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Banach-Tarski and the Paradox of Infinite Cloning

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Re: Banach-Tarski and the Paradox of Infinite Cloning

#31

Post by tsimionescu »

ithkuil wrote: post #14
tsimionescu wrote: post #6

Earlier quoted context omitted.

Well, infinity is inherently unscientific, as there is no way to scientifically differentiate between an infinite quantity and a really huge quantity (same for infinitesimals), in finite time. What this means is that a universe that contains infinities is, even in theory, entirely indistinguishable (in finite time) from an universe that contains really large/small but finite quantities.

What about negative numbers? Or complex numbers? They are only tools, which can be quite useful to build models of the world with predictive powers but shouldn't be confused for the underlying reality. Even whole numbers are an abstraction that makes sense only when you can clearly define what is the thing you're counting.

Whole numbers can be defined and proven to be necessary to describe the world pretty easily. From there, rational numbers are trivial to define. Negative numbers are somewhat more abstract, but they have very intuitive definitions in many domains, such as accounting. It may be possible to avoid them in a theory of physics, though.

The complex numbers (well, at least those with a rational imaginary part and a rational real part) have been recently proven to be necessary to describe the universe[0] (assuming quantum theory is correct).

The irrational numbers are then are the only numbers that are harder to pin down, and I'm not sure that there is a way to prove that any physical quantity has an irrational value, vs a rational value that is arbitrarily close to that irrational value.

[0] https://arxiv.org/abs/2101.10873

Re: Banach-Tarski and the Paradox of Infinite Cloning

#32

Post by roywiggins »

paulpauper wrote: post #15

I don't understand the paradox. Obviously if you dissaemble or scamble something u can reararange it?

It's like taking a bed apart and rearranging it into two beds, each identical to the original bed, without adding any more material.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#34

Post by danparsonson »

jhgb wrote: post #22
desperate wrote: post #3

To me this is proof that infinity is something only present in our math and not in the universe. Infinity is a nice approximation but it feels like wishful thinking that our universe or anything in it is infinite. Happy to hear disagreements tho.

> infinity is something only present in our math and not in the universe This is true of all mathematical objects. The number 7 doesn't exist in the universe either. It's not a physical object.

OP didn't argue that finite numbers are physical objects, they said that infinities are not present in the universe. For example, I could in theory hand you 7 electrons but there are not infinity electrons for me to hand to you.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#36

Post by BeetleB »

Sinidir wrote: post #27

Can someone correct me if im wrong? What i see here is a splitting of the set of points in the sphere? However the set of points in the sphere is not really the sphere. A point has no volume so no matter how many you add together you don't get something with a volume. This seems more akin to splitting the natural numbers into odd and even numbers which are all equally large. The language that i see in this article an…

> A point has no volume so no matter how many you add together you don't get something with a volume.

You're on the right track.

The Banach-Tarski paradox requires accepting that non-measurable sets[1] exist. A non-measurable set is a set with a an inspecifiable volume. Note: That's non-measurable - not 0. It means you have a quantity of something, whose volume is not 0, but it's also not any other number.

Once I realized that the paradox requires it, all the WTF aspect went away. Of course - if you can accept quantities for which you cannot specify a volume, you can probably accept about anything.

[1] https://en.wikipedia.org/wiki/Non-measurable_set

Re: Banach-Tarski and the Paradox of Infinite Cloning

#37

Post by snet0 »

bobthechef wrote: post #19
robot_no_419 wrote: post #8

Earlier quoted context omitted.

Mathematics doesn't really "exist" in the first place. It's more of a language that's rich enough and with enough logic to describe/approximate the laws of physics that actually do exist.

So quantity, structure, formal necessity don't exist? Mathematics has nothing to do with laws of physics. Even if the laws of physics[0] were different, these mathematical[1] truths would remain the same. [0] Laws of physics don't actually exist. They're shorthand generalizations about features of particulars. The notion of some kind of abstract disembodied "laws" that somehow "govern" everything is absurd. [1] For c…

Laws in the physics sense don't govern, though, they describe. It's akin to saying moral laws don't decide what's immoral, they simply describe it.

Given that, "laws of physics" are certainly describable. We simply write the formulae that tell us what the next state of the dynamical system is. They are ways of delimiting what is physically possible, given the current state of the art.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#38

Post by ouid »

Sinidir wrote: post #27

Can someone correct me if im wrong? What i see here is a splitting of the set of points in the sphere? However the set of points in the sphere is not really the sphere. A point has no volume so no matter how many you add together you don't get something with a volume. This seems more akin to splitting the natural numbers into odd and even numbers which are all equally large. The language that i see in this article an…

"no matter how many you add together", is where this argument breaks down in ZFC. The sphere is indeed the union of all of the singletons consisting of its points, all of which are measure zero. Banach-Tarski is mainly considered "weird" because it describes a partition into so few pieces, and they are rearranged via rigid motions only. It is trivial to come up with bijections between compact finite dimensional manifolds, (https://en.wikipedia.org/wiki/Space-filling_curve). For another example of the axiom of choice wreaking havoc on the notion of measure, see https://en.wikipedia.org/wiki/Vitali_set .

Re: Banach-Tarski and the Paradox of Infinite Cloning

#39

Post by thaumasiotes »

Sinidir wrote: post #27

Can someone correct me if im wrong? What i see here is a splitting of the set of points in the sphere? However the set of points in the sphere is not really the sphere. A point has no volume so no matter how many you add together you don't get something with a volume. This seems more akin to splitting the natural numbers into odd and even numbers which are all equally large. The language that i see in this article an…

> The language that i see in this article and elsewhere however is suggesting that we actually duplicated the sphere (doubled the volume).

> This seems incorrect.

It isn't incorrect. You're right that the number of points in the sphere does not equate to the volume of the sphere. But the Banach-Tarski theorem does in fact let you double the volume. It is considered to be of interest because it does the following:

1. You have a ball.

2. You cut the ball into 5 pieces in a very clever way.

3. You move the pieces around.

4. Now you have two balls, each the same size as the first.

The key, interesting part of this is in step 3, where we only use translations and rotations. Those preserve volume. (By contrast, it's easy to scale a ball of radius 2 to become a ball of radius 3, but that's not a volume-preserving transformation.) The part of the process that doesn't preserve volume is actually step 2, where we cut the ball into pieces. People find it unintuitive that this step doesn't preserve volume.

You can also cut your ball into several pieces and move the pieces around such that you end up with a much larger ball.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#40

Post by paulpauper »

roywiggins wrote: post #32
paulpauper wrote: post #15

I don't understand the paradox. Obviously if you dissaemble or scamble something u can reararange it?

It's like taking a bed apart and rearranging it into two beds, each identical to the original bed, without adding any more material.

Oh that is pretty cool. I wonder if it could be proved with differential geometry for the sphere, which has a simple paramertization
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