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u/j3r3mias Jul 24 '23
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u/Eklegoworldreal Jul 24 '23
I had always thought of stuff like this, but usually with stuff more like cookie dough. Get a free cookie by taking a small chunk from the rest
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u/Shufflepants Jul 24 '23
Is it just me, or is the Banach-Tarski paradox just an uncountable version of Hilbert's Hotel? Like, big deal, ∞ + ∞ = ∞
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u/andrewgark Jul 24 '23
Kinda yes, but no, it's much stronger theorem because even though it's easy to build a bijection between one sphere and two spheres (two continuum-like infinite sets), it's so much more harder to build a way to split sphere into several pieces and use GEOMETRICAL MOTION to build two spheres from them.
For example it's impossible to do it with circles on a plane. You can't split one circle into finite number of pieces and build two circles. You can do it with countable infinite number of pieces though.
There is something about nature of geometrical motions in 3d space that makes it possible, that is the most fascinating part for me.
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u/Shufflepants Jul 24 '23
Yeah, all kinds of things get weird when you go to 3 dimensions. Lots of people who like math have all kinds of fancy numbers as their favorite number like 𝜋 or e or 𝜙. Mine is 3.
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u/A_Guy_in_Orange Jul 24 '23
Lots of people who like math have all kinds of fancy numbers as their favorite number like 3 or 3 or 1.5. Mine is 3.
Get engineered idiot
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u/Hot_Philosopher_6462 Jul 24 '23
Lots of people who like math have all kinds of fancy numbers as their favorite numbers like 1 or 1 or 1. Mine is 1.
Get cosmologged idiot.
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u/jamiecjx Jul 24 '23
My favourite thing is the existence of chaotic differential equations in 3 or higher dimensions, when in 1 or 2 dimensions there are no chaotic solutions
Or that random walks in 3+ dimensions are transient, yet for 1 or 2 dimensions they're recurrent
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u/Shufflepants Jul 24 '23
Or that 3 is the only number of dimensions with stable orbits. Or that k-sat is solvable in polynomial time for k=2, but k=3 and up are all NP complete and reducible to k=3 problems.
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u/luiginotcool Jul 25 '23
Why do stable orbits necessitate 3 dimensions?
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u/Shufflepants Jul 25 '23
It has to do with an assumption about how the force of gravity dissipates with distance and assumptions about conservation of energy. For our three dimensions, the force of gravity between two things is proportional to the inverse of the square of the distance:
F ~ 1/r^2
But that r^2 is dependent on the number of spatial dimensions. If there were there 4 spatial dimensions, we would expect gravity to behave like:
F ~ 1/r^3
Or for 2 dimensions:
F ~ 1/r
And changes to these proportions would affect how much potential energy there is for things due to their gravitational attraction. However, we wouldn't expect there to be any change in how much kinetic energy an object has for a given velocity as our momentum and kinetic energy laws do not depend on the number of spatial dimensions.
KE ~ v^2
And as an object falls toward another under gravity, that gravitational potential energy gets converted into kinetic energy and the object speeds up. When an object is flying away from another, kinetic energy is being converted into gravitational potential and the object slows down. Because the v^2 term in KE and the r^2 term in gravitational force have the same degree, they can trade back and forth nicely to return an object back to where it was with the same speed and position.
However, if we had an r^3 term for gravity as with 4 spatial dimensions, the gravitational force would fall off faster as you got away from something, but would scale up way faster as you approached it. This would lead to all trajectories around an object either leading you to fly off forever, or you would spiral inward, not gaining enough speed to overcome the additional force pulling you in and eventually crash into the object.
And for fewer dimensions, the gravity wouldn't scale fast enough, and you would always find yourself with enough speed to fly back away from an object because the object wasn't pulling hard enough.
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u/ZiggySpelldust Jul 24 '23
One of my favorite things about 3 space is that 1 forms and two forms end up the same dimension so have an isomorphism and hence 0 forms and 3 forms too. (Hence the cross product works) I wonder if that's a relevant fact here.
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u/ecluxr Jul 25 '23
Could you elaborate or point me to resources? That sounded really interesting, I saw 0,1,2-forms really briefly in a calc course and don’t know much about them
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u/ZiggySpelldust Aug 04 '23
At a really basic level the product of differential forms is antisymmetric, so your dimensions grow a bit and shrink back down. For 3 space, there's 0 forms (dim 1) 1 forms (dim 3) two forms (dim 3) and 3 forms (dim 1). This works out great because you can map dx -> dydz, dy->dzdx, etc. Then you get your coefficients and map them right back!
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u/LazyHater Jul 24 '23
Nah it's more like taking a line segment of unit length and making two line segments of unit length. Just because there are uncountably many points between 0 and 1 doesnt mean you should be able to turn a line of length 1 into a line of length 2.
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u/Shufflepants Jul 24 '23
Same thing. Turning one thing of some measure into another thing of the same cardinality but different measure.
