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- เผยแพร่เมื่อ 24 ธ.ค. 2024
- The Banach-Tarski theorem says we can duplicate a solid, 3D ball. An extension of the theorem generalizes it to transforming any 3D shape into any new 3D shape of a new shape and size. It's a beautiful piece of math and demonstrates how infinity can have strange results.
Made for #some3 challenge - big thanks to Grant Sanderson (@3blue1brown )
Hopefully you at least learned... something? Even if you don't quite understand the paradox in full, I hope you at least learned a thing or two about infinity or about set theory.
Well, this was my first video. I’d love for you to tell me what I did well and what I should work on improving. I was definitely rushed for time for this video. I just didn’t realize how long animation can take. Because of this, I had to cut out several sections which would have made the proof more approachable and understandable, and the ending section is incomplete. Maybe I’ll make a follow-up, I don’t know… I do want to send massive thanks to Grant Sanderson, 3blue1brown, for organizing and hosting this Summer of Math Exposition. Though a tiny bit stressful because of my underestimated scope, this was fun and helpful for my brain. I might never have taken my hand at explaining stuff were it not for him so thanks!
TO LEARN MORE:
The book where I got a majority of this information is called The Pea and The Sun by Leonard M. Wapner. I could not recommend it enough if you're interested more in this topic. It assumes no background knowledge and explains everything.
Vsauce has made a good video on the topic here: • The Banach-Tarski Paradox
Everything in this video is my own: my own script, my own music and my own animations using blender.
Finally, a new concise and digestable take that is more rigorous than Vsause's video of this subject.
Vsause is trash
Yeah the banach-tarski episode was definitely one of the ones were bi sauce got a little more in-depth than what they used to be but definitely wasn't a full-blown explanation
Vsauce only spouts nonsense. literally everything is more concise and digestible than their videos.
@@sumdumbmick nah I just think Michael got lost, started as look at this cool stuff type videos but turned into a weird pseudo educational platform. That being said I think vsauce2 and vsauce3 actually have their own unique thing and are pretty interesting.
@@chicken29843bi sauce?
This guy knew what he was doing with that thumbnail lol
Edit: he changed it. Never forget what was taken bros.
im glad im not the only one haha
What?
@@PeriwinkleaccountI believe it changed now. it used to be one dot with an equals sign and two dots, forming a shape that looked a bit like a penis
@@KingJellyfishIIworst moment in the history of TH-cam
@@KingJellyfishIInever forget what was taken from us
What is an anagram of the Banach Tarski Paradox?
The Banach-Banach Tarski-Tarski Paradox Paradox.
I'll get me coat
i like math jokes like this one. one of them which is like this is "the b in benoit b. mandelbrot stands for benoit b. mandelbrot"
You should submit this for the #SoME3 competition. Its a great video I always wondered how the Banach-Tarski theorem was proved but was too lazy to research it.
Thanks, and yep; I did make it for SoME3!
@@mathpuppy314 You should probably change the title to inculde "#SoME3" at the end.
It's wild to see such a good video come across my feed and discover that it's the first video ever on the channel, only 4 days old. Commenting to help tell the algorithm that more people should see this video. Good luck!
Really great explanation. I hadn't truly understood the theorem until watching this. +1 subscriber!
A suggestion for your next videos: please increase your sound volume, this video was one of the quietest ones I've watched in a long time. Had to boost the volume like 5x so I could hear it at 70% volume in my PC 😅
Supplementing this with vsauce, i was finally able to understand the theorem in a way that i could explain it to someone else even with my limited knowledge of set theory and mathematics, great work!
Really great work. Commenting for the algorithm.
Please keep making videos.
This is great, the excerpt about the free group helped me realise what was lacking from Vsauce's explanation years ago. I wasn't understanding that it was the combination of actions and their inverses together that are at the root of this paradox. That's the single thing that was missing for me.
Mathematicians and their esoteric work really are quite remarkable.
