite number of pieces and moving them around to get a strange result.

Not an infinite number of pieces. Five pieces.

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Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby ConMan » Thu Aug 10, 2017 7:21 am UTC

Eebster the Great wrote:
ConMan wrote:B-T is about breaking a thing up into an infinite number of pieces and moving them around to get a strange result.

Not an infinite number of pieces. Five pieces.

Sorry, you're right. Five pieces, but not particularly obviously shaped ones.
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Absolutely not! Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby morriswalters » Thu Aug 10, 2017 12:38 pm UTC

The longer I edit the worse it gets. Remember the brain damaged child.

I'm being told two things by two different sources. On one hand
Eebster the Great wrote: I don't know what a point cloud is, but I have never seen that term in connection with the Banach-Tarski paradox.
And then this.
The Banach–Tarski paradox is a theorem in set-theoretic geometry, ..... However, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points. The reconstruction can work with as few as five pieces.[1]
While my math skills are suspect my reading comprehension skills are not. Is there a way to read that where the pieces are anywhere near a conventional solid?
Eebster the Great wrote:Again, can you explain how your proof (which I still do not understand at all) would not apply equally well to anything smaller than the universe? For instance, why can't I use the same logic to prove that a pizza box has no edge?
I just restated something familiar in an unfamiliar way. Edges assume you can see them. If the universe exists the way the current theory suggests, then haven't cosmologists told us the space is expanding and saying that something exists outside it isn't meaningful? Doesn't that translate into, there is no edge because everything exists inside the universe? Because an edge only exists if there is another side?
Eebster the Great wrote:Well, points don't have "shape." Points have no properties except location. The 3-ball has a shape, but it turns out that the shape actually doesn't matter. The theorem can be extended to all bounded solids with nonempty interior (so, for instance, it doesn't apply to spheres, but it does apply to spherical shells with finite thickness, or to cubes, or to busts of Pallas). But it is true that any decomposition of any solid demonstrating this theorem must involve nonmeasurable sets, i.e. sets that don't have a defined volume. (It's not that they have zero volume; they do not have a volume at all.)
On the first highlighted area, just no. They don't have a location. They are defined by the geometry of the shape they take. If the points are dimensionless, obviously they have no volume nor do they have an absolute location.

To continue, a sphere composed of infinite points is equivilant to a spherical shell composed of an infinite number of points because volume is undefined. On the second highlighted area, you do realize that that the origin is the reason a they talk of shells instead of spheres, right? They actually cite that case in a proof for creating infinite sets of spherical point clouds. And devolve it into a case of a sphere with no origin.
If I define a dimensionless sphere, I in fact define a point, for r sufficiently small. So any given sphere, for r sufficiently close to the origin a spherical shell of finite thickness is equal to the sphere. The is limit of a sphere as r approaches 0.

Returning to the point. In space at the orbital shell representing the orbit of Earth, when I look at the Sun from that point, what I see is a disc in a plane. I see that disc from any point anywhere on the shell. And those points are connected to those discs by a sight line. Along that sight line are an infinite number of places that I can see that disc. Each one of those places represent a point, both on the shell and in the volume. When you pick a radius you define the volume. All those places you can look at still exist. When I define an sphere of dimensionless points haven't I, in fact, created a special case of a solid, one with no voids, but with infinite spaces?
Eebster the Great wrote:It's sentences like this where I get lost. what does "he moved through space through a series of discrete axis" mean?
This is what happens when you discuss this without rigor. I let things get too scattered and then, well... Anyway to the point. I picked an origin and defined an axis about that point in three space. Each axis about that point is orthogonal to all other axis. When I walked my avatar around the cube I chose his motion so that he moved through each distinct axis about that origin smoothly and continuously. In three space, I would be ascending the z axis in a spiral. In n-space I moved around and through all axis without ever repeating any axis. With the conditions as stated in the examples I have given you.

