• The Banach–Tarski Paradox - Vsauce
    53 replies, posted
[media]https://www.youtube.com/watch?v=s86-Z-CbaHA[/media]
Damn what program was he using for those visualizations, they were great. I'm surprised he didn't mention the fact that the cardinality ("size") of the rational numbers (fractions) is equal to the cardinality of the natural numbers.
by far the most confusing and interesting and longest vsauce yet. god damn
[QUOTE=Panthereye;48344749]by far the most confusing and interesting and longest vsauce yet. god damn[/QUOTE] And probably my favorite now.
[QUOTE=Panthereye;48344749]by far the most confusing and interesting and longest vsauce yet. god damn[/QUOTE] it reminded me of a documentary i watched recently (it has more about cantor in it) you might like this [media]https://www.youtube.com/watch?v=iQlg86lsy-o[/media]
This is the first VSauce I had an issue with because I straight up didn't understand it very well. I'm not good a math so I was lost from beginning to end.
I was fine up till he started building the second sphere. The fact it's 3:50AM and I got up at 10:00AM on 5 hours of sleep probably isn't helping with that, though. :v:
He draws his 8's weird.
The obvious problem is that real spheres don't have an infinite amount of points, they are made of a finite number of molecules
[QUOTE=JakeIsWin;48346801]He draws his 8's weird.[/QUOTE] this bugged me the whole video who draws 8s as two separate circles? seriously
i really feel like vsauce has sort of become the spiritual successor to bill nye for people who grew up with bill nye
[QUOTE=Instant Mix;48347062]this bugged me the whole video who draws 8s as two separate circles? seriously[/QUOTE] I've seen people draw a small "X" then draw 2 small arcs on the top and bottom of it to make the "8" Don't know if it's worse than 2 circles.
[QUOTE=Laserbeams;48346834]The obvious problem is that real spheres don't have an infinite amount of points, they are made of a finite number of molecules[/QUOTE] In that case, "real spheres" don't exist at all. They don't fit the definition. But you don't to make a sphere out of material objects for them to have some kind of real existence. How about "the set of all locations in space in a space equidistant from this electron"? Good on Vsauce for tackling a really difficult mathematical problem. I never expected to see a decent visualization of the Banach-Tarski paradox.
[QUOTE=JohnnyMo1;48347204]How about "the set of all locations in space in a space equidistant from this electron"? [/QUOTE] Definitely. But that takes us back to the abstract world, working with the description of a position rather than something physical. So no infinite supply of Lindt orbs :< [img]http://i.imgur.com/9WcraZm.jpg?1[/img] Excellent episode, I like that focus was on one actual scientific theorem. I haven't been following vsauce for a while since a few of the episodes went really silly consisting of a string of tangents rather than focusing on one subject, but this was really interesting.
[QUOTE=Im Crimson;48347321]Definitely. But that takes us back to the abstract world, working with the description of a position rather than something physical. So no infinite supply of Lindt orbs :< [img]http://i.imgur.com/9WcraZm.jpg?1[/img] Excellent episode, I like that focus was on one actual scientific theorem. I haven't been following vsauce for a while since a few of the episodes went really silly consisting of a string of tangents rather than focusing on one subject, but this was really interesting.[/QUOTE] Are positions not physical? They certainly seem to be in physics.
My eyes started glossing over. He explained this as simply as possible and it was still far, far over my head.
I've only had a short class of set theory, but somehow when he covered the Hyperwebster where Volume A contained itself as well as all other volumes it felt like it fell into Russell's Paradox somehow...
[QUOTE=LoneWolf_Recon;48347669]I've only had a short class of set theory, but somehow when he covered the Hyperwebster where Volume A contained itself as well as all other volumes it felt like it fell into Russell's Paradox somehow...[/QUOTE] Nah, I'm pretty sure there's nothing wrong with that construction in standard set theory. Notice, it does not [I]really[/I] contain itself. A set "A" which contains itself must have an [I]element[/I] "A". It's not good enough to just be able to delete things and get the set back. Set theory guarantees that a set which actually contains itself does not exist.