I like the way you outline this proof. One reason in particular is I think it highlights well why someone would choose to reject the Axiom of Choice. I had a graduate professor long ago who needed this axiom to do a lot of hefty lifting in his course (and for his research), and thus he passed it off as such an “obviously true” thing in that sly, hand-wavy sort of way that actually makes you suspicious it’s not as obvious as he made it sound. Upon thinking about it more, it is at least a bit troublesome to considering picking one item each out of an uncountable number of non-empty sets as you cannot even list the sets you want to pick from, so how could you actually direct the choosing task to be performed? As mentioned in the proof here, you also have to locate an uncountable number of starting points to perform rotations on without any actual directions of how to find any of the points in question and no way to create any such list. The Axiom of Choice supposedly saves the day in these situations, but it simply says you can do the thing that, well, you can’t actually find any way to do, so it more defines into existence a bypass the user really wants but can’t actually create properly. Given all of this, I dislike it, but as the video pointed out, strange things can occur with infinity even ignoring the Axiom of Choice. As such, I also question the use of infinity at all, but that is an argument for another time.
As a semi-joking answer: perform the choosing in a parallel manner with an uncountable infinite number of choosers :)))
There are many ways to define math and we only set up axioms because otherwise we would have no notion of abstract truth in mathematics. Axioms are usually made to be things that must obviously be true in the euclidean world around us, but such geometric axioms break down in spherical or hyperbolic space. In my eyes, whatever way you choose to define math is valid in some right, and any theorem that follows from some axiom is valid in that mathematical space. If you choose to reject infinity and continuity because we have no concrete notion of the infinite size or infinite detail in our real world, then I think that is a valid way to view math. You just gotta live with the rules you give yourself.
Now in calculus, infinity is usually expressed in the form of a limit, demonstrating what would happen as a variable reached an arbitrarily large size, and it doesn't necessarily refer directly to an "infinity" existing somewhere out "at the end" of the number line. What I've done in this video is I assumed that such an infinity simply exists somewhere out there, rather than just exploring what would happen as the size it got arbitrarily large. I think assuming Choice is very similar to assuming that such an actual infinity does exist and I personally accept both.
By the same logic, infinite sets are problematic because you cannot list their elements either.
math is the subject in which what we are talking about may or may not be true, nor do we care about it. we just care that our reasoning is sound given the said rules, which maybe meaningless.
The Axiom of Choice is not "obviously true" (if it was, It wouldn't even make sense to be able to deny it).
It's something that we choose to be true because it's useful (the same way that we choose to be true that we can always make the union of an arbitrary family of sets, or the powerset of any set).
Wonderful video and great editing! Surprised this channel is so small! Keep at it!
Great content. Finally I really understand whats going on after years! Please keep doing more
Very good video! I liked the much more mathematically based starting point of the free group illustrating the seemingly recursive nature of many infinite sets.
I liked the animations too - the visuals on the sphere really helped visualize what was going on. I particularly liked the animation of the sigma branch replacing the wholr at 5:14 - soooo smooth! It looks cleaner than would be expected with manim, and indeed you say you used blender. Props to you for not using the same-ol library - it gives most math videos a particular feel and I definitely get a different vibe from your video.
the part with the words and free group reminds me of a similar thing where you have a book of all "words" ever. (each "word" is a string of letters with no length limit, so like "a", "aa", "aaa" ... then "ab", "aba", "abaa"... "b", "ba", "baa" ... and it keeps going on)
but then you have an idea to separate the book out into 26 books, each book being all "words" with that starting letter, and because it all starts with the same letter, you decide to take out the first letter in each "word" in the book.
turns out each of those separate books are the same as what you started with.
i'm not sure where i've heard this from, might be numberphile or something
you have definitely heard this from vsauce's video on the exact same theorem
Great video! Just a technical suggestion, though: you probably should check your compression settings. Even on 1080p, the quality is looking pretty rough, in particular on the 3D animated bits. Take a look at 7:40 for instance!
Agreed. Also not sure if it's just me, but the audio volume seems very low, compared to most videos on the platform; would consider boosting it.
@@JadeVanadiumResearch Oh yea, too low even on full blast.
@@JadeVanadiumResearch was just about to say that. I had to install a browser extension just to watch this video.