Aside

This is one part SF, one part astronomy, and a case of a linear equation, in n different variables. Terry Pratchett and Steven Barnes wrote a novel called The Long Earth which uses this conceit. In their universe they portray the each world as different by my axis. They create a narrative hook to move from place to distinct place. I simply took this and got rid of the deceit and said if you followed the sight line point to point, you could posit a universe that was unbounded and could where you could go everywhere, depending on your choice of sight line. And given that the scope, all cases can be covered by allowing discontinuities. Places the sight line could go, that you can't. An alternative way of thinking of the edge, in this case, is a space with no sight lines in at least one axis.

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Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby gmalivuk » Thu Aug 10, 2017 1:55 pm UTC

The Long Earth series depends on at least one additional dimension along which people can "step" into parallel worlds. It still has nothing to do with Banach or Tarski and can be thought of in relatively simple geometric terms (even if canonically it's not quite so simple).

You seem confused about the difference between how many points are in a set and how much space that set takes up (i.e. its measure). The surface of a sphere has the same number of points as the interior, but the surface has zero volume. (And a circle has the same number of points but zero area.)

A crucial point of Banach-Tarski is that some sets don't have a measure. Not that a set of points has zero volume, but that it has no well-defined number that corresponds to volume at all. (And what makes it a "paradox" is that one sphere can become two through purely rigid motions. There's nothing too special about the fact that you can take one sphere and create a function that moves the points around to make two. Obviously you can change the measure of a set of points by deforming it and stretching it. Doing so by just sliding and rotating subsets is what's paradoxical.)
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Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby doogly » Thu Aug 10, 2017 1:57 pm UTC

An edge doesn't imply something on the other side of it. It's common when doing geometry to embed everything into some higher dimensional space, because of our experiences with drawing shapes on a paper. You want to look at a circle? Draw your lil S^1 on a paper, which is R^2. You want to draw a disk? Draw your D^2 by filling in your circle. It's on a paper, it lives in R^2. Nice. We got our notebook open on the desk, we are learning, together.

But it doesn't have to be that way. The disk ain't need to live in your notebook. It has an edge, and on the paper, outside of that edge there's all this graph paper keep going, but who needs that? Disk ain't need that. Try to think like the disk, instead of the student. The disk definitely has an edge -- and meaningfully so! That edge will determine how differential equations on that disk be like, which is not how things would be like if instead it were something else. You should imagine, what if I were a disk, and what if there were differential equations on me?

Imagination is the most important thing in math, and sci fi is really like training wheels for imagination. By which I mean, it thinks it is helping, but is really not. You should just practice bicycling by rolling down a hill at a high speed and just be sure to wear a helmet, it'll work itself out. [disclaimer: I haven't ridden a bicycle since like 5th grade, and I wasn't good, but I can do math.]

Also a point definitely has a location if you want to start using that point in some topology. "Topology" is the right scope of discussion for B-T. It is important to look at some different math clubs, like, what goes on in set theory club? What goes on in topology club? What goes on in geometry club? They are different clubs but they all play meet up for lunch and exchange interesting ideas. This is why "What properties does a point have" is maybe not completely obvious. A point can have some coordinates, but then you are definitely doing geometry. It can live in like, some kind of a neighborhood, maybe of another point. Topology club. No properties, it's just part of this list. Set theory. Set theory is boring but you should eat your wheaties. They're good for you.
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Re: Absolutely not! Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby Soupspoon » Thu Aug 10, 2017 2:28 pm UTC

morriswalters wrote:Terry Pratchett and Steven Barnes wrote a novel called The Long Earth which uses this conceit.

Barnes? Baxter.

And the conceit in TLE never really gets explained in these compatible terms. I suspected something Hilbertian, or perhaps with a Cantor Pairing path, especially as Baxter had form in this area, but it really wasn't specified any which way. Being mid-way through TLM (so would rather not have spoilers go in either(/any) direction - I've only got so far as the inscrutable monoliths bit), I am not entirely disabused of that basic notion, but obviously realise it to be more complicated than my original assumption. Even assuming it is intended to be a fully-formed theory, and not just a particularly reinforced peg to hang a huge Mcguffin upon.

