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Banach-Tarski and the Paradox of Infinite Cloning
- dcminter 5y agoIf I remember rightly there'a a Feynman anecdote where he points out that as the real universe is quantised this is a purely mathematical notion. I used to riff with a friend that we were "the two members of the Banach-Tarski quartet." :)
- tsimionescu 5y agoCurrent physical models don't have the universe itself (space-time) quantized - only matter is quantized. Even the planck time and planck length only represent minimal measurable distances/durations - the maths still assume that two things can be separated by fractional multiples of these. That's not to say that physics requires infinities, but current models also don't disallow infinity. Of course, actual infinity is outside the purview of science - there is no way to differentiate between infinity and something too big/small to measure, even in principle. Apparent paradoxes related to infinity, such as Banach-Tarski, don't change this, as they also require infinite precision to realize, making them impossible to test as well - even if a sphere is indeed made up of an infinity of space-time points, and even if we could manipulate those, we wouldn't be able, in finite time, to extract the necessary infinite subsets of points to create the two spheres from one.
- rob_c 5y agoPhysics doesn't require infinities, but it's scary how well QED/QFT approximates the g-factor (https://en.m.wikipedia.org/wiki/G-factor_(physics) https://en.m.wikipedia.org/wiki/G-factor_(physics) ) for electrons given the amount of renormalization (cancelling of infinities) needed to estimate the true value in nature.
- dcminter 5y agoI didn't remember quite correctly - here it is, from the section "A Different Box of Tools" in "Surely you're joking Mr Feynman". He doesn't state it explicitly, but I think it's clear they must have been talking about Banach-Tarski: --- ...It often went like this: They would explain to me, "You've got an orange, OK? Now you cut the orange into a finite number of pieces, put it back together, and it's as big as the sun. True or false?" "No holes?" "No holes." "Impossible! There ain't no such thing." "Ha! We got him! Everybody gather around! It's So-and-so's theorem of immeasurable measure!" Just when they think they've got me, I remind them, "But you said an orange! You can't cut the orange peel any thinner than the atoms." "But we have the condition of continuity: We can keep on cutting!" "No, you said an orange, so I assumed that you meant a real orange."
- desperate 5y agoTo me this is proof that infinity is something only present in our math and not in the universe. Infinity is a nice approximation but it feels like wishful thinking that our universe or anything in it is infinite. Happy to hear disagreements tho.
- jhgb 5y ago> infinity is something only present in our math and not in the universe This is true of all mathematical objects. The number 7 doesn't exist in the universe either. It's not a physical object.
- deleted 5y ago[deleted]
- danparsonson 5y agoOP didn't argue that finite numbers are physical objects, they said that infinities are not present in the universe. For example, I could in theory hand you 7 electrons but there are not infinity electrons for me to hand to you.
- jhgb 5y agoThat sounds like a weird interpretation of "to be present in the universe" to me. Also I was under the impression that it's unknown whether the universe contains an infinite number of electrons or not.
- PeterisP 5y agoIt's certainly known that the observable universe does not contain an infinite number of electrons, as it has a finite size and finite mass. And it's rather moot to talk about the space beyond the observable universe that can never affect us or anything we can observe in any way whatsoever, so any other statements about it are inherently unfalsifiable, so all the science of physics is relevant only w.r.t. the (finite) observable universe.
- mugwumprk 5y agoI do enjoy articles like this. They are such good ways to make math compelling for laymen such as myself.
- Ericson2314 5y agoI highly recommend https://twitter.com/andrejbauer/status/1428471658088738818 https://twitter.com/andrejbauer/status/1428471658088738818 and follow ups. Yes, this stuff is fishy, and yes we can blame ZFC which is a bad formalization in comparison to what we've developed since. But the real scandal is why does our definition of geometry "leak" the underlying set theory it's built atop so much? Surely it's bad to have such a leaky abstraction in pure math! The series goes on to show that by abandoning "points" — which pull all the funny set theory stuff into geometry/topology/whatever is the topic at hand, one can still have a classical foundation — e.g. with the axiom of choice and law of excluded middle — that makes mathematicians feel at ease, but also purge this Banach–Tarski gobbledygook.