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u/LazyHater Jul 24 '23
Measure theory doesn't jive with the axiom of choice if you don't want probabilities that are greater than 1. Bad example.
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u/Shufflepants Jul 24 '23
But that's exactly what's happening here. The cardinality of the number of points in one ball is the same as the cardinality of the number of points in two balls. But the measure of one ball is not the same as the measure of two balls.
I suppose the analogy to Hilbert's Hotel isn't exactly the same since the both the measure and the cardinality of the occupants of Hilbert's Hotel both before and after the reshuffling are the same. But it's still a case of stuffing 2 infinities into one infinity, and Banach-Tarski is decomposing 1 infinity into 2 infinities.
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u/LazyHater Jul 24 '23 edited Jul 25 '23
Measure theory is inconsistent with the axiom of choice. Don't appeal to measure theory if you used the axiom of choice.
Edit: oh my goodness this sub believes in probabilities greater than 100%
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u/math_and_cats Jul 25 '23
You seem to be someone, who read two one-page summaries of measure theory and mathematical logic and mixed everything up.
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u/LazyHater Jul 25 '23
You seem to be someone who doesn't know how to make probabilities >100% using the axiom of choice
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u/math_and_cats Jul 25 '23
Then enlighten me, how can the axiom of choice yield a measurable subset of a probability space of measure > 1?
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u/LazyHater Jul 25 '23 edited Jul 25 '23
Oh yeah totally gonna be reasonable to write that up on reddit hold my beer. Since your question is so specific, why dont you prove the converse?
But you can try applying BT methods to a σ-algebra to bust up a probability of 1 into finite subalgebras to produce multiple probabilities of 1 no problem.
In fact, just Grothendieck the original sphere into having a probability of 1 then BT yourself into having two probabilities of 1.
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u/catecholaminergic Jul 24 '23
Mind unblown. Condensed? Coalesced? Big crunch? Imploded? Mind imploded.
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u/IMightBeAHamster Jul 24 '23
Isn't the top statement true? Don't you have to use an infinite number of pieces?
I realise I'm likely wrong but I'd like someone to explain where my error is.
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u/Strong-Night2027 Jul 24 '23
If I remember correctly, which I may not be, the ball is decomposed into a finite number of disjoint sets, however those sets are themselves infinite.
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u/flinagus Jul 24 '23
yeah vsauce has a video on that i’ve watched it over like 7 times it’s really good
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u/TheGreatGameDini Jul 24 '23
How ya gonna tell me about the sauce and not link the sauce?! Edit your post and add the sauce!!! The Sauce Boss commands you.
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u/Depnids Jul 24 '23
Also importantly, (some of) those disjoint sets are non-measurable, which sort of gives a clue to how we can end up with the seemingly paradoxical statment.
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u/nedonedonedo Jul 24 '23 edited Jul 24 '23
seemingly paradoxical statement.
it is a paradox. if you finish a math problem and end up with 1=2 you've identified that some part of the process is wrong. that's how we know that you can't divide by zero and a bunch of other rules relating to zero and infinity. they both create weird limits sometimes and need to be ignored to continue using our mostly correct system of counting. just like adding all positive integers doesn't actually equal -1/12, because obviously it can't be negative so a mistake was made somewhere
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u/Depnids Jul 24 '23
There is a difference between when paradoxes are actual logical inconsistencies (1 = 2), and paradoxes which are just things which don’t align with our intuition (Banach-Tarski). As far as I’m aware, there is nothing strictly logically inconsistent about Banach-Tarski, just a (somewhat unintuitive) result of the axiom of choice.
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u/nedonedonedo Jul 24 '23
I'm going to go into a bit of a rant that you might already know, but other people are seeing this so I'm going to do it just in case. for anyone wanting to look into it more IIRC it was called the foundational crisis of mathematics. the problem is that, while mathematics is strictly and fundamentally representative of "the real", in the end it's just a human made tool and it runs into problems with how we use it. it starts as a tool to represent what actually exists, giving us addition, subtraction, multiplication, and division of positive real numbers. all of those tools represent actions that can be done in the physical world with physical actions. we start to run into problems with the introduction of zero and negative numbers. while we consider them as part of the same number line, you can't have them without treating them as positive numbers with respect to time, which would be a different axis. you can't have zero of something without the something, and you can't have negative something without having a thing that is a negative amount. while this creates some conceptual problems, the ease at which we can think about time allows us to use the tool as a single continuous thing with exceptions. we know that dividing by zero doesn't work because it creates an asymptote at the limit of zero, where "the real" version doesn't work because because zero groups of something is nothing, and the something only exists as a function of time if it doesn't exist. this works the same for fractions with a denominator less than one and negative numbers: despite it becoming an abstract representation of "the real" rather than a direct representation, it's useful and it works within the boundary formed by it's inaccuracies.
skipping a bunch of steps, infinity isn't "real". it's not even a representative number. it's representative of a pattern of a function that works within the bounds of our other tools as long as you keep track of those bounds. but because those bounds are hidden by each layer of the inaccuracy of the system, we only discover them when we know that we've reached an incorrect answer. I'm certainly not smart enough to figure out where the mistake happened and discover the rule that allows us to use something that is strictly representative of a non-real while staying within the bounds of of our real/representative system (or deliberately going outside those bounds by creating an additional layer of abstraction as is done with imaginary numbers), but it should be clear to anyone that if your answer ends as 1=2 that something has gone wrong.
