April 1st, 2017, 07:20 PM  #11  
Senior Member Joined: Aug 2012 Posts: 2,424 Thanks: 759  Quote:
Quote:
So ok here is the deal. We all agree that the length of the line segment from 2 to 5 is 3; and that the area of a rectangle with length 2 and width 6 is 12; and that the volume of a rectangular parellelopiped (aka shoebox) with length l, width w, and height h is lwh. Using calculus we can determine the length/area/volume of many other shapes. Archimedes determined the area of a circle 2000 years ago using methods that would be perfectly recognizable to a calculus student today. Now mathematicians began to wonder: Is there a way to assign a number, called a measure, to any set of real numbers (or subsets of the plane, or 3space, etc.) whatsoever? It turns out that the answer is no, because of the fact that arbitrary sets of real numbers can be very weird. Some sets of reals are so strange that there is no sensible way to assign them a size at all in a way that's compatible with lengths of intervals. Here is a slightly more technical discussion. If you've ever run across the standard axioms of probability this will be familiar. What do we mean by a measure? Given a set $X \subset \mathbb R$ we want to assign a number $\mu(X)$, called the "measure of $X$", that has the following properties: * For any set $X \subset \mathbb R$, we have $\mu(X) \geq 0$. Certainly we don't want to allow negative lengths or areas. * If $X$ and $Y$ are disjoint sets of reals (meaning they have no elements in common) then $\mu(X \cup Y) = \mu(X) + \mu(Y)$. In oher words if the measure of the interval $[2,5]$ is $3$, and the measure of the interval $[10,20]$ is $10$, then the measure of the union of those sets is $13$. Perfectly sensible. * We want to go a little farther with the disjoint union idea. Suppose we have a countably infinite collection of intervals like $[0,\frac{1}{2}), ~~[\frac{1}{2}, \frac{3}{4}), ~~[\frac{3}{4}, \frac{7}{8})$, etc. The union of these sets is the interval $[0, 1)$, which we know has measure (or length) $1$. The missing point at the end doesn't make any difference when we are determining the length of an interval. So we want our measure to obey the rule of countably additivity: If $I_n : n = 1, 2, 3, \dots$ is a countably infinite collection of pairwise disjoint sets, then $\mu(\cup_{n \in \mathbb N}) = \sum_{n \in \mathbb N} \mu(I_n)$. "The measure of a countable disjoint union is the sum of the measures." * Now if we have some set $X$ and it has some measure, we want its measure to stay the same if we slide $X$ along the real line by a constant. In other words for any real number $r$, we have $\mu(X) = \mu\{x + r : x \in X\}$. This requirement is called uniformity or translation invariance. * Finally, we insist that the measure of the unit interval is $1$. Now it turns out that there is no function defined on all subsets of reals that satisfies all of these requireents! If we want to define a measure on subsets of reals, we either have to throw out one or more of these requirements, or else give up on the hope that we can satisfy them for all possible subsets of reals. So when we generalize the idea of the length of an interval to a measure satisfying all these requirements, some sets are measurable and some simply are not. That's a nonmeasurable set. In 1905 an Italian mathematician named Giuseppe Vitali constructed a specific example of a set of real numbers that can not be assigned a measure. His example is now called the Vitali set. The world "constructed" is used somewhat loosely. The construction depends on the Axiom of Choice and the set can't be explicitly visualized. But the way the Vitali set is defined is perfectly clear according to the rules of set theory. The uniformity property deserves special mention. It says that rigid motion preserves measure. That's why the BanachTarski paradox is so surprising. We can cut a sphere into five pieces and move the pieces around using rigid motions, and end up changing the volume of the sphere. Therefore it must be the case that at least one of the pieces is nonmeasurable. See also https://en.wikipedia.org/wiki/Measure_(mathematics) Last edited by Maschke; April 1st, 2017 at 07:24 PM.  
April 1st, 2017, 07:32 PM  #12 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 472 Thanks: 29 Math Focus: Number theory 
Is this leading to volume being paradoxical somehow? Go ahead, if you wish, y'all. I have a visitor. Will catch up tomorrow? The Planck length and other Planck units rely solely on the fundamental, radiative constants G, h and c (sometimes kappa, e, etc.). They have as much to do with the observer than the observed. Radiative constants are often set to zero for simplicity of calculating. 
April 1st, 2017, 07:42 PM  #13  
Senior Member Joined: Aug 2012 Posts: 2,424 Thanks: 759  Yes, The BT paradox has a lot of different parts, each one understandable on its own. If you want to see the outline of the proof, please read the Wiki page you linked. It's a very good exposition. There are several different parts that each need to be explained and then put together. You did ask what's a nonmeasurable set. The BT paradox depends on certain sets being nonmeasurable. Quote:
The BT paradox applies to ZFC set theory. It has nothing to do with the real world. There is no evidence at all that set theory has anything to do with the real world. Nothing at all to do with physics or the Planck length. This is purely a mathematical theorem.  
April 1st, 2017, 08:00 PM  #14  
Senior Member Joined: May 2015 From: Arlington, VA Posts: 472 Thanks: 29 Math Focus: Number theory 
Maschke, Quote:
I will now step aside to try the wiki website I started with, and your next to last post. (I liked zylo's paradox of the sum of infinite sets, re. the limitation of volume.) Is there any observation on a paradox of a paradox being nonparadoxical?  
