Author Topic: The Banach傍arski Paradox  (Read 362 times)

Offline Vetus Ordo

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The Banach傍arski Paradox
« on: March 26, 2019, 02:39:10 PM »
The Banach傍arski Paradox

The Banach傍arski 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.

A stronger form of the theorem implies that given any two "reasonable" solid objects (such as a small ball and a huge ball), the cut pieces of either one can be reassembled into the other. This is often stated informally as "a pea can be chopped up and reassembled into the Sun" and called the "pea and the Sun paradox".

The reason the Banach傍arski theorem is called a paradox is that it contradicts basic geometric intuition. "Doubling the ball" by dividing it into parts and moving them around by rotations and translations, without any stretching, bending, or adding new points, seems to be impossible, since all these operations ought, intuitively speaking, to preserve the volume. The intuition that such operations preserve volumes is not mathematically absurd and it is even included in the formal definition of volumes. However, this is not applicable here because in this case it is impossible to define the volumes of the considered subsets. Reassembling them reproduces a volume, which happens to be different from the volume at the start.

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 non-measurable 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.

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.


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Offline cgraye

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Re: The Banach傍arski Paradox
« Reply #1 on: March 26, 2019, 08:56:40 PM »
Yes, this is an interesting piece of mathematics.  Do you have any thoughts on it?
 
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Offline Vetus Ordo

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Re: The Banach傍arski Paradox
« Reply #2 on: March 26, 2019, 09:28:15 PM »
Yes, this is an interesting piece of mathematics.  Do you have any thoughts on it?

It's interesting, hence why I shared the video.

In any case, we're talking about infinite sets so I'm not sure how applicable this can be to reality.
DISPOSE OUR DAYS IN THY PEACE, AND COMMAND US TO BE DELIVERED FROM ETERNAL DAMNATION, AND TO BE NUMBERED IN THE FLOCK OF THINE ELECT.
 

Offline Non Nobis

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Re: The Banach傍arski Paradox
« Reply #3 on: March 27, 2019, 12:14:05 AM »
I have a brother who's fascinated with mathematics, and he'll love this (maybe he's already seen it).  I will forward it to others too, just in case.
[Matthew 8:26]  And Jesus saith to them: Why are you fearful, O ye of little faith? Then rising up he commanded the winds, and the sea, and there came a great calm.

[Job  38:1-5]  Then the Lord answered Job out of a whirlwind, and said: [2] Who is this that wrappeth up sentences in unskillful words? [3] Gird up thy loins like a man: I will ask thee, and answer thou me. [4] Where wast thou when I laid up the foundations of the earth? tell me if thou hast understanding. [5] Who hath laid the measures thereof, if thou knowest? or who hath stretched the line upon it?
 

Offline Non Nobis

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Re: The Banach傍arski Paradox
« Reply #4 on: March 27, 2019, 06:38:52 PM »
I have a brother who's fascinated with mathematics, and he'll love this (maybe he's already seen it).  I will forward it to others too, just in case.

My brother had this to say:

Quote
It has something to do with the drawbacks of Euclidean three-dimensional space as the standard scheme for analyzing spatial objects. And it has a connection with the Axiom of Choice, another thing I don稚 quite understand. (Notice the play between
田hoice of axioms and 殿xiom of choice).  It is  part of set theory. It is the C in ZFC, the standard axiomatic foundation of Set Theory, in other words, the standard axiomatic foundation of standard mathematics. It shows the weakness of that foundation in some sense, showing that standard mathematics is not simply synonymous with mathematics.

 

Gdel showed that you cannot have an axiomatic system that will make all of mathematics roll out (there will always be unprovable true propositions in your formal system; Tarski showed that there is an axiomatic system that will make all of Euclidean geometry roll out.

I have a simple prejudice in favor of Euclidean geometry as being the most sane, but I sure don't understand all these things.
[Matthew 8:26]  And Jesus saith to them: Why are you fearful, O ye of little faith? Then rising up he commanded the winds, and the sea, and there came a great calm.

[Job  38:1-5]  Then the Lord answered Job out of a whirlwind, and said: [2] Who is this that wrappeth up sentences in unskillful words? [3] Gird up thy loins like a man: I will ask thee, and answer thou me. [4] Where wast thou when I laid up the foundations of the earth? tell me if thou hast understanding. [5] Who hath laid the measures thereof, if thou knowest? or who hath stretched the line upon it?
 

Offline Davis Blank - EG

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Re: The Banach傍arski Paradox
« Reply #5 on: March 27, 2019, 07:59:59 PM »
This seems like another form of Zeno's paradoxes and never being able to reach a wall that is a finite distance away because it has an infinite number of spaces between yourself and it.

I think that these paradoxes exist shows that we have a colossal misunderstanding of logic.  Or of reality.  Or perhaps the innate inability to properly understand it at all.
 

Offline Kreuzritter

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Re: The Banach傍arski Paradox
« Reply #6 on: March 28, 2019, 11:49:52 AM »
What a formalistic construct of the human mind has to do with reality is anyone痴 guess. Higher mathematics is ingenious and a great exercise in following through some set of formal definitions of concepts (aka 殿xioms) to their formalistic logical implications, but what else? Sure, sometimes these have practical applications when we can express the formalism through physical phenomena instead of abstractly through symbols, but even then, it痴 not reality and it痴 not some secret code of reality, but a decent approximation of some narrow aspect of it. What do mathematical infinities have to do with a real infinity? What does something like the real number line have to do with a real extended physical space? Not much, besides being attempts to represent phenomenal realities through conceptual schemes.
 
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