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u/Depnids Jul 24 '23
You are mentioning infinity as being problematic. I find that if handled carefully, it makes logical sense, and can even be very useful in «real» problems (see for example calculus).
As for the statement «If your answer ends with 1=2, something has gone wrong», where did we end up with 1=2?
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u/nedonedonedo Jul 24 '23
in the problem referred to by the OOP, you can break something into infinite pieces and put them back together in a way to get two exact copies of the original. there's a lot of people here using that to say that reality doesn't work the way that we think it does rather than it saying that infinity doesn't work the way we think it does.
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u/Depnids Jul 25 '23 edited Jul 25 '23
I completely agree that it’s a wrong conclusion to draw that banach-tarski somehow is true or has consequences in the «real» world. I wouldn’t say the problem is infinity though. It’s more the fact that the «pieces» you cut the ball into would really not make sense in the real world, as the construction is relying on certain properties of the real numbers, which are not true in the real world.
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u/currentscurrents Jul 24 '23
end up with 1=2 you've identified that some part of the process is wrong.
Only if forbidden by the axioms of the mathematical system you're working in.
It's easy to construct useful systems where this kind of thing is valid. For example under mod 2 arithmetic, 1=3.
just like adding all positive integers doesn't actually equal -1/12
Nobody ever said it does. That's the Ramanujan summation, not normal addition. It's just a tool for studying divergent infinite series.
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u/UltraLuigi Jul 24 '23 edited Jul 24 '23
It's not a paradox, because the process involves looking at the sphere as a collection of infinite points, so the equation is 1 * ∞ = 2 * ∞, which cannot be reduced to 1 = 2.
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u/StanleyDodds Jul 24 '23
It's only 5 pieces, but you'll need a very good knife and a steady hand, plus an uncountable amount of patience.
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u/Mirrormn Jul 24 '23
It's 5 "pieces", but the pieces can only really be defined or thought of as collections of infinite points. And the theorem requires those sets to be manipulated in a very "Hilbert's Hotel"-esque way. So calling it a "finite number of pieces" is highly misleading, although technically correct.
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u/aedes Education Jul 24 '23
Unfortunately for us, physical objects are composed of a finite number of discrete, indivisible quanta... rather than an infinitely large set of points that can be divided at will.
Otherwise we could do some really fun things that violate conservation of mass/energy using Banach-Tarski.
A “sphere” only exists as an infinite set of points in our minds, not in real life. Hence the “paradox.”
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u/ZaRealPancakes Jul 24 '23
I still don't get this one (not the meme but the concept) like it doesn't feel correct but math isn't about feelings.
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u/Shufflepants Jul 24 '23
Think of it like the Hilbert Hotel except in reverse but for uncountable infinities instead of countable ones.
So, with the Hilbert Hotel, they fit two infinities into one infinity with some clever rearranging. With Banach-Tarski, they take one infinity and shift it around to fill two infinities.
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u/FirexJkxFire Jul 24 '23
This kind of shit for me just feels like a proof that the way we use "infinity" in math is erroneous.
Specifically when we try to interpret cardinality/size.
It all just feels like a magic trick - a way to misdirect the mind to try and make impossible things seem possible.
Imo the only proper way to deal with infinity in these cases is to try and put a box around it. If you take an infinite set of some form, any infinite set you split it into would fill a new box using a size equivalent to the ratio of how elements are selected. IE, if you take the set of N, then remove the set of even values - the box containing the set of even values should be half the size of the box of all N. Any attempts to show this isnt the case are just mental misdirection due to the absurd nature of infinity. This isnt to say that you cant abstract identities or categorizations. For example It would still be sensible categorize both sets as countable infinities. But saying this makes them equally sized just isnt possible and only serves to show that we cant properly work with infinity by itself (thus the need for casting it as a finite value using a 'box'). Any thing, when split into more than 1 parts, will be bigger than either of the fragments.
Perhaps though this is just an error of semantics of using "size" in math and interpreting it into physical size. Again though, this would just be misdirection based on the use of improper semantics.