I was wondering why I get a recommendation for a well made video by a channel with 122 subs. The description clears it up. Well done.
I think you may want to fiddle with your video settings though, in some places (like the rational numbers grid) there were some pixelation artifacts
Well done. I'll look forward to seeing more content from you
✅️ • Rigorous.
✅️ • Simple.
✅️ • Understandable.
I think I've found a gold mine with this channel... ✨️
continue man it will be worthwhile
Please turn the volume up
Well turn it yourself it works very well, but I agree the video is too quiet
@@lienardgael808 even when i turn it to 100% it's not loud enough for me i wish they just reuploaded it
@@axog9776 ah that's sad it works for me :/
did you turn up the youtube built in volume?
yes@@handleAlreadyUsed
Excellent work!
The axiom of choice is definitely weird, but the discussion over it has gone very far.
You may think that duplicating a sphere is impossible, since there should be some kind of "volume" that is preserved when rotating and translating pieces.
This idea leads to measure theory, which can assign a "volume" (=measure) to an enormous collection of sets of points.
However the axiom of choice allows (and is needed) to create weird sets that are non-measurable.
Why needed? Well Solovay showed that the statement that "all sets are measurable" is consistent with set theory.
(The actual result is more complicated and needs a discussion about inaccessible ordinals)
This is an interesting point - nobody brings up the fact that although we're mapping points that were originally on a sphere, it's really just a set (without corresponding understanding of measure). In that sense splitting one thing into two parts seems perfectly valid, much like how one can split the naturals into the evens and the odds and get two countably infinite sets.
Definitely a appreciable video considering this is your first video, all the best for your future videos
Some points about the Axiom of Choice. As far as I know the following are true:
- ZF + AC (Zermelo-Fraenkel Set Theory assuming the Axiom of Choice) is consistent (you can never prove a contradiction, that is p and not p for some proposition p)
- ZF + (-AC) (Zermelo-Fraenkel Set Theory rejecting the Axiom of Choice) is also consistent.
- AC is not only consistent, but actually provable (hence true) in DTT (dependent type theory)
The last point gives me all the reason to believe that AC is okay, because to me DTT sounds like the most accurate axiomatic system to describe the world of everything. I wish DTT was more popular than ZF as a foundation of mathematics
I and my friend are both engineering students, although my friend is more practically minded. When I attempted to explain the paradox to him, his argument was that this theorem plays around too loosely with infinity.
Infinity is the reason this theorem works, but in my eyes, it's a necessary part of math. All of calculus revolves around infinity and this is just extending that intuition to sets, which necessarily allows the duplication.
Math that works in math doesn't always have an isomorph in physics/engineering. For example, if you tried to build a duplicate bridge, you'd discover that the foundation was based on a house of cards.
@@Galahad54 LOL
Great video, but I especially smiled on the arccos(1/3) part. Angle units may differ from person to person, but their cosine is the same :^)
This was a really good explanation and you have a very nice voice
Thank you so much!
Dude, the duplication is always a result of the axiom of choice, if your conclusion is that it does not depend from it please check it again on wikipedia
Many people get the image that the axiom of choice is really that controversial, but they need to see what consequences exist without it, for example the Cartesian product of two non-empty sets being empty or the epsilon-delta definition of continuity not being equivalent to the sequence definition
well done !
according to the planck length , objects are not infinitly subdivideable so that infinite sets do not apply here
Ah, yes! Just the video I needed to melt my brain and prove I'm too dumb for this.
2:52 irrational numbers have to have infinite digits, because if they had a finite number of digits instead, then we can move the point n decimal places to the right, where n is the amount of decimal digits, then divide by 10^n. We get the number back, but now we have it written as a fraction of integers, which is what a rational number is. If we had a number with infinite decimal digits, though, we have a problem. n is infinitely large, which means moving the decimal point infinitely far away would give us infinity. And even then, 10^n would also be infinite, so that part fails as well.
4:58 by " *extending to words of ANY length* ", do you mean that we can have an uncountable length too, like for example Aronszajn trees have?