One might as well ask Elon Musk Reid Malenfant about the nature of our universe. (And I really need to read more of the Manifold series than I already have, too...)

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Re: Absolutely not! Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby arbiteroftruth » Thu Aug 10, 2017 3:39 pm UTC

morriswalters wrote:
Eebster the Great wrote:Well, points don't have "shape." Points have no properties except location. The 3-ball has a shape, but it turns out that the shape actually doesn't matter. The theorem can be extended to all bounded solids with nonempty interior (so, for instance, it doesn't apply to spheres, but it does apply to spherical shells with finite thickness, or to cubes, or to busts of Pallas). But it is true that any decomposition of any solid demonstrating this theorem must involve nonmeasurable sets, i.e. sets that don't have a defined volume. (It's not that they have zero volume; they do not have a volume at all.)
On the first highlighted area, just no. They don't have a location. They are defined by the geometry of the shape they take. If the points are dimensionless, obviously they have no volume nor do they have an absolute location.


You have that exactly backwards. A point being dimensionless means it has no shape. In order to have a shape it has to have things like length, width, angles, curvature, something along those lines. A dimensionless point can't have those things. But just because the point itself is dimensionless doesn't mean the space it exists in is also dimensionless. A dimensionless point can quite easily exist within a 3-dimensional space, and when it does, it's quite easy to talk about its location within that space. That's what means, for example, for a point to be the center of a sphere. The central point is dimensionless, and has a location at the center of some sphere.

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Not just no but h*ll no!Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby morriswalters » Thu Aug 10, 2017 7:21 pm UTC

gmalivuk wrote:A crucial point of Banach-Tarski is that some sets don't have a measure.
So I gathered from reading the related material. Shape doesn't have measure.
gmalivuk wrote:There's nothing too special about the fact that you can take one sphere and create a function that moves the points around to make two.
Well that's a mouthful. I suggest you try to do it without leaving the shape you defined.
Spoiler:
Here's how. That shape is your set. Rotate the pieces in such a way as to never leave that geometry. He chose n greater or equal to 3. n=1 is trivial, the sphere is itself. n=2 is a hemisphere, you can rotate along the surface, but not translate. The hemisphere falls outside the shape. The sets after that point, as you have pointed out, are trivial. It is any group of objects that are symmetrical to each other, having at least 1 point on the surface and one at the center. 3 and 4 each require two shapes, a 12-gon for 4, 12 things, 3 at a time, and a sexahedron for 3, 9 things, 3 at a time. 2 things 2 at a time gives you the sphere. Now make those out of points defined as having no dimension so you can rotate and translate. A modification, late in the editing. Looks to me that is doesn't like odd numbers, so divisible by two. Cool. Is that right.

What makes the problem interesting is the way the problem is constrained. The n=1 case is non distinct, you simply twirled the globe, obviously the sphere is symmetrical to itself, n=2 isn't symmetrical in the sphere. All the other cases are unique ones. Symmetrical to the center and each other. Take the limit of those shapes as n goes to infinity and those shapes become rays orthogonal from the center. For n greater than or equal to 3 tending towards infinity. That's as rigorous as I can make it. Feel free to shoot holes in the math. I would expect no less. This last result especially.

However each one of those spheres produced by those pieces is unique and represents an infinite set, of the ways those pieces can be rearranged
Read the title text over my post. I've already conceded that game. I have heard the consensus.

@Soupspoon

I read it. That I got the other guys name wrong should give you some idea of what I thought about it. It was interesting, but didn't set my pants on fire. It's a much overworked conceit. Faster Than Light travel in n space rather than 3 space. Heinlein once put a pocket universe in the back seat of a car that opened into OZ. In the current reality keep watching TV in 2018. This conceit will show up again. Like my old pappy told me, it ain't about the trip, it's about the destination. I'm a 3 space kind of a guy. Let me phrase it another way. A view port has no mass and no dimensions. Doesn't cost me anything and has zero calories. All I need is a frame of reference. All I need do is insert it where I please. And there I am. Very RNG like.