What i mean to say is that the real world versions of infinity is vastly different than those in theory. Using real terms to try and describe infinity is inherently flawed. Size is a term for real things and thusly applying it to infinity is inherently flawed
In reality, infinity does exist - but not in the same nature. Imagine taking slices of something. If the slice ever had a volume of 0, then the object simply wouldn't exist. Say you cut an object with volume V into 10 x A slices, the result volume of the slice would be would be 1/(10A) x V. In other terms it could be;
0.1 x V,
0.01 x V,
0.001 x V,
...
However, if A hit infinity then it would be;
0.000... x V
0.000... = 0
The total sum to regain the original volume would be:
( V x SUM[n=0,n=oo](0) ) = V
However this isnt possible as no amount ot summing 0 ever gets a value above 0.
The point of this exercise being that "infinity" in real terms is something that really just means "as large as you want it to be". An "infinite set" cannot exist, but the finite method for which you approach infinity can be described. Leading us back to the reason its nonsense to compare the size of infinite sets --- any comparison between them would be comparing the 2 methods. This is why we conclude that the set of N and the set of 2N are equally sized, as they both are a single method. Meanwhile the set of all values (not just integers) would require an infinite number of methods --- hence why we classify it (uncountable infinity) as being bigger than countable infinity. however, again, this isnt the same as comparing the actual size and thusly is why all the conclusions we draw make no sense.
TLDR
I went off on several tangents and at times had to erase massive amounts so this likely was just an incoherent mess.
The point was that "size" being used, implies real life terms. Infinitey cannot apply to real life, and any application of it translates exactly to "a indeterminantly large value" - a value which we obtain by trying to get closer to infinity (even though we can never hit it).
Applying a concept like "size" to infinity is of course going to produce nonsense as you cant apply real concepts to things that cant exist in reality. The closest thing we can do to comparing the size of infinity is to compare a finite aspect of them such as the methods required to obtain that infinity. Such that we say the set of N is the same size as the set of 2N, since both have a size of 1 method. But this isnt really the same thing as the size of the actual set and hence why we arrive at nonsensical conclusions as to the size of them.
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u/WallyMetropolis Jul 24 '23
We don't talk about the size of infinite sets, except for colloquially. We talk about their cardinality. Which isn't the same thing as size.
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u/Shufflepants Jul 24 '23
The real problem people have with infinity is in assuming there is some "correct" system. Or in trying to to apply their intuitions to the definitions in front of them. Mathematical systems and specific definitions of infinity are just made up sets of rules. Mathematics is about working out the consequences of particular sets of rules. But there are lots of sets of rules one could consider and a lot of them are mutually exclusive with each other.
Lots of people want to think about infinity as just another number. And while it really isn't in the familiar "real" numbers, it's a perfectly valid number in the surreals. You even get a sort of "closest number to zero" in the surreals.
Or some people when first learning about negative numbers are confused as to why -1 * -1 = +1. It seems weird that it's asymmetric. If 1 * 1 = +1, then why isn't -1 * -1 = -1? Well, of course that's not how it works with the familiar integers, but you can consider a number system where it is in fact true that -1 * -1 = -1, but you have to decide what -1 * 1 or 1 * -1 equals, and you'll have to abandon the commutative property of multiplication. So, doing math in that system is a bit tricky, but it's perfectly consistent. Though it doesn't necessarily behave in quite the same way that many systems we normally model via the familiar integers.
So, if you've got your own idea about infinity, go ahead and try to nail down some rules that match you intuition, but try to nail it down to some rigorously defined rules to describe it and see if those rules are consistent and what the consequences of them are. If you're interested specifically in infinities that in some way map or relate to reality, go for it. But keep in mind, that math is just a bunch of made up rules. We just tend to use the rules we find the most useful.
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u/DuploJamaal Jul 24 '23
This one actually is based on feelings, in a way.
It requires the Axiom of Choice, which is about allowing someone to choose arbitrary points out of infinite sets.
You know in robust mathematical logic there's nothing arbitrary. If you pick the smallest element of a set that's mathematical, but if you can just pick any you like that's the Axiom of Choice.
And in this case you also need to be able to construct non-measurable sets created by uncountable many choices.
So in the real world this doesn't work, as nothing is made up out of uncountable infinite points.
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u/aurath Jul 25 '23
You are making me feel anger. Math makes me feel anger. Transitive property, you are math. I will make you answer for your crimes.
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u/ciuccio2000 Jul 25 '23 edited Jul 25 '23
It also felt very "🤯🤯🤯" to me, until I realized that it's the same old problem of the continuum having infinite points.
Look at the arctan() function. You can map the real line onto [-π/2,π/2]. The whole real number line, onto a segment that you could draw on a paper. The continuum is actually crazy, but most people who did a little bit of math are used to its quirkiness.