April 1st, 2017, 08:11 PM  #15  
Senior Member Joined: Aug 2012 Posts: 2,424 Thanks: 759  Quote:
Quote:
Likewise my post. All you need to know is that there is no way to assign a sensible measure to every set of real numbers or every set of points in three space. Yes. The word paradox means several different things. In this case it's being used to mean "counterintuitive." But there's no actual paradox. I think someone already made that point. It's just a counterintuitive theorem that depends on the axiom of choice, which is the heart of many counterintuitive theorems. Note that if we banish the axiom of choice and instead assume its negation, we also get many counterintuitive results, such as a vector space that has no basis. Weirdness with Choice, weirdness without it. And that's weird!  
April 1st, 2017, 09:43 PM  #16  
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126  Quote:
If it's not a real ball it is not a paradox. Just a contradiction in terms due to imprecise or contradictory definitions. You can imagine and abstract innumerable undefined mathematical things and layers of definitions and arrive somewhere as a result of rules. My mind doesn't work that way, I'm no good at it, and have no interest in it, other than possibly a game of wits, which is only fun if everybody is working with the same transparent rules and things. Like being challenged to a game of chess without being told what the rules or pieces are. Or giving the explanation of the "paradox" as ZFC plus some definitions. I'm afraid that any rational questions like how do you divide the ball up would just lead to more undefined jargon, frustration, and annoyance. That said, my apologies for intruding on your discussion. PS "the Planck length, denoted ℓP, is a unit of length, equal to 1.61622938×10−35 metres." wiki It is not the absolute measure of length, oddly enough, now based on the speed of light, 300,000,000m/sec (involving time?). But obviously my comment about absolute length is totally irrelevant to this discussion.  
April 1st, 2017, 10:57 PM  #17 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126 
"The easiest decomposition ``paradox" was observed first by Hausdorff: The unit interval can be cut up into countably many pieces which, by translation only, can be reassembled into the interval of length 2. This result is, nowadays, trivial, and is the standard example of a nonmeasurable set, taught in a beginning graduate class on measure theory." From https://cs.uwaterloo.ca/~alopezo/mathfaq/node70.html How? 
April 1st, 2017, 11:39 PM  #18  
Senior Member Joined: Aug 2012 Posts: 2,424 Thanks: 759  Quote:
 
April 2nd, 2017, 11:18 AM  #19 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 472 Thanks: 29 Math Focus: Number theory 
I'd like to see your "undergraduate" version, Maschke. As you indicate, I may grasp at least part of it. At least those who understand formal set theory will appreciate your intermediate version. My first perusal of the Wiki page left many blanks. I seem to realize Cantor better. 
April 2nd, 2017, 02:28 PM  #20  
Senior Member Joined: Aug 2012 Posts: 2,424 Thanks: 759  Quote:
I'd enjoy walking through the Wiki page and trying to answer questions about it and raise some of my own as we go. A process like that would last as along as people have interest. I haven't got five minutes worth on this though. I do have five seconds worth.: 5 second explanation: One or more of the pieces must be nonmeasurable. Nonmeasurable sets violate our intuition that rigid motions preserve measure. Now as I say I haven't got a five minute explanation. To go any deeper, you'll have to ask specific questions and I'll do my best to answer them. I'm no expert, but I do understand most of the major parts of the proof. I've never been through it in detail and this would be one of my motivations in working through the Wiki page. But it's a process. I haven't got one short killer post that will be any better than the 5 second explanation. That said, if you can work hard on the free group on 2 letters, along with the incredibly insightful picture, once you understand it, then we can begin to get an intuitive handle on geometric paradoxes. I don't think there's an easy way through this proof for either of us and anyone else who's interested in the subject. I think that Wiki page proof is pretty good (the proof sketch, not so much the rest of the page). If someone wants to understand the proof of BT, Wiki's proof sketch is the place to start. Quote:
You don't need to know much to work through this proof. The trick is that it's not an argument like Cantor's diagonal argument, which can be presented in a few minutes. The BT proof is one little thing after another, but a lot of different things. Each one is perfectly understandable in isolation. You just have to be prepared to spend some time on it. I've been chipping away at this proof for years. I think we could walk through the Wiki proof in a few weeks at a leisurely pace. I'd certainly appreciate the motivation to go through this myself. So please ask questions, and meanwhile let's all try to grok $F_2$, because that's the heart of the matter. It's the Library of Babel with an alphabet of 2 letters that can be written or erased*. If you ever saw any group theory, it's the group you get by taking all expressions of two variables and their inverses without any other restrictions. That's where to start. (*)  You can erase a letter before you write it! So $b^{1}a$ is a word that starts with erasing $b$ then writing $a$. Then if you had the word $b$ and you concatenated $b^{1}a$ onto it, you'd end up with $b(b^{1}a) = (bb^{1})a= a$. I swear, if you can understand that you can understand the proof of the BanachTarski paradox. Last edited by Maschke; April 2nd, 2017 at 02:41 PM.  

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