Alt-Txt

While I had heard of something remotely like Hilbert Space, it could've mugged me in broad daylight and I couldn't have given a description. My mother taught me to never look at strange spaces since they don't have edges, and I try to be a good boy for mamma. However I'm curious and read the Wikipedia entry, and thought I would choke. Thank you. :lol: So many spaces and not enough portals.

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Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby gmalivuk » Thu Aug 10, 2017 8:52 pm UTC

"Shape doesn't have measure" is a meaningless statement, as far as I can tell.

The sets in B-T aren't really complete "shapes" at all. They're sets of points that don't have a volume, though they do still "fill" the sphere (in the same sense that rationals "fill" the number line despite leaving out most of the points).

As for turning one sphere into two without restricting ourselves to rigid motions, take the sphere in the center (with half the radius of the original) and move it three units to the right, then double the radial distance of every point. Then double the distance from the surface of every point in the remaining shell. (One of the resulting spheres will lack a center point or the other will lack its surface, but both of those sets have zero volume so it's not really important. If the original sphere also lacks one or the other, we can do this rigorously.)

Even less interesting is if all we want to do is double the volume of a set of points, without worrying whether we actually duplicate a shape. Just pick an axis and double all the coordinates along that axis.

B-T allows us to double the volume *without* stretching anything. We just shift and rotate some subsets of a sphere and then we get two spheres.
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Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby Zamfir » Thu Aug 10, 2017 9:03 pm UTC

disclaimer: I haven't ridden a bicycle since like 5th grade

Of all the concepts in this thread, I found this the hardest to wrap my head around.

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Re: Not just no but h*ll no!Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby Soupspoon » Thu Aug 10, 2017 10:02 pm UTC

morriswalters wrote:@Soupspoon

I read it. That I got the other guys name wrong should give you some idea of what I thought about it. It was interesting, but didn't set my pants on fire. It's a much overworked conceit.
Well, I know Baxter from some of his other works (particularly Voyage, about the alt-history of the US Space Program, had things gone differently). I've met him, as well, though at the time I didn't know I already knew (about) him. Go figure.

I would describe the Long Foo books as "gentle". There's obviously a longer progressive arc to the tale, perhaps to be resolved in Utopia, and (apart from sparks of action, even in the largely misnomered War book) it seems like it's supposed to be gradually revelatory, not conclusive. I'd rate the Red/Green/Blue Mars trilogy as better progression. And that series barely breaks out of 'hard and current' science territory. No crazy-physics Mcguffins except those explainable by centuries of human technical progress. Anyway, this isn't about book critique, but about modelling the universe.

Faster Than Light travel in n space rather than 3 space. Heinlein once put a pocket universe in the back seat of a car that opened into OZ. In the current reality keep watching TV in 2018. This conceit will show up again. Like my old pappy told me, it ain't about the trip, it's about the destination. I'm a 3 space kind of a guy. Let me phrase it another way. A view port has no mass and no dimensions. Doesn't cost me anything and has zero calories. All I need is a frame of reference. All I need do is insert it where I please. And there I am. Very RNG like.
I get where you're starting from, but I'm not sure where you're going.

But I think my response to that is that you're going about it the wrong way. You can state a theory and then set your (fictional) universe up to use that theory, but that doesn't always relate well to the Real World in any imaginable way. I don't know if you're familiar with Christopher Priest's novel The Inverted World? The subjective conceit is understandable, though I'm not entirely sure that one would see a sun shaped like the see their sun, and the mismatch between viewpoints (of the naturalised denizens of the city and those that have been passed) strains my credulity. As a model, I don't see it's use (slipping back to literary critique, it's an interesting style and a not sub-par version of a Hero's Revelation), but seems more connectable than trying to link the BTP with anything, unless you work on the idea a lot more, to try to make it relevent.

And there aint no FTL (yet!) in the Long Universe, so I don't know where that bit comes from.

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Re: Absolutely not! Re: Does the Banach–Tarski Paradox prove that you can't see the edge of the universe?