- boxfire 5y ago> But who's going to work without AC (other than crazy HoTT people)? Yeah... Those crazy HoTT people, trying to actualize the goal of putting mathematics on an actually firm foundation and removing the rest of the gobblygook handwaved into the religion of math as opposed to the pure logic it represents... Also you can use HoTT WITH AC / law of excluded middle... It's just not there by default and there are some really nice things you get without it, so it's pretty much only the lazy crutch of mathematics since forever. If you see proof via excluded middle, consider it a code smell (and recall by the Curry-Howard correspondence the proof is essentially code)
- Ericson2314 5y agoAndreij Bauer is one of those HoTT people, so this is quite tongue-in-cheek.
- creata 5y agoYou're exaggerating the significance of HoTT. Mathematics is already on a pretty firm foundation. And I somehow doubt Bauer meant any ill intent with that line...[0][1] [0]: https://en.wikipedia.org/wiki/Homotopy_type_theory#Special_Year_on_Univalent_Foundations_of_Mathematics https://en.wikipedia.org/wiki/Homotopy_type_theory#Special_Y... [1]: http://math.andrej.com/2016/10/10/five-stages-of-accepting-constructive-mathematics/ http://math.andrej.com/2016/10/10/five-stages-of-accepting-c...
- paulpauper 5y agoI don't understand the paradox. Obviously if you dissaemble or scamble something u can reararange it?
- bobthechef 5y agoWhen's the last time you came across something that you could disassemble and could then reassemble into two things identical to the first thing you disassembled? It is natural to suspect that foundational axioms are somewhere flawed.
- x3n0ph3n3 5y agoOne thing worth pointing out is that our universe operates on the integers rather than the real numbers, and the Banach-Tarski requires operating on the reals.
- Smaug123 5y agoIs that known? It's an appealing idea, but bearing in mind that general relativity is very resistant to quantisation, I'm not sure I'd be comfortable to declare it as fact.
- roywiggins 5y agoIt's like taking a bed apart and rearranging it into two beds, each identical to the original bed, without adding any more material.
- smiley1437 5y agoIs this basically the same premise as Zeno’s arrow, but with more steps?
- derbOac 5y agoI had the exact same thought -- it seems like some multidimensional version of a Zeno's paradox, with all the attendant issues.
- Viliam1234 5y agoAn interesting difference is that the Banach-Tarski trick works in 3D, but not in 2D or 1D.
- lostmsu 5y agoIn 1D there are much simpler tricks.
- Sinidir 5y agoCan someone correct me if im wrong? What i see here is a splitting of the set of points in the sphere? However the set of points in the sphere is not really the sphere. A point has no volume so no matter how many you add together you don't get something with a volume. This seems more akin to splitting the natural numbers into odd and even numbers which are all equally large. The language that i see in this article and elsewhere however is suggesting that we actually duplicated the sphere (doubled the volume). This seems incorrect.
- pacifist 5y agoA sphere in the mathematical sense is the 2D shell on the surface of the sphere you're thinking of. So, no height, no volume.
- pacifist 5y agoMy bad. Should have RTFA. We're talking about 3D here.
- BeetleB 5y ago> A point has no volume so no matter how many you add together you don't get something with a volume. You're on the right track. The Banach-Tarski paradox requires accepting that non-measurable sets[1] exist. A non-measurable set is a set with a an inspecifiable volume. Note: That's non-measurable - not 0. It means you have a quantity of something, whose volume is not 0, but it's also not any other number. Once I realized that the paradox requires it, all the WTF aspect went away. Of course - if you can accept quantities for which you cannot specify a volume, you can probably accept about anything. [1] https://en.wikipedia.org/wiki/Non-measurable_set https://en.wikipedia.org/wiki/Non-measurable_set
- ouid 5y ago"no matter how many you add together", is where this argument breaks down in ZFC. The sphere is indeed the union of all of the singletons consisting of its points, all of which are measure zero. Banach-Tarski is mainly considered "weird" because it describes a partition into so few pieces, and they are rearranged via rigid motions only. It is trivial to come up with bijections between compact finite dimensional manifolds, (https://en.wikipedia.org/wiki/Space-filling_curve https://en.wikipedia.org/wiki/Space-filling_curve). For another example of the axiom of choice wreaking havoc on the notion of measure, see https://en.wikipedia.org/wiki/Vitali_set https://en.wikipedia.org/wiki/Vitali_set .
- golemotron 5y agoSo much becomes easier to see when you see infinity not as a thing but as an algorithm.
- carnitine 5y agoAlgorithms are not things?
- golemotron 5y agoYou just restated the Banach-Tarski Paradox.