[QUOTE=Laserbeams;48346834]The obvious problem is that real spheres don't have an infinite amount of points, they are made of a finite number of molecules[/QUOTE] the probability cloud of electrons isn't
:snip:
[QUOTE=JohnnyMo1;48347204]In that case, "real spheres" don't exist at all. They don't fit the definition.[/QUOTE] Every point on the surface of a real sphere is roughly at the same distance from its center, so it does fit the definition. It doesn't have to be perfect because there are no perfect things in real life anyway [QUOTE=JohnnyMo1;48347204]But you don't to make a sphere out of material objects for them to have some kind of real existence. How about "the set of all locations in space in a space equidistant from this electron"?[/QUOTE] Unless there actually is something measurable in this set of locations, there is no reason to consider it real [QUOTE=Killuah;48347743]the probability cloud of electrons isn't[/QUOTE] You can't prove that it actually is a perfect mathematical sphere with infinite points. The electron cloud model claims it is, but is it really?
[QUOTE=Laserbeams;48348376]You can't prove that it actually is a perfect mathematical sphere with infinite points. The electron cloud model claims it is, but is it really?[/QUOTE] There isn't really anything to show that a probability cloud has a finite amount of points either. Why would it be?
[QUOTE=Laserbeams;48348376] You can't prove that it actually is a perfect mathematical sphere with infinite points. The electron cloud model claims it is, but is it really?[/QUOTE] Considering there's no position in space an electron [I]can't[/I] be (unless its already occupied) but just a infinitesimally small percent chance it could take. I don't see why it couldn't be counted as an infinite point sphere. The probability density function may have practical zero percent chances where an electron, for instance, could be on the other side of the galaxy rather than in proximity to the nucleus, but mathematically and physically its not [I]exactly[/I] zero.
[QUOTE=Laserbeams;48348376]Every point on the surface of a real sphere is roughly at the same distance from its center, so it does fit the definition. It doesn't have to be perfect because there are no perfect things in real life anyway[/QUOTE] "Almost a sphere" does not fit the mathematical definition of "sphere." [QUOTE=Laserbeams;48348376]Unless there actually is something measurable in this set of locations, there is no reason to consider it real[/QUOTE] I can measure a distance of 1 meter from a point, so it's totally measurable. Distances are measurable. [QUOTE=Laserbeams;48348376]You can't prove that it actually is a perfect mathematical sphere with infinite points. The electron cloud model claims it is, but is it really?[/QUOTE] This is silly. One side has evidence. Do you have a more accurate model? This is an unjustified appeal to skepticism.
It doesn't even need to be a sphere. I didn't have a deeper look into this but it looks like it only needs to be a set like shown that allows a certain amount of transformations which is in this case rotations that don't break symmetry. It was also just an example from me. And so far it worked, it was only to show that the "we never gun use dat silly fundamental researchers" is as stupid and debateable as always.
[QUOTE=JohnnyMo1;48348537]"Almost a sphere" does not fit the mathematical definition of "sphere."[/QUOTE] No, but the real world definition and the mathematical definition have to be different, because mathematical spheres are impossible in the real world [QUOTE=JohnnyMo1;48348537]I can measure a distance of 1 meter from a point, so it's totally measurable. Distances are measurable.[/QUOTE] No matter how precise your measuring of distance is, it will still be off to some degree because every measuring tool has an observational error. Whatever you do, you will still end up with an "almost perfect" sphere in the real world [QUOTE=JohnnyMo1;48348537]This is silly. One side has evidence. Do you have a more accurate model? This is an unjustified appeal to skepticism.[/QUOTE] You can't take a random point of the cloud and say for sure that the electron will be there at some point in time. It [i]might[/i] be, but it won't (assuming we even know where the electron actually is, which is impossible)
Itll never happen, but Im still waiting for the day that they do some kind of ridiculous video game theory that could be proven true, and it starts with "Vinesauce, Vinny here".
Question for folks. The claim is made that the infinity between 0 an 1 is larger, infinitely so, than the infinity of all natural numbers. My question is, why can't the work that is done with the Infinte-Hotels example and/or the sphere example of 'shifting one over' prove that they are all equally infinite? So, for example, if you do the diagonal test, why couldn't you just shift all your numberes back one to cover your bases?
:mindblown: too confusing
[QUOTE=ForgotPassword;48353240]You're thinking of the diagonal test with only a single number being produced. Imagine an infinite number of new numbers coming from that, then you'd have to shift for infinity. The infinity between 0 and 1 in this aspect is what makes it "bigger" than the natural numbers.[/QUOTE] Right but why can't you just keep shifting?
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