Postby Eebster the Great » Thu Aug 10, 2017 10:40 pm UTC

morriswalters wrote:The longer I edit the worse it gets. Remember the brain damaged child.

I'm being told two things by two different sources. On one hand
Eebster the Great wrote: I don't know what a point cloud is, but I have never seen that term in connection with the Banach-Tarski paradox.
And then this.
The Banach–Tarski paradox is a theorem in set-theoretic geometry, ..... However, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points. The reconstruction can work with as few as five pieces.[1]
While my math skills are suspect my reading comprehension skills are not. Is there a way to read that where the pieces are anywhere near a conventional solid?

Correct, and if by "point cloud" you mean the same thing I meant by my informal term "infinite scattering," then yeah, that makes sense. The pieces are definitely not solids. A better description of the pieces is "non-measurable subsets of R3." However, the balls themselves are solids.

Eebster the Great wrote:Again, can you explain how your proof (which I still do not understand at all) would not apply equally well to anything smaller than the universe? For instance, why can't I use the same logic to prove that a pizza box has no edge?
I just restated something familiar in an unfamiliar way. Edges assume you can see them. If the universe exists the way the current theory suggests, then haven't cosmologists told us the space is expanding and saying that something exists outside it isn't meaningful? Doesn't that translate into, there is no edge because everything exists inside the universe? Because an edge only exists if there is another side?

This has already been answered, but even if cosmology does prove that the universe has no edge (and I certainly expect it has no edge), it does not do so by invoking the Banach-Tarski paradox.

If the points are dimensionless, obviously they have no volume nor do they have an absolute location.

"Dimensionless" in this context means having no extent, not existing in no dimensions. In geometry, a point is dimensionless by definition, but it is still embedded in some higher dimensional space. That's in the same way that the one-dimensional curve defined by y=x2 can be embedded in R2.

To continue, a sphere composed of infinite points is equivilant to a spherical shell composed of an infinite number of points because volume is undefined. On the second highlighted area, you do realize that that the origin is the reason a they talk of shells instead of spheres, right? They actually cite that case in a proof for creating infinite sets of spherical point clouds. And devolve it into a case of a sphere with no origin.

A sphere does have defined volume: its volume is zero. A "spherical shell of finite width" is just the solid of rotation of an annulus and so definitely has volume. The volume of a spherical shell of inner radius r1 and outer radius r2 is V=4/3π(r22-r12), simply the difference of the volumes of the interiors of the outer sphere and the inner sphere. Note that in this sense, you can treat the sphere as a special case of a spherical shell when the inner and outer radii are equal. The reason for the distinction between balls and balls deprived of the origin is that they are topologically distinct, while a ball deprived of the origin is not topologically distinct from a spherical shell (in the usual topology).

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No!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Really!

Postby morriswalters » Fri Aug 11, 2017 2:02 am UTC

Soupspoon wrote:And there aint no FTL (yet!) in the Long Universe, so I don't know where that bit comes from.
My humor is never appreciated. I think I was born without half of it. And my imagination doesn't need the real world. What is Long Mars. Long Mars is The Martian, without space ships. Long Earth is billions of copies for the best place known for man. That's boring because the inevitable fact is that anything close to us looks like us, boring. Well kinda. Mars is novelty.
Eebster the Great wrote:This has already been answered, but even if cosmology does prove that the universe has no edge (and I certainly expect it has no edge), it does not do so by invoking the Banach-Tarski paradox.
Yes, look at the title text of my last few post's. I ceded that point.
Eebster the Great wrote:"Dimensionless" in this context means having no extent, not existing in no dimensions. In geometry, a point is dimensionless by definition, but it is still embedded in some higher dimensional space. That's in the same way that the one-dimensional curve defined by y=x2 can be embedded in R2.
Ok, I read this as, it's defined by the curve that contains it. Yes? No?
Eebster the Great wrote:The reason for the distinction between balls and balls deprived of the origin is that they are topologically distinct, while a ball deprived of the origin is not topologically distinct from a spherical shell (in the usual topology).
Ok. This is why I stated that the pieces had to rotate and translate about the origin. And why each sphere is unique. And why the original proof bounded n at 3 or greater.
Eebster the Great wrote:A sphere does have defined volume: its volume is zero.
Ok. I've used that property any number of times in this discussion by defining the sphere in precisely that way. Several statements I have made depend on that property. Anyway thanks again to everyone who responded.