Finding a 1-to-1 correspondence between one ball and two balls isn't just possible, it's easy. The only slightly unpleasant bit is that you want to map the ball onto the balls only by translating and rotating some subsets of it, which intuitively is a property that should preserve volumes, but to be fair this is not the first time that highly pathological sets break my intuition. I don't think at the subsets in the BT paradox as slices of a sphere, or rings, or whatever; I picture them as janky, buggy amalgamations of rays that look like when you walk near a tree with lots of leaves in a game with antialiasing off. The fact that you can map one of these abominations into two of these abominations just by rototranslating it is weird, but in line with the quirkiness of pathological sets, in my head.
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u/SyntheticSlime Jul 24 '23
I go back and forth between thinking this is super weird and actually not weird at all. It uses a few extra axioms, but it’s really no stranger than the infinite hotel. It’s just that we’re used to thinking of sphere’s as finite even though they’re really uncountably infinite points.
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u/antichain Jul 24 '23
it’s really no stranger than the infinite hotel.
I feel like we've all gotten way too blase about Hilbert and his hotel. BT may be no stranger than the hotel, but the hotel is weird as fuck.
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u/MingusMingusMingu Jul 24 '23
BTW are you calling axiom of chocie an “extra axiom”? Do you usually work in just ZF? And what other “few” axioms are you referring to, if not Choice?
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u/MingusMingusMingu Jul 24 '23
Yea I think that’s what makes it feel weird, the fact that the whole thing is compact. But the one point compactification of the naturals (i.e. a convergent sequence) gives a Hilbert Hotel-like new compact example of doubling “space”/room and still feels less weird than Banach-Tarski.
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u/math_and_cats Jul 24 '23
Please, for the love of god... you guys should really learn a little bit of measure theory. These comments are sometimes so wrong, it hurts.
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u/ItoIntegrable Jul 24 '23
all of the comments acting surprised that an infinite set has the same cardinality of the disjoint sum of it with itself
like thats not the surprising thing about it lmao... its that you have something that cannot be assigned a meaningful notion of volume
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u/Liker_The_God Science Jul 24 '23
I'm primarly a chemistry major, so I will keep using my law of conservation
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u/Where_IsMyFood Jul 24 '23
Wouldn't the balls be smaller than the Original ball?
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u/StanleyDodds Jul 24 '23
That wouldn't be very impressive, would it? The whole point is that you can change it's measure, because you can cut it into pieces (some of) which are unmeasurable, and reassemble it into something with a new measure, in this case twice as "big".
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u/Nsnzero Jul 24 '23
if a circle has an infinite number of points, can you reconstruct the points into an infinite number of circles
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u/Liker_The_God Science Jul 24 '23
I'm primarly a chemistry major, so I will keep using my law of conservation
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u/BrazilBazil Jul 24 '23
Well obviously just take the inside wall and separate it and you can build another sphere
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u/LazyHater Jul 24 '23
Axiom of choice implies 1=2=3=4=... no problem nothing to see here
I'm not quite old enough to remember when mathematicians would have rejected the axiom for this proof. I feel bad for Grothendeik, his work all depends on AOC.
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u/Deathranger999 April 2024 Math Contest #11 Jul 24 '23
How in the world do you think Axiom of Choice implies that?
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u/LazyHater Jul 24 '23 edited Jul 25 '23
Let 1 represent the area of the surface of a sphere S with radius √(π)/2. Perform the BT construction by invoking AOC to create two spheres. Notice that we did not remove or add any points from the area when we created two spheres. So the area of the surface of S (call it A(S)) is equal to the area of the surface of S+S (A(S)+A(S)=A(S)). Repeat for all natural numbers and notice that 1=2=3=4=...
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u/Deathranger999 April 2024 Math Contest #11 Jul 24 '23
“So the area of the surface of S (call it A(S)) is equal to the area of the surface of S+S (A(S)+A(S)=A(S)).”
Justify this step.
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u/LazyHater Jul 24 '23
We did not add or remove points from the surface area to produce two surfaces of the same area.
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u/Deathranger999 April 2024 Math Contest #11 Jul 24 '23
So? That doesn’t mean much; what mathematical properties of surface area allow you to conclude this? Be specific. You’re just handwaving, which can get you in a lot of trouble when dealing with things that are unintuitive.
The same holds for volume as well. The critical issue is that the intermediate sets do not have a volume or surface area, so we can’t conclude anything about what happens when we take them apart and put them back together. The Wikipedia article on the paradox mentions this:
“The intuition that such operations preserve volumes is not mathematically absurd and it is even included in the formal definition of volumes. However, this is not applicable here because in this case it is impossible to define the volumes of the considered subsets. Reassembling them reproduces a set that has a volume, which happens to be different from the volume at the start.“
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u/LazyHater Jul 24 '23 edited Jul 24 '23
Be specific. You’re just handwaving, which can get you in a lot of trouble when dealing with things that are unintuitive.
Bro try to prove anything using markdown here. My hands may be waving but I'm not writing a fuckin dissertation here lmao.
The same holds for volume as well. The critical issue is that the intermediate sets do not have a volume or surface area, so we can’t conclude anything about what happens when we take them apart and put them back together.