Or it supposed to be a tree of countable height here?
I just want to be sure that these can be uncountable, as I didn't learned about this, sorry if I sound too nit-picky 😉
Each word would be countable in length because you could "count" their letters, starting from the beginning and moving right. This property means we can carry out the rotations in their specific order (because rotation order is not commutative). I'm not too familiar with Aronszajn trees, so I'm not sure if this group of words would be an example of one, but upon a quick wikipedia read, I don't think so.
Don't be sorry about being nit-picky with different cardinalities - an uncountable infinity is completely different from a countable one!
Great video. I would love to see your new posts
Intriguing!!
Really good video with good explanations, but for the volume was maybe a little low for me. Good work!
very nice
is it just me or is the audio very very quiet
Very cool video! Can’t believe I found it before it becomes popular.
Can you explain, why sigma is a rotation by arccos(1/3)? Does it ensure that the points won’t collide?
Yes; in short, arccos(1/3) is chosen so that points from different starting points won't touch overlap with each other. If points were counted twice then it would become trivial to show twice as many points as you started with. Rather every point is counted only once.
Also, on a side note, if different rotation axes were chosen, other than the x and y axes, then different angles would be required to ensure that points aren't counted twice.
That's the angle between the corners of a cube, seen from the center. Equivalently, the angle between axes of a regular tetrahedron.
Make a video on the axiom of choice please, you are good :D
Nice!
The real reason this works, is because a) Cantorianism is too flawed to deal with continua, and b) the roots of the paradox go all the way to Euclidean Geometry; it doesn't and will never make sense to assume a "large" set of distinct zero-dimensional entities can form a positive-dimensional entity. If we well-ordered the unit interval, thanks to AC, and added slowly points to our collection, at what... point would we form a set of positive measure? We don't even know what the cardinality of the continuum is, only that some cardinals are forbidden, and for every single one of the others there exists a model of ZFC s.t. they have continuum's cardinality! (Cohen's theorem).
That's why measure theory works better, it doesn't treat subsets of R as "collections of points" but as separate entities that we assign numerical values to them.
The Banach-Tarski proves there are irreconcilable differences between viewing continua as... simply themselves, and as "point-sets".
In conclusion, treating continua as sets which themselves are "collections of distinct points", ignoring that almost all of them won't ever even have a name, is a terrible idea, and is a good indication we need better foundations of Mathematics, like Type Theory for example, or other examples of Constructivism.
But this shouldn't be surprising, right?
I mean, the the interval "[0, 1]" has te same cardinality as the literal whole "ℝ" set, so you clearly should be able to establish a bijection between "one ball" and "two balls" (but I don't see where the Axiom of Choice is used). 👀
Hmmm didnt the Russell paradox change how we define to be a set? Not just a collection of things
They are still collections, but there are restrictions on what those collections can be
They are still collections, but there are restrictions on what those collections can be
You are correct, but I chose to sacrifice some mathematical technicalities for simplicity. Restrictions brought about by Russell's Paradox do not change the content of this video.
@@mathpuppy314 great video
One minor gripe, you need to equalize the audio... I have you set to max on my speakers and can barely hear you.
Volume?
Nice! This video skips over the problems with the poles though e.g. the points on the sphere that stay the same under rotation around x or y axis.
Yeah you're correct; the original plan was to go much more in-depth on the technicalities, but I realized soon before the SoME competition deadline that video making was hard... I just didn't have enough time. Maybe I'll remake this video in the future!
Yes, poles would be missing from one of the spheres, but they are just points. You can just utilize the trick mentioned in the Vsauce video on the paradox: Draw some circle that contains at least one missing point. There are uncountably many points on this circle, at least one of which is "missing". The cardinality of points on the complete circle is the same as the cardinality of the circle with finitely many points missing, thus allowing us to map the points to the complete circle in some way, thus filling the gap. Repeat this procedure for each missing point.
