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Mathematics is Inconsistent!


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We have the Banach-Tarski paradox that says: 1 sphere = 2 spheres or 1 = 2. Since it can also be proven that 1 ~= 2, we have an inconsistency.

Actually it says a sphere is equidecomposable into 2 spheres. It is then follows that: 1 = 2.

To exclude this we have to make the following operation invalid: fill a hole in a line by shifting the line up to infinity.

Edited by Willem F Esterhuyse
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1 hour ago, Willem F Esterhuyse said:

We have the Banach-Tarski paradox that says: 1 sphere = 2 spheres or 1 = 2. Since it can also be proven that 1 ~= 2, we have an inconsistency.

Actually it says a sphere is equidecomposable into 2 spheres. It is then follows that: 1 = 2.

To exclude this we have to make the following operation invalid: fill a hole in a line by shifting the line up to infinity.

What was the question ?

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2 hours ago, Willem F Esterhuyse said:

It is then follows that: 1 = 2.

No, it does not follow. The paradox does not say 1 ball = 2 balls any more than chopping a ball in half to give you 2 pieces says that 1 = 2.

 

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4 hours ago, Willem F Esterhuyse said:

We have the Banach-Tarski paradox that says: 1 sphere = 2 spheres

No, it doesn’t. 

What it does say is that you can decompose a Euclidean 3-volume into a finite number of subsets, each of which is itself a non-measurable collection of infinitely many points, and then reassemble these subsets in a new way. The crucial point here is that you cannot uniquely and self-consistently define the notion of ‘spatial volume’ for a non-measurable infinite collection of individual points, so this decomposition does not preserve the original volume, contrary to naive intuition. It’s a subtle ‘trick’ of sorts to do with Lebesgue and Banach measures.

IOW, the Banach-Tarski paradox breaks down and reassembles a 3-volume in a way that does not itself preserve the original volume. Thus it is hardly surprising that you can turn a ball into two balls in this manner - in fact you could turn a ball into anything at all in this manner, no matter how big or small. It isn’t a true paradox, and most certainly not an inconsistency in mathematics.

Also, don’t forget that unfortunately we do not really live in an infinitely sub-divisible 3-dimensional Euclidean world where such a procedure could in fact be implemented - it would be a neat little trick with lots of interesting applications!

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4 hours ago, Markus Hanke said:

No, it doesn’t. 

What it does say is that you can decompose a Euclidean 3-volume into a finite number of subsets, each of which is itself a non-measurable collection of infinitely many points, and then reassemble these subsets in a new way. The crucial point here is that you cannot uniquely and self-consistently define the notion of ‘spatial volume’ for a non-measurable infinite collection of individual points, so this decomposition does not preserve the original volume, contrary to naive intuition. It’s a subtle ‘trick’ of sorts to do with Lebesgue and Banach measures.

IOW, the Banach-Tarski paradox breaks down and reassembles a 3-volume in a way that does not itself preserve the original volume. Thus it is hardly surprising that you can turn a ball into two balls in this manner - in fact you could turn a ball into anything at all in this manner, no matter how big or small. It isn’t a true paradox, and most certainly not an inconsistency in mathematics.

Also, don’t forget that unfortunately we do not really live in an infinitely sub-divisible 3-dimensional Euclidean world where such a procedure could in fact be implemented - it would be a neat little trick with lots of interesting applications!

Yes +1

I would just lik to add to this part "in fact you could turn a ball into anything at all in this manner, no matter how big or small. " the following in response to the OP's assertion.

8 hours ago, Willem F Esterhuyse said:

To exclude this we have to make the following operation invalid: fill a hole in a line by shifting the line up to infinity.

You are right the the proposition works the other way round as well, but adding to Markus comment that it anything at all does not have to have the same shape as a ball and could have 'holes' in it. So no you don't have to exclude turning a pincushion into a pumpkin.
It is true however that doing this topologically (ie continuously) means you can't make a doughnut into an apple.

 

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25 minutes ago, mathematic said:

The paradox is in pure mathematics.  A physical object cannot be decomposed into non-measurable pieces.