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Re: No!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Really!

Postby Eebster the Great » Fri Aug 11, 2017 6:40 am UTC

morriswalters wrote:
Eebster the Great wrote:The reason for the distinction between balls and balls deprived of the origin is that they are topologically distinct, while a ball deprived of the origin is not topologically distinct from a spherical shell (in the usual topology).
Ok. This is why I stated that the pieces had to rotate and translate about the origin. And why each sphere is unique. And why the original proof bounded n at 3 or greater.
Eebster the Great wrote:A sphere does have defined volume: its volume is zero.
Ok. I've used that property any number of times in this discussion by defining the sphere in precisely that way. Several statements I have made depend on that property. Anyway thanks again to everyone who responded.

You can say your posts depended on these facts, but that doesn't explain why you directly contradicted them in your last post. For instance, if your proof depends on the fact that spheres have zero volume, why did you say "a sphere composed of infinite points is equivilant to a spherical shell composed of an infinite number of points because volume is undefined"? Either volume is defined or it isn't.

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Re: No!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Really!

Postby Soupspoon » Fri Aug 11, 2017 9:03 am UTC

morriswalters wrote:Yes, look at the title text of my last few post's. I ceded that point.

Hint: Make changes like that to your OP and I might have noticed it. I never saw that it had changed, or even propagated onto my own direct reply. Not sure about anybody else. (And *cough*apostrophe*/cough*, while I'm trying to help...)

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Foolish question debunked. Move on, nothing to see.

Postby morriswalters » Fri Aug 11, 2017 12:28 pm UTC

@Ebster the Great

Specifically to your point, outside of a rigorous definition, undefined and zero are easy to confuse when you're using rhetoric to describe them. Sometimes I miss it. Even when I'm talking about it. And I told you as much when I said I was as rigorous as I was able to be. You, in effect, at this moment, are asking me why I asked a stupid question. I'm telling you that the question was as not stupid as I could make it. If I could ask not stupid questions, I wouldn't ask them here. I would publish. :| trending towards :roll:

People have consistently referred to what I said as a proof, when it could be no such thing. It has no rigor. I suggested that the proof, which is rigorous, says something about reality. I pointed to sets, that I thought belong to the class of sets that the proof represents. In particular EM radiation and photons.

If you consider a edge of the universe as a point where two spheres touch at precisely one point. And if I use the Bertrand Russell relation, I can say that however close you look, I can posit a point which is always smaller than you can see. Inside that point there is nothing to reflect back from, so you can't establish a sight line. Therefore you can't see the outside the universe. I infer this because in discussions elsewhere the paradox is described as a hotel where can always make room for another guest by moving the guests to the room to the right of the room they are in. So I blame Bertrand Russell and the internet. :| trending to :wink:

More seriously, if I can answer the question without resorting to testing my knowledge against someone with greater knowledge, I do so. It's easier and less stressful.


Soupspoon wrote:Hint: Make changes like that to your OP and I might have noticed it. I never saw that it had changed, or even propagated onto my own direct reply. Not sure about anybody else. (And *cough*apostrophe*/cough*, while I'm trying to help...)
Don't assume that I understand forums as well as you. To give you some indication of my nativity, I never considered that I could change the title of my original post and that it would propagate. However, if the occasion ever arises after this, I'll do so, and I have done so to this topic. I'll also simply shut up, which is probably the wiser choice and better solution. I am a victim of my own mouth. No body can know you're ignorant if you don't announce it. One final point. How are FTL, dimensional travel, Holodecks, and Cowboys on ponies, alike?


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