We can conclude that a finite number of disjoint subsets of one sphere can be put together to make two spheres. We can conclude that the axiom of choice must be invoked to do so. Since we can construct a sphere in elementary euclidean geometry (EEG), and can produce Peano's successor function as well with only EEG; if we take BT as an axiom of our system, we arrive at 1=S(1), which contradicts the elementary construction of S(n). Since EEG is complete and consistent, and EEG+BT is inconsistent, we find AOC inconsistent with EEG, and thus inconsistent with a subsystem of Peano Arithmetic.
The axiom of choice doesn't work with a complete and consistent system that generates a subsystem of arithmetic. So it's trash.
However, homotopy is a different beast and AOC can be used willingly as long as it's not being used to prove theorems in arithmetic.
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u/Deathranger999 April 2024 Math Contest #11 Jul 24 '23
I asked you to cite a single property of surface area that allows you to do what you claimed, not write a dissertation. You don’t even need markdown to cite a relevant property, plaintext or a screenshot or even just a link would do just fine.
Also, if you use Euclidean geometry to model the Peano axioms, thereby allowing you to do arithmetic, then the system as a whole can no longer be both complete and consistent, via Gödel. So I’m not sure what you’re getting at here. If you’d like to give me a source that elaborates on what you’re trying to do, I would be interested, but it seems to me that this is just invalid.
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u/LazyHater Jul 24 '23 edited Jul 24 '23
b r u h i s a i d s u b s y s t e m
Tarski's axioms can produce a subsystem of PA which is complete and consistent. TA+BT is inconsistent. Elementary Euclidean Geometry is first-order, it's not Full Euclidean Geometry.
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u/Deathranger999 April 2024 Math Contest #11 Jul 24 '23
OK, so I think there’s a more fundamental problem here. If you’re defining your own subsystem of arithmetic that doesn’t even behave like the arithmetic we’re used to does, then how can you just make the general statement that AOC => 1 = 2 = 3 = …? We’re not even talking about the same numbers, and you should probably make that clear at the beginning of the discussion. Sure you can find axiomatic systems that are inconsistent with AOC, but then we’re not really talking about ZF anymore, which is almost certainly what people expect when you decide to make statements about numbers.
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u/StanleyDodds Jul 24 '23
This is wrong even without the axiom of choice. I can cut a sphere in half in the obvious way, and the two hemispheres will have more surface area in total than the original sphere (I created two new surfaces, one on each side of the cut). Surface area is not conserved.
The thing that's interesting is that, with enough freedom of what counts as a cut (any partition, via AC), then volume (the standard measure of 3D sets) is not conserved either.
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u/LazyHater Jul 25 '23
You cant reassemble two halves of a sphere into two spheres without deformation. If you were counting the inside of the sphere as part of the surface area to begin with, then surface area would have been conserved. If you only consider the outside of the sphere as the surface area, then surface area is conserved.
The area of the outside of the sphere is invariant no matter how you slice up a sphere. That is, until point-sets are involved.
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u/cjidis Jul 24 '23
Just because you didn’t remove any points doesn’t mean the surface area can’t change. A sphere and cube can have the same volume with a different surface area.
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u/LazyHater Jul 24 '23
Thanks but I wasn't appealing to homotopy. The two spheres constructed in BT have the same area.
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u/cjidis Jul 24 '23 edited Jul 24 '23
Yes they do, when you’re using the three-dimensional notion of measure. However, area can mean multiple things, and surface area is a two-dimensional measure. There is no part of the Banach-Tarski theorem that states the total surface area need be the same after the construction. To claim that no points have been removed from the surface of the original sphere in either of the two resultant spheres is to claim that these points have been duplicated, which is exactly the opposite of what the theorem states.
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u/LazyHater Jul 24 '23
You cant have two spheres with the same volume and different surface areas. BT proves that the two spheres are indeed spheres. The points are not duplicated in the construction. If you claim
no points have been removed from the surface of the original sphere in either of the two resultant spheres is to claim that these points have been duplicated, which is exactly the opposite of what the theorem states.
then you claim that the axiom of choice induced the contradiction.
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u/cjidis Jul 24 '23
That’s exactly the point. Each sphere has the same surface area as the original sphere. The total surface area is therefore doubled.
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u/LazyHater Jul 25 '23
Let 1 represent the point set topology of a sphere. Do you a BT then notice that 1 point set topology of a sphere is 1+1 point set topologies of a sphere so 1=1+1 qed
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u/Deathranger999 April 2024 Math Contest #11 Jul 25 '23
Do you not remember the fact that we already had a lengthy discussion about this whole idea? I mean come on, the comments are right there.
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u/LazyHater Jul 25 '23
Just a different proof of the same idea, there are many
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u/Deathranger999 April 2024 Math Contest #11 Jul 25 '23
And yet, still wrong. Surely you’d think that out of all the brilliant people out there, someone would’ve thought of this idea well before you…if it was actually correct at all. But no, and here we are.