That’s the salient point. The theorem only applies to the mathematical construct and does not apply to a physical object. It’s math, but not physics.

Which is why it seems paradoxical - it’s contrary to experience and expectations. (Neo can do it in the Matrix, but not in the real world. There is no spoon)

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21 hours ago, Markus Hanke said:

in a way that does not itself preserve the original volume

The paradox specifies a sphere not a solid ball: it has no volume.

23 hours ago, swansont said:

chopping a ball in half to give you 2 pieces says that 1 = 2.

This comparison is invalid, irrelevant.

1 sphere equidecomposable into 2 spheres                                                   Premise

1 object = 1 object + 1 object                                                                           Abstraction

1 = 1 + 1                                                                                                              Existential Specification

1 = 2                                                                                                                    Sum

Edited by Willem F Esterhuyse
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1 hour ago, Willem F Esterhuyse said:

The paradox specifies a sphere not a solid ball: it has no volume.

I think Markus meant "measure" in general. There are many cute puzzles like this that are similar and involve other kinds of measures, like Hilbert's curve covering a patch of plane.

On the other hand, mathematics based on natural numbers is known to be incomplete, and no theory of this kind can prove its own consistency, so I would relax about the whole thing.

Besides, there is no unified set of axioms for all of mathematics, as far as I know. You can relate chunks of it, but not all.

I would relax even more.

I may be wrong, but I think in this post-Bourbaki era mathematicians tend to be more freely constructive and more deeply involved in guesswork. This has proved to be very fruitful for both physics and mathematics.

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1 hour ago, Willem F Esterhuyse said:

The paradox specifies a sphere not a solid ball: it has no volume.

It says a sphere, though - a mathematical structure - and not a ball, i.e. a physical object.

And it specifies a solid 

“Given any two bounded subsets A and B of a Euclidean space in at least three dimensions, both of which have a nonempty interior…”

https://en.wikipedia.org/wiki/Banach–Tarski_paradox

1 hour ago, Willem F Esterhuyse said:

This comparison is invalid, irrelevant.

Because you say so? Because it’s inconvenient?

1 hour ago, Willem F Esterhuyse said:

1 sphere equidecomposable into 2 spheres                                                   Premise

1 object = 1 object + 1 object                                                                           Abstraction

1 = 1 + 1                                                                                                              Existential Specification

1 = 2                                                                                                                    Sum

 

I object cut in half to become 2 objects

1 object = 1 object + 1 object                                                                           

1 = 1 + 1                                                                                                              

1 = 2                                   

I agree it’s invalid, but it’s the same argument

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32 minutes ago, swansont said:

And it specifies a solid 

It doesn't need to be a solid; the "solid sphere" version of the theorem is proven based on the "hollow sphere" version of it, as shown in the Wiki page:

3. Use the paradoxical decomposition of that group and the axiom of choice to produce a paradoxical decomposition of the hollow unit sphere.

4. Extend this decomposition of the sphere to a decomposition of the solid unit ball.

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10 minutes ago, Willem F Esterhuyse said:

"cut in half" does not have an "equi-" prefix, therefore not the same argument.

Does 1 + 1 = 2 require the items being counted to be identical? If I have a nickel and a dime, do I not have 2 coins? Even though they are not identical?

And we can call cutting in half being equidivided.

Is it really your objection that you can create identical spheres, rather than creating 2 of them?

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4 hours ago, Willem F Esterhuyse said:

The paradox specifies a sphere not a solid ball: it has no volume.

The original Banach-Tarski “paradox” explicitly concerns a solid ball, not a sphere, in Euclidean 3-space (see first sentence):

https://en.wikipedia.org/wiki/Banach–Tarski_paradox

Not that this really makes any difference, because any 2D surface element of a sphere can likewise be considered a non-measurable ensemble of points, in which case the surface area of that subset wouldn’t be self-consistently defined either. So you could - using a similar process as in the original “paradox” - deconstruct a sphere and re-construct it in some other way, without the original surface area being conserved. 

There still isn’t an inconsistency, because this has to do with measures.