- carnitine 5y agoNo, I definitely didn’t.
- mineOther 5y agoInfinity is the axiom of paradox. Does the inclusion Infinity complete an otherwise incomplete set of axioms? It solves the halting problem for a finite Turing Machine. I don't buy the diagonalization proof as anything more than the Pythagoreom Theorom. You have infinite rows, and infinite columns. Infinity is Schrodinger's Cat. Once you check in on the state (nth row by mth column) the only thing you can say about the diagonal number is that is hasn't occurred in the rows up to that point, not beyond, nor in the columns (if n > m). Ergo, Infinity is a paradox, and only mathematical in the absurd.
- Smaug123 5y agoFrom your description, I fear you don't understand the usual diagonalisation proof that constructs an uncounted real from any attempt to count the reals. Why should "the longest" diagonal have anything to do with it?
- ishtanbul 5y agoVsauce made a great video about this https://youtu.be/s86-Z-CbaHA https://youtu.be/s86-Z-CbaHA
- karmakaze 5y agoI recall watching the video and not being surprised by its paradox. The set of starting points is uncountably infinite (R2), and since each starting point leads to a countably infinite number of L/R/U/D-rotation-ending sets, each of those L/R/U/D sets has the same cardinality as that for starting points. And so on. In the end, what I took away from this was similar to saying the interval [0.0, 0.5] has the same cardinality as [0.0, 1.0] albeit in a higher number of dimensions. It would be surprising if an uncountably infinite set in a lower dimension could fill in a higher one, but uncountably infinities in the same number of dimensions doesn't seem like a paradox that needs this sphere, rotation, and dictionaries to demonstrate. In reading the comments for the video, I got the sense that this is different and that I was missing something but couldn't come close to guessing what that was.
- KirillPanov 5y agoSpoiler: the cut line between the two apple-halves is fractal in shape, with infinite surface area and taking forever to cut at any finite cutting speed. It's sort of hilarious to see a physics site mention the Banach-Tarski paradox. It is, after all, the most obvious hole poked in the most basic working assumption used by physicists: that space and time are measured with real numbers. I've seen physicists go to pretty absurd extremes to avoid thinking about the problems this creates. Fixing it properly is not easy: simply dropping the axiom of choice leaves you unable to do useful physics. Getting back to a useful state, making all sets Lebesgue, can only be done with large cardinals: https://www.jstor.org/stable/1970696 https://www.jstor.org/stable/1970696 Large cardinals are pretty exotic even by the standards of mathematicians. In many departments they are in fact the domain of logicians. In fact, the existence of certain classes of Woodin cardinals is equivalent to the Axiom of Determinacy (AD), which is the "mathematically respectable" way of investigating logics with infinitary conjunction/disjunction. In fact, AD is precisely the Law of Excluded Middle (A or not-A) for logics with infinitely-long conjunctions. Quite odd that something so ethereal would be connected to a tangible act like cutting an apple in half.
- cupcake-unicorn 5y agoGreat video on this: https://www.youtube.com/watch?v=s86-Z-CbaHA https://www.youtube.com/watch?v=s86-Z-CbaHA
- inetsee 5y agoCan someone explain the flaw in my reasoning here? Assume I have a sphere made of pure iron. I divide the sphere into individual iron atoms. I divide this group of atoms into two groups of atoms. I take each of those groups of atoms and form them into 2 spheres. How is it that these two new spheres are not either less dense or smaller that the original sphere?
- Tomte 5y ago> I divide the sphere into individual iron atoms. You have highly restricted the act of choosing sets of points here. B-T doesn't say that any "division" results in that unintuitive outcome. Note that points are infinitesimally small and infinitely many, and atoms in your iron sphere are neither.
- hodgesrm 5y agoYour sets are finite. B-T depends on properties of infinite sets.
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- ncmncm 5y agoI was initially appalled by Banach-Tarski. Looking into it more closely, it turned out to be both trivial and not notably meaningful, like most surprising results involving uncountable infinity. Nothing that affects us involves actual infinities, so infinities are just a convenient approximation that often produces correct-enough answers. Anything infinities imply that seems crazy trivially is.