(And please correct me if I misunderstood how to fix it, I'm not a professional mathematician or anything like)
I'm used to tau meaning a full rotation around a circle so it was a little bizarre seeing the way you used it here 😅
Yeah, you just have to get used to different letters having a variety of meanings. At this point in my math journey, you could set up a system of equations using the variables pi and epsilon and I'd figure it out... 😆
At 6:19, everything gets a little fuzzy, and your voice sounds suddenly distant , which was kinda distracting. Otherwise, this is a great video and the content is great, I like how you started with the algebra!
Thanks for the complement! I was making this video for SoME3 and I'd only finished about half of it several days before the deadline, so the second half was rushed and incomplete. I might remake this video at some point in the future.
You should mention that the ALGEBRAIC numbers are also countable.
Surely the paradox in Banarch-Tarski is in the assumption that a surface can be made up of an infinite number of points at all. Since the individual points have an undefined area, their combined area can't be properly defined either.
EDIT: According to what I've just read the axiom of choice is actually what they use to create these "solid-like" infinite sets of points in the first place (quite literally, the axiom of choice states that you can have sets without a measurable volume), so I'd guess you're a tad off the mark to suggest that you don't need the axiom of choice for any duplication at all?
Good video though, it really helped put the simpler parts of the paradox into perspective.
Thanks for the compliment!
About your question, measurement of infinite point sets is a well-studied topic, and if you're interested, I would recommend researching the Lebesgue Measurement system. It provides a clear numeric measure for line segments, areas, volumes, etc. The Axiom of Choice when used with infinite sets can result in sets which cannot be assigned a Lebesgue measure, and this is the reason Banach-Tarski can work.
When I duplicate the first group of points without using the Axiom of Choice, I do it in a way which does not change its Lebesgue measure (it remains at 0 because of the discreteness of a countable set). Hope this clears things up!
I love Math
1) start with axiom
2) find ”paradox”
3) don’t spend a second to think if the axiom might be the problem 😂
The paradox exists in some for with or without the axiom of choice, and other weird shit occurs when you remove it.
Is the Banach-Tarski Theorem really all that surprising?
E.g. If we have a line segment from 0 to 1, we could multiply all the points by 2 and get a new line twice the length.
So it seems to me if you're allowed to move infinite points in a real space you can make whatever you want if you're clever about it.
The core concept which I didn't find time to animate into this video is the idea of piecewise congruence, or equidecomposibility. When making any transformations, only transformations are allowed - this means no streatching, squishing, or reflecting. With Banach-Tarski, points are not stretched apart so the analogy of stretching [0, 1] to fill [0, 2] is not similar.
However, the BT theorem does fall under the axioms we have set up so it really should not be surprising. You are correct that moving an infinite number of points around in space could, under our axioms, make whatever you wanted.
I'm a master of mathematics and I have never met a mathematician who does not believe in the axiom of choice.
Isn't this just a set containing itself, like a shortcut linking to the folder it's in?
If it was then it would be incompatible with set theory as it would break the axiom of regularity and axiom of pairing. But the axiom of choice is orthogonal to set theory, it does not contradict anything in it.
wouldn't that be restated as demonstrably invalid --or "DUH"
I cho choo choose u pikachu! Now i have 2 Pikachus! Now 4! ... ❤
I have no issue with the audio, maybe it's because I have headphones idk
That preview is questionable
Nice video though!
yeah.... should have probably added a disclaimer there lol
I've only ever seen the vsauce video about this and I barely understood what the hell he was talking about
Infinite sphere glitch
8:00 tautautau
It looks very interesting but I don't understand anything of what you're saying 😂 I see it, but I cannot comprehend 😂
Wow
the ultimate source of the paradox is identical to an unrecognized paradox in the definition of the Set of Reals.
firstly, the Reals are said to be both continuous and subject to the Archimedean Property. already this is interesting, because every subset of the Reals which is subject to the Archimedean Property is discontinuous, such as the Rationals.
secondly, absolute infinity and its inverse are not Real by definition.