 

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On 11/24/2022 at 3:40 AM, Willem F Esterhuyse said:

We have the Banach-Tarski paradox that says: 1 sphere = 2 spheres or 1 = 2. Since it can also be proven that 1 ~= 2, we have an inconsistency.

Actually it says a sphere is equidecomposable into 2 spheres. It is then follows that: 1 = 2.

To exclude this we have to make the following operation invalid: fill a hole in a line by shifting the line up to infinity.

Using measure theory with fractals one can assert the claims being made here. Therefore, I believe that we are over simplifying Willems claim. However I am stumped on the level of complexity needed to understand this further. Can you please explain further @Willem F Esterhuyse

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17 minutes ago, Willem F Esterhuyse said:

I don't know much about measures and fractals. The sphere is constructed by labelling each point on the sphere and doing a shift operation to the points . 

https://en.wikipedia.org/wiki/Measure_(mathematics)

https://en.wikipedia.org/wiki/Fractal

Hausdorff measure (fractals):

https://en.wikipedia.org/wiki/Hausdorff_measure

I hope that helps.

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On 11/25/2022 at 6:33 AM, ALine said:

Using measure theory with fractals one can assert the claims being made here.

Measure theory and fractals are not involved in the proof of Banach-Tarski. It's true that the pieces of the decomposition are nonmeasurable, but the existence of nonmeasurable sets is not difficult to prove without invoking any measure theory beyond the definition of a measure. At the very least, one need not study or look up measure theory in order to follow the proof of Banach-Tarski. Fractals don't enter into it at all AFAIK.

The Vsauce video on Banach-Tarski is very good, as is the Wikipedia page on the subject. In particular, the proof outline on Wikipedia is straightforward, and not all that difficult to follow, if one is willing to patiently work through it. The proof has a lot of moving parts, but each part is relatively simple.

The essence of the proof is that there's a paradoxical decomposition of the free group on two letters. This does not involve the axiom of choice or nonmeasurable sets. I can't find a clean, elementary discussion of this online and it's a bit lengthy but perhaps I can write something about it later. There's a brief description here. https://en.wikipedia.org/wiki/Paradoxical_set

The bottom line is that Banach-Tarski is a theorem that does not contradict any other mathematics.

Vsauce vid: https://www.youtube.com/watch?v=s86-Z-CbaHA

https://en.wikipedia.org/wiki/Free_group

ps -- I see that Step 1 of the proof sketch on Wikipedia has a pretty decent explanation of the paradoxical decomposition of the free group on two letters. 

https://en.wikipedia.org/wiki/Banach–Tarski_paradox#Step_1

Once that's established, the next step is to show that the group of rigid motions of Euclidean 3-space contains a copy of the free group on two letters, and therefore has a paradoxical decomposition. After that, it's just a technical matter of applying this decomposition to the unit ball in 3-space.

Edited by wtf
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22 hours ago, Willem F Esterhuyse said:

I don't know much about measures and fractals. The sphere is constructed by labelling each point on the sphere and doing a shift operation to the points . 

I can't possibly imagine why this post would have offended anyone, so I am adding a counterbalancing+1.

 

Willem has actually told us about one of his limitations and, i believe, asked for help.

 

I am not exactly sure what is meant by a shift operation, but I can guess. That would certainly be one way of putting.

Not my personal favourite however because I am used to the shift operator being something from numerical analysis, not something defined by a programmer.

But if my guess is correct then W is right.

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6 hours ago, studiot said:

I am not exactly sure what is meant by a shift operation

There's a point in the proof of the paradoxical decomposition of the free group on two letters where you have a decomposition into five pieces, but one of the pieces is a single point that appears in both copies of the decomposition. You have to get rid of it by "Hilbert shifting" as in Hilbert's hotel. You move every point "up one room" to remove the point from one of the copies without losing any points. That's what I thought OP meant, but I don't know if that's actually what he had in mind. 

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On 11/26/2022 at 3:54 PM, wtf said:

Measure theory and fractals are not involved in the proof of Banach-Tarski

Never made this claim. Using it as a potential method for analysis.

 

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