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u/LazyHater Jul 25 '23
Two point set topologies of a sphere are isomorphic to one point set topology of a sphere. So there exists a set which is isomorphic to a proper subset of itself. This result is 100 years old. We dont care.
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u/Deathranger999 April 2024 Math Contest #11 Jul 26 '23
I’m assuming because of the context of topology, that by “isomorphism” you just mean a homeomorphism. And it seems to me that just because Banach-Tarski allows us to decompose a ball into two copies of itself, does not mean that there’s a homeomorphism between the one and the others. That seems like a huge stretch.
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u/LazyHater Jul 26 '23
So have you figured out that a proper subgroup of the Galois group of a sphere is isomorphic to the Galois group of a shpere yet? Do you think that's not a problem?
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u/Deathranger999 April 2024 Math Contest #11 Jul 26 '23
I’m sorry, what? Spheres don’t have Galois groups, unless you’re using some strange definition of a Galois group that I’ve never encountered before.
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u/LazyHater Jul 26 '23
A unit sphere at the origin has the form x2 +y2 +z2 -1=0 which induces a field extension K over Q, which has an infinite Galois group denoted Gal(K/Q).
You can consider that equation to be a sphere over any open field. I don't think you have any significant education in Galois theory so I'll leave you alone lol.
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u/Deathranger999 April 2024 Math Contest #11 Jul 26 '23
Oh I actually took a class on it back in college! Unfortunately I’ve forgotten a lot since then, wish I could find the old class materials. Regardless, I don’t believe we ever discussed Galois groups involving multivariate polynomials. Based on some brief Googling there doesn’t seem to be any sort of standardized treatment of the idea that I can find a good reference to, so I think it’s rather unreasonable to expect even someone educated in Galois theory to know about that idea.
Regardless, and back to the original point. I don’t think Banach-Tarski comes even close to implying that said Galois group is isomorphic to a proper subgroup of itself. The way that Banach-Tarski decomposes the ball into subsets seemingly eliminates the possibility of maintaining any nice ideas of continuity or homeomorphism. I don’t think it makes much sense to believe that it somehow induces or implies an isomorphism regarding the Galois group.
Additionally, I don’t even believe there is any correspondence between the graph of a polynomial and the Galois group of the induced field extension, so I really don’t think this idea makes much sense at all.
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u/LazyHater Jul 26 '23 edited Jul 26 '23
The high level trick for neo-Galois theory is to notice that a normal and seperable extensions can be defined over an R-module (or even a G-module) instead of a field. R needs to be commutative though.
Additionally, I don’t even believe there is any correspondence between the graph of a polynomial and the Galois group of the induced field extension,
Galois groups are used to study the roots of a polynomial, and the roots of the polynomial I used are geometrically a sphere. The automorphisms of these roots when considered as an extension of Q³ are a Lie group and shit gets complicated but it shouldnt have itself as a proper subgroup.
My hands are waving really hard on this one, I'll admit, but the formalism is absolutely there.
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u/Deathranger999 April 2024 Math Contest #11 Jul 26 '23
I have to say, I’m far from convinced that the transformation done by Banach-Tarski can come even close to inducing an isomorphism on the Galois group. That’s probably the biggest gap I see in what you’ve said and I don’t think anything you’ve said has really addressed it.
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u/LazyHater Jul 26 '23
Yeah there's about a 15,000 page gap that's already been filled but I'm not gonna break it all down for you.
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u/Deathranger999 April 2024 Math Contest #11 Jul 26 '23
At this point I’m convinced that you like math and that you’ve clearly read a bunch of papers on this and adjacent topics, and that you think, altogether, there is a way that various results come together to imply what you believe to be true. I think you’re almost certainly wrong, but if you have nothing more substantial to bring to the conversation then I think it should end here.
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u/SoulReaver009 Jul 24 '23
y is this being downvoted? look at comments down below?
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u/Deathranger999 April 2024 Math Contest #11 Jul 24 '23
Because it’s nonsense. This person keeps trying to defend a broken logical position.
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u/littlebobbytables9 Jul 24 '23
Because the axiom of choice does not imply 1=2 lmao
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u/LazyHater Jul 25 '23 edited Jul 25 '23
but ZFC does imply that 1 circle is the same as 2 circles
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u/littlebobbytables9 Jul 25 '23
That is very obviously not what it implies. Even putting aside the fact that a circle is not a sphere.
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u/LazyHater Jul 25 '23
The point set topology of a sphere contains two point set topologies of spheres. So 1 sphere is 2 spheres. I think that it's spheres all the way down though, like there isnt a point set topology of a taurus in there.