the paradox arises when we realize that mathematical analysis requires us to utilize the inverse of absolute infinity when evaluating limits. for instance if f(x) = 1/x then:
lim x->0+ f(x) = positive infinity
f(0) = undefinable
lim x->0- f(x) = negative infinity
the limits are evaluated at the immediate neighbor to 0 on the positive and negative sides. which means that they must exist on the Real Axis, further, being the immediate neighbors to 0 they prove this axis to be continuous, but by virtue of being the immediate neighbors to 0 they cannot possibly be subject to the Archimedean Property. further, since their inverses are signed versions of absolute infinity, they are also definitely not Real values.
thus, we can see that the Real Axis is in fact continuous, but it is continuous specifically because there are gaps between the Reals. a fact which should have been predicted by the fact that the Reals are an Archimedean Group, but somehow nobody noticed.
now, what this means is that no Real is actually referencing a point on the Real Axis, but instead references a smeared out region of points. since we can flip between saying that the limit of a Real value is identical to that Real value, or acknowledging that it isn't. and this is all standard practice at least as far back as Dirichlet.
so what ends up happening is that you can break up the Reals into any number of sets that are identical to the Reals, and you can do it in many different ways. and the kicker that the axioms of conventional mathematics render the magic trick you performed completely mysterious, because the problem is with the axioms themselves, and within an axiomatic theory the axioms are unquestionable by definition.
consider a circle of area 10. conventionally removing any point from it, or adding any point to it cannot possibly change its area, because points have no area.
but now consider if we construct this circle via discs which have their central 1/10 removed. the largest disc thus has an outer circumference which matches the target circle, but because the central 1/10 is missing, it only has area 9. within the hole we place a scaled down copy of the disc, whose area is 0.9. by doing this infinitely we arrive at an area constructed by the discs of 9.9R (where R indicates the decimal expansion repeats infinitely). the limit of this is 10, a claim which I have absolutely no problem with at all. however, the fact that the limit is 10 is almost universally accepted as proof that 9.9R = 10, and this is a claim which I do have an objection to.
see, this nesting of discs can absolutely never fill the central point of the circle. so, the area covered by the discs is necessarily less than the area of the circle. so if the area of the discs is 9.9R, and the area of the circle is 10, then 9.9R < 10. this is necessarily the case from a Set Theoretic standpoint, because that central point is included in the value 10, but not in the value 9.9R, thus the sets are distinct, and if area is taken to be related to the set referenced by the name of the value, then the areas must also be distinct. and this distinctiveness is specifically of a type which renders the values 9.9R and 10 strictly ordered, as 10 includes all of 9.9R plus points which 9.9R does not, thus making 10 greater than 9.9R in any reasonable sense.
worse, we can very easily modify the construction so that we get any number of missing points, which means that 9.9R < 9.9R. so even the decimal expansion 9.9R doesn't name a specific value.
and to top it off we can actually soundly show that 9.9R = 10 by noting that 1/3 = 0.3R, and 30/3 = 10, so 9.9R = 10.
thus we end up with 9.9R < 9.9R < 9.9R = 10. and we can show exactly the same thing in the other direction by adding random isolated points to the circle, so we end up with 9.9R < 9.9R < 9.9R = 10 = 10.0R < 10.0R < 10.0R < 10.0R. and since the Reals cannot distinguish between any of these, and claims to be continuous, what happens is the Real value 10 necessarily includes all of this, abstracting it to a single value.
and since the string of inequalities extends infinitely in both directions away from the central Integer value 10, we can thus subdivide the Reals into any number of copies of the Reals without any problem whatsoever, and the conventional axiomatic rules are utterly incapable of deciphering why.
the tragedy is that Banach-Tarski does not fully understand this. in it, the copies are not full copies, because, wrongfully, it assumes that the Reals reference points, not smears of infinite points surrounding central points.
a stronger form of Banach-Tarski is thus trivially possible if we simply recognize the paradox of the Reals, but keep our solution couched in conventional axiomatic set theory, like ZF(C), wherein discontinuity of each copy cannot be perceived, due specifically to the fact that the discontinuity of the Reals is axiomatically denied.
I've never thought about this that way. It's an interesting concept. In my eyes, removing one point from an uncountably infinite set of points is subtracting a value of 0 measure, and doing this any finite number of times does not change the size of the set, and so in fact 9.9R = 10 because 10 - 0.0R = 10 - 0 = 10. I'll keep thinking about this though...
wtf did I learn today?