- perl4ever 5y agoUntil recently I never questioned the idea that, say, the positive integers and the odd positive integers are equivalent because they can be paired, but this cloning thing seems like something that falls out of that. And it seems like that view of infinity isn't actually necessary if Cantor style cardinality is not the last word. In the paragraph on nonstandard analysis in the Wikipedia page on infinity, it says: "The infinities in this sense are part of a hyperreal field; there is no equivalence between them as with the Cantorian transfinites. For example, if H is an infinite number in this sense, then H + H = 2H and H + 1 are distinct infinite numbers" https://en.wikipedia.org/wiki/Infinity https://en.wikipedia.org/wiki/Infinity I can't say anything precise or mathematical, but after I read the above, I have an "obvious in hindsight" feeling. If H=inf is different from H + 1, how much different is it? 1/inf or an infinitesimal amount! And an infinitesimal is not nothing. The quanta article says "You can add or subtract any finite number to infinity and the result is still the same infinity you started with" but this seems like just a dogma for non mathematicians?
- DerekL 5y ago> Until recently I never questioned the idea that, say, the positive integers and the odd positive integers are equivalent because they can be paired, but this cloning thing seems like something that falls out of that. They really aren’t connected. The first statement (the positive integers can be partitioned into two sets, each of which has the same size as the original set) follows from the usual axioms of set theory (ZF), while the Banach–Tarski paradox cannot be proven to work without the Axiom of Choice or a similar axiom.
- nwallin 5y agoIsn't that just because Hilbert's Hotel is a property of the natural numbers (well ordered) while Banach-Tarski is a property of the reals? (not well ordered without AoC) We can split the natural numbers into odd and even groups by starting with 1 and iterating on the odds, and starting with 2 and iterating on the evens. But because the reals are not well ordered, the step in Banach-Tarski where we pick an arbitrary point that hasn't already been grouped into a set is impossible. The natural numbers (and therefore Hilbert's Hotel) provide a natural way to say "whatever, just pick one" but we need to invoke the well-ordering theorem (which is equivalent to the Axiom of Choice) make the same "whatever, just pick one" statement about the reals. (and therefore Banach-Tarski)
- trhway 5y ago> How can you double the volume of an object just by decomposing and rearranging it? That part is easy - for each point on a unitary sphere move it to a point at position 2x ( ie. to a corresponding location on the sphere of the 2 units radius) - you've just doubled the volume, i.e. you've just built a 2 units radius sphere out of the points belonging to 1 unit radius sphere. Banach-Tarski of course more fun and illustrates much more than just volume.
- alkyon 5y agoAll of this is still less crazy than quantum mechanics itself. Some people may cringe, but the proof is valid anyway (not sure what in principle might be wrong with ZFC?). I wouldn't be surprised if it revealed some hidden aspects of reality we still aren't aware of.
- Nevermark 5y agoAfter reading the approach it seems like too much work! Anyone willing to critique my suggested simpler proof (which didn’t occur to me until after reading the article). TLDR; It is basically the same as proofs that all countable sets have the same cardinality. (TLDR of that: map set of positive integers x to the even numbers by doubling, and the odd numbers by doubling and subtracting 1. Take the union of even and odd and you end up with the set you started with, the positive integers x). For a circle: We can identify all the points on a circle as the points p associated with the [x,y] coordinates of the complex numbers p = e^(2.c.i.pi), where 0 <= c < 1. (And . is multiply.) If we take each of those points p and rotate it by doubling its c, we now have the same points represented by the expression p = e^(2.c.i.pi), where x <= 0 < 2. So the same number of points, but two passes around the circle, 0 <= c < 1 and 1 <= c < 2. We can move the second set of points in the x positive direction by 2 or more to avoid the overlap. We have now rearranged points of one circle into two. For the surface of a sphere: We simply divide a sphere up into points defined by a stack of circles at real-valued vertical z positions, z <= -1 <= 1. And their real [x,y] points are the real and imaginary parts of each circle e^(2.c.i.pi).circumference(z), where 0 <= c < 1, and circumference(z) is the cos(z). Again, rotate the points by doubling c, so that they are now located at c, where 0 <= c < 2. There are now two overlapping sphere surfaces. We can move the second in any direction by 2 to avoid the overlap. Similar generalizations work for including the volume. Anyone understand why this simpler proof is wrong, or why the more complex proof in the article does something better?
- Smaug123 5y agoI didn't really read your proof, because it's late and I'm tired, but there is no Banach-Tarski construction in 1 or 2 dimensions (whether or not you like the axiom of choice). The third dimension is crucial. So if your proof doesn't somehow intrinsically rely on "n >= 3", it can't be right.