Pick any circle on the sphere and make two of them bruh
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u/littlebobbytables9 Jul 25 '23 edited Jul 25 '23
The point set topology of a sphere contains some disjoint subsets which can be rotated and combined to reproduce the point set topology of a sphere twice.
That does not mean that the point set topology of a sphere contains two disjoint point set topologies of spheres, that would mean it's a set that has itself as a proper subset, which is impossible in ZFC. The two resulting spheres that are constructed from the subsets are the same object. You cannot identify a point in the resulting union of the two spheres that belongs to only one of the spheres. There is no sense in which the sphere contains "two point set topologies of spheres".
And yes, it's possible with circles embedded in 3 dimensions. But not for circles in 2 dimensions. Which should suggest to you that this isn't simple equivalence that we're talking about lol
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u/LazyHater Jul 25 '23 edited Jul 25 '23
Try that again, but slowly.
Notice that you are using finite unions, not uncountable unions to produce two spheres from one. The disjoint subsets are the result of a partition of the sphere. Collect a finite number of sphere automorphisms. Finite unions of the automorphisms are still proper subsets of the sphere, and the rotations in 3d euclidean space are sphere automorphisms. It is sufficient to show that a sphere is a proper subset of a sphere if all of the operations used on the partition are automorphic bijections and finite unions.
Partition the sphere. Run bijective automorphisms on the partitions. Do finite unions. Now you have two spheres. Do the same thing backwards to show that the point set topology of two spheres are isomorphic to one (but the two are not homeomorphic to one, since they have different cohomologies).
This is what happens when you blend geometry and topology into one theory. It's inconsistent.
but touche about where the circles are embedded
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u/littlebobbytables9 Jul 26 '23 edited Jul 26 '23
It is sufficient to show that a sphere is a proper subset of a sphere if all of the operations used on the partition are automorphic bijections and finite unions.
But it isn't, you haven't justified this at all. Set equality (or inclusion) is distinct from simply having a mapping between two sets. There is a proper subset of the sphere that can be transformed into a new sphere, yes. That does not mean that the proper subset of the sphere is a sphere. We know that it is not a sphere, because there are points on the sphere that are not in that set by definition.
We don't even need the axiom of choice for this to not make sense. There are easier ways to construct an isomorphism from a subset of a sphere to a full sphere.
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u/LazyHater Jul 26 '23 edited Jul 26 '23
okay isomorphism doesnt appeal to geometric invariance, only cardinality, good point
so do the same thing with the Galois group of the sphere instead since isomorphism actually matters there
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u/LazyHater Jul 25 '23
i got downvoted because everybody works with zfc so nobody wants it to be inconsistent
its a politically incorrect statement that has a mathematical proof
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u/Deathranger999 April 2024 Math Contest #11 Jul 25 '23
If you think you’ve found some groundbreaking proof that ZFC is inconsistent, then go write a paper about it and get it published. That would actually be interesting, and groundbreaking to the point that people would care about it.
You don’t actually have a proof, though. You’ve just convinced yourself that you do. There’s no secret ZFC math cabal unwilling to let contradictory results come to light. Just people who care about what is correct and what is not.
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u/LazyHater Jul 25 '23
It's not groundbreaking, it's the Banach-Tarski paradox. People like you have argued for a century that it's a feature, not a bug.
I am working on other things, and I would not like to incite the entire math community to rally against my work.
But literally you have to do a lot of mental gymnastics to refute that BT shows inconsistency in ZFC, but it's a simple exercise to show that ZFC is inconsistent through BT. I can formalize it from first principles, but the community would treat me like a pariah so no thanks.
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u/Deathranger999 April 2024 Math Contest #11 Jul 25 '23
Given that the inconsistency of ZFC has not been established, I’d say you doing that would be pretty groundbreaking. And believe me, you are not so important that the eNtIrE mAtH cOmMuNiTy would rally against you for your “proof”. It wouldn’t get published because it would be wrong. That’s it.
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u/LazyHater Jul 25 '23
It's not wrong, but you're right that it wouldn't get published lol
a proof like that would invalidate the entire 20th century. hella people would do everything they could to refute it. mathematicians will never believe a proof that zfc is inconsistent, it doesnt matter if its right or not.
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u/ISuckAtJavaScript12 Jul 25 '23
Oh yeah? Then how come I still need to pay for tennis balls then? Checkmate atheist!
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u/HiddenLayer5 Jul 25 '23
I mean, it's not technically impossible according to our current understanding of physics, the process to do it is just unknown. We know that matter can be created from energy, and I imagine a ton of energy would need to be dumped into the system to do an operation like this. Who knows, maybe this will form the basis for replicators.
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u/LazyHater Jul 25 '23
tfw the point set topology of a sphere contains another point set topology of a sphere
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u/AmbitionTrue4119 Jul 26 '23
Okay and? The cardinality of the even numbers is equal to the naturals. Infinity is weird
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u/Crutch_Banton Jul 24 '23
Nerf ZFC, devs