Cool video but i dont agree with the characterization in the beginning. No mathematician is debating the proof. Just some mathematicians dont accept the Axiom of Choice as valid and thus dont accept the assumptions used in the proof. The proof itself is not debated at all.
Opinions -
The bad:
1- The animations for the main subject only appear at the beginning
2 - Besides de underlying truth of the concepts leading to explain the contradictory nature; they didn't work out as an effective tool for proof
3 - The last minute could have been used to show correlated subjects or even show additional animations for the Banach-Tarski itself, instead of an blank image
The good:
4 - Nice, concice explanations for the set theory and abstract algebra parts
Thank you so much!
If it's debated and potentially false then why is it already a theroem?
It's a theorem in the most common version of set theory (called ZFC), which includes the axiom of choice.
Depending on the axiomatic system used, the proof might not exist, but most people agree on the ZFC system in which this theorem does work. The 'C' in ZFC stands for the axiom of "choice" and the alternative that many suggest is just ZF axioms (same except for choice).
I think it's invalid for a different reason: it formalises the task in a way that has two flaws and exploits one of them (the allowance for unmeasurable sets). You can find the other flaw (which you should only fix after having fixed the one abused here) when trying to formalise a normal result, such as that you can turn any polygon into any polygon of the same area: you can only give each point on the surface of the cut to one piece, so another construct that may be not very obvious is needed
I don't quite see where you think the points on the line of the cut are... You are correct that we have to be very careful when working with piecewise congruence with polygons, but when defining the sets of the bt theorem we explicitly define every point individually to a piece so we don't need to worry about points "along a cut" as there are no cuts. Otherwise, yes, this does mean that each set is immeasurable which you may find invalid.
@@mathpuppy314 In the second sentence, I wasn't talking about BTP. I just pointed out that a different problem that may look similar doesn't work out as nicely with the same formalisation of the concept of cutting a shape into pieces
@@orisphera So, in other words, the ability to arbitrarily double the points of the sphere relies on them being not just uncountably infinite in number, and infinitely choosable, but *also* infinitely _differentiable._
It's essentially a clash between Set Theory and Calculus: is the derivative of the surface of the sphere sufficiently differentiable to accommodate dicing it that finely into single points? Or, alternatively, is it _not_ sufficiently differentiable and, even if you created two distinct, discret, individual sets _with_ all the points sufficient to describe the same sphere; those points still cannot be "linked up" into a sphere "because Calculus". Or, to put it another way, since you took half of the uncountably infinite points of the sphere away, each set is now a *discontiguous* collection of points and, as such, according to Calculus, can no longer be re-integrated back into a complete sphere because you can't "jump holes", so to speak. And the surface of the "pseudo-sphere" described in this set has an _infinite_ number of point-holes accounting for *exactly* half it's surface area.
why does it sound like youre talking into a cup
Not every mathematician accepts "infinity". There are mathematicians only work with countable infinities, others only with finite objects. Set theory "axioms" contain an assumption that infinite sets exist, no no for them. Also "real" numbers are not in their vocabulary. These researchers also reject the axiom of choice.
It's valid to work with a set of set theory axioms that don't include infinite sets and see what you can discover. But to claim that one set of non-contradicting axioms is more valid than another, and thus to claim that the matter of discourse of several fields of mathematics doesn't exist, is childish.
I reject the existence of real numbers. What we think of as a real number is not a number at all, but actually an interval between two rational numbers.
Uncountable infinity should not exist. If we ever end up with a duplication glitch like this, we've clearly gone wrong somewhere leading up to it.
So which interval is π then?
@@ribozyme2899 My trick is that there isn't one single interval equal to π. There are countably infinitely many rational approximations less than π, countably infinitely many rational approximations greater than π, countably infinitely many intervals containing π.
@@natsuzkan So isn't that basically the same concept as the construction of real numbers as Dedekind cuts? ( en.wikipedia.org/wiki/Dedekind_cut )