B Banach-Tarski

Hi! Does anyone think Banach-Tarski's paradox needs reworking? I first came across it in a video by Vsauce. I've been told that things might be reworked as to avoid the paradox, just as set theory was fixed so as to avoid Russell's paradox. How to make sense of a smallest unit of space is what it's about for me. If you take the smallest length and make it the two shorter sizes of a right triangle, than half the hypotenuse would be the true "smallest", and the process goes on unto infinity. The ultimate mathematical magic trick would if someone used the Banach-Tarski paradox to mathematically get a bigger object out of one with smaller measure.
 

fresh_42

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Hi! Does anyone think Banach-Tarski's paradox needs reworking?
No.
The ultimate mathematical magic trick would if someone used the Banach-Tarski paradox to mathematically
get a bigger object out of one with smaller measure.
You don't need Banach-Tarski for that. It can always easily be done in topology. "bigger" requires a volume, topology does not.

I once listened to a lecture about it where the professor emphasized his view, that our understanding of the concept "point" is to blame here rather than AC. We have similar paradoxa, e.g. the Hilbert curve which also indicate that topological dimensions don't quite match intuition. However, this does not need fixing, or as another professor of mine used to say: The real world is discrete. And despite of it, we calculate well with continuous or even smooth functions and our bridges and buildings don't collapse.
 
My understanding of B/T was that it worked through the uncountable infinity of points within the original object. This is reconcilable with the object being discrete?
 

fresh_42

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My understanding of B/T was that it worked through the uncountable infinity of points within the original object. This is reconcilable with the object being discrete?
Yes, and that's why the dimensions are crucial, in this case zero dimensional points. A real life object would have finitely many molecules, a mathematical ball has uncountably many points. This is why it would never work in reality. That's what I meant: we calculate with a continuum but we can only ever align finitely many things.

Mathematical concepts and real life objects are two entirely different things. We cannot even "produce" a circle! Not if we allow an electron microscope to investigate them. Therefore there are some mathematical concepts which are counterintuitive. A point is one of them.
 

mathman

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B/T is based upon dividing a measurable set into a finite number of non-measurable sets and then putting them together differently.
 

S.G. Janssens

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I've been told that things might be reworked as to avoid the paradox, just as set theory was fixed so as to avoid Russell's paradox.
You are right that they might be "reworked", but by "reworking" (= avoiding the existence of non-measurable sets) you may lose more than you will gain. Perhaps most notably, it is possible to prove that you will have to give up on the Hahn-Banach extension theorem. That theorem is central in analysis.

So, you could indeed lay the foundations such that you avoid the paradox (which is - by nature of every paradox - only apparent anyway), but you will be short on bricks to build the rest of the house.
 

WWGD

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A necessary assumption is that you have infinitely-many atoms to do these many partitions. In actuality, the number of atoms in the universe is bounded/finite.
 
Is a non-measurable part the same as a "simple substance", to use an old scholastic way to say it? This solution seemed to me to be saying that 0 and 0 can equal 1 (or a size). It seems the more I look, the more counter-intuitive is truth
 

fresh_42

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Is a non-measurable part the same as a "simple substance", to use an old scholastic way to say it? This solution seemed to me to be saying that 0 and 0 can equal 1 (or a size). It seems the more I look, the more counter-intuitive is truth
No, ##0+0\neq 1##. But uncountably many ##0## can add up to ##1##: ##[0,1] = \cup \{\,\{\,\frac{r}{s}\}\,|\,r,s\in \mathbb{Z}, r\leq s\,\}##. And again, it is the concept of a point which is counterintuitive, since they can add up to a line although they are dimensionless.
 
Finding where exactly something becomes that zero tangles me into knots, probably because I struggle with infinity. Aristotle said something can potentially be divided into an infinity, but not actually. I think that the object has the same measure whether divided or not. Math then seems to be getting something from infinite nothingness, or having something both finite and infinite in cardinality. The circumference of a circle is a factor of endless pie, as pointed out in the Vsauce video
 

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