The Point of the Banach Tarski Theorem
The Banach-Tarski theorem challenges common sense by showing a solid ball can be split into pieces to form two identical balls in 3D space. It questions measurement principles and the Axiom of Choice's role in resolving mathematical paradoxes.
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The Banach-Tarski theorem, a perplexing result in measure theory, reveals the limitations of what is possible in mathematical measurements. It demonstrates that certain intuitive expectations we have for measures, such as additivity and invariance under isometries, cannot all be satisfied simultaneously. The theorem states that in three-dimensional space, a solid ball can be divided into pieces that can be rearranged to form two identical solid balls, defying common sense. This impossibility highlights the intricacies of defining measures accurately across different dimensions. The theorem's implications challenge fundamental assumptions about measurement and underscore the importance of concepts like the Axiom of Choice in resolving mathematical paradoxes. By relaxing certain requirements for measures, such as moving from countable additivity to finite additivity, mathematicians can navigate around the limitations posed by the Banach-Tarski theorem in lower dimensions but face insurmountable obstacles in three dimensions. The theorem prompts deeper reflections on the nature of measurement and the role of foundational principles like the Axiom of Choice in mathematical reasoning.
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13 commentsHere's the basic idea ...
In Classical Euclidean Geometry there are five axioms, and while the first four seem clear and obvious, the fifth seems a little contrived. So for centuries people tried to prove that the fifth was unnecessary and could be proven from the other four.
These attempts all failed, and we can show that they must fail, because there are systems that satisfy the first four, but do not satisfy the fifth. Hence the fifth cannot be a consequence of the first four. Such systems are (for obvious reasons) called Non-Euclidean Geometries.
So we can use explicit examples to demonstrate that certain proofs are impossible, and the Banach-Tarski Theorem is a result that proves that a "Measure"[1] cannot have all four obviously desirable characteristics.
That's the basic idea ... if you want more details, click through to the post. It's intended to be readable, but the topic is inherently complex, so it may need more than one read through. If you're interested.
[0] https://news.ycombinator.com/item?id=40797598
[1] Technical term for a function that takes an object and returns a concept of its size. For lines it's length, for planar objects it's area, for 3D objects it's volume, and so on.
In an intuitionist framework none of this applies.
It relies on the fact that not only does it not provide a constructive framework to produce such a division, but also that no such constructive framework is even theoretically possible! So this theorem tells us nothing about the nature of three dimensional objects or our ability to "measure" objects.
Doesn't it require fractal cuts? Seems like a loophole to me. It only seems paradoxical because you assume the resulting pieces are smooth at some scale, like real cuts are.
Matematikrevyen: Matematikrevyen 2011: Banach-Tarski (3m42s) [2011-12-30]: https://www.youtube.com/watch?v=uFvokQUHh08 (music video)
The Point of the Banach-Tarski Theorem - https://news.ycombinator.com/item?id=34482226 - Jan 2023 (105 comments)
The Point of the Banach-Tarski Theorem – not just a curiosity - https://news.ycombinator.com/item?id=9674286 - June 2015 (91 comments)
But I wonder, isn't there an implicit 5th property that could be relaxed? That's the property that the codomain of mu is the reals. Is it viable to use, say, the hyperreals instead, or some other exotic extension that would allow us to name the (nonzero and nonreal!) number mu(V) such that a countable sum of mu(V) comes out to 1?
1. original "terrain" sphere
2. modeled version of the sphere. the virtual "map" sphere.
but because the abstractions are so thick (so to say, pardon the poetic language) — or the recursion so recursive, the "map" of the sphere accounts for it being a map by producing two duplicates virtual copies, one intended to reflect the terrain and the other the map (but both are virtual maps, but this is really hard to 'perceive'/'say' within the formalisms)
Perhaps zero, one, more or all of the following:
- duplication of volume seems to duplicate mass and energy, which is impossible
- the Axiom of Choice is false in the real world
- the real world is not based on real numbers (ironic)
- mathematical measures are not related to real world measure(ment)s
- there are no realizable infinities or infinitesimals
- the real world is ultimately discrete at the lowest level
145 points 1 year ago 105 comments https://news.ycombinator.com/item?id=34482226
19 points 4 hours ago 5 comments https://news.ycombinator.com/item?id=40798224
Further suggest that HTML query params be disallowed in submissions; if URLs with params are relevant they can be added as comment(s).
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