> Mathematics need not directly model the 'real' world or match our physical intuition
There's so many other ways to demonstrate this. For example, it's possible to map the integers to the rational numbers one-to-one, which is a good definition for "as many as" when dealing with infinite sets, even though there are infinitely many rational numbers between any two rational numbers. Moreover, there are as many integer multiples of a trillion as there are integers total.
Thank you for the summary and the link. I'd also add that our physical intuition also need not match the "real world", it's a model "good enough" for evolutionary purposes that fails in myriad ways.
Axiom of choice, the well-ordering principle and Zorn's lemma are equivalent statements (any one proves the other two). But each has a very different "believability" feel to it.
The Axiom of choice has never felt completely self-evident to me. E.g.: what if you have sets where the elements are non-computable? How do you "choose" objects that cannot even be named?
Something like: "the set of all programs that cannot be proven to halt" and the like can be used to create pathological sets where the set itself obviously exists, but you cannot name any of the members.
Actually, an ever better example is: "The set of reals that are not the solution to any equation that can be written with a finite number of symbols." -- an infinite set that has no nameable members!
> the set of all programs that cannot be proven to halt
That's easy: pick the TM which is minimal according to some lexicographic ordering on the specification of TM's. That's not computable (obviously) but it's perfectly well defined.
> The set of reals that are not the solution to any equation that can be written with a finite number of symbols
Yeah, that one is harder :-)
(I would simply say "The set of numbers that cannot be described by any finite number of symbols" in order to short-circuit arguments about what constitutes an "equation" and whether or not Chaitin's constant is the solution to some equation.)
Those are just subsets of R though (though I think your last example would take more work to make rigorous, if it's even possible).
The weirdness of the axiom of choice only really comes through when you consider bigger and bigger collections of sets.
For example:
- sets indexed by the natural numbers: S_1, S_2, ...
It seems totally reasonable that you should be able to make a new set by picking something from the first set, something from the second set, etc
- sets indexed by continuous time (i.e. real numbers). Here it's a bit less 'obvious'. If I have sets S_t for _every_ time t > 0, can I really make choices 'fast' enough? What if the sets are so unstructured that I'm forced to stop and look at each set in turn to make my choice?
- sets indexed by the power set of the real numbers. If you weren't convinced that I'd struggle to pick elements of S_t for all t > 0, what if I had to make a choice for every _possible combination_ of real numbers, infinite or otherwise?
I feel like the last example demonstrates how powerful the full axiom of choice actually is.
NB - I'm a dilettante rather than an actual logician, so there may be mathematical inaccuracies here.
> Those are just subsets of R though (though I think your last example would take more work to make rigorous, if it's even possible).
You could form e.g. the non-algebraic reals, which is almost all of the reals.
> sets indexed by the power set of the real numbers. If you weren't convinced that I'd struggle to pick elements of S_t for all t > 0, what if I had to make a choice for every _possible combination_ of real numbers, infinite or otherwise?
To the extent to which you can form that indexed collection of sets in the first place, surely you can form a similarly indexed collection of elements of them the same way. How can you say you've formed a non-empty set if you can't select an element of it? If we permit ourselves to form this indexed collection "lazily", surely we can do the choice "lazily" as well. (Just my intuition about these things)
It sounds like you're adding an additional constraint though. By requesting all the members of each set to be able to be named, it seems like you're restricting the sets to be countable.
> E.g.: what if you have sets where the elements are non-computable?
I think your second set doesn't exist (exactly like the set of all sets doesn't exist) under the axiom of choice, for I can use the axiom of choice to select an element, hence giving it a name :). I took some liberty with the allowable names, of course.
> Actually, an ever better example is: "The set of reals that are not the solution to any equation that can be written with a finite number of symbols." -- an infinite set that has no nameable members!
That's not a good example; the problem you're creating is due to sloppy use of language, not any cleverness in the definition.
All real numbers, and all numbers of any other variety, can be written with a finite number of symbols. That's what it means to give something a name.
> All real numbers, and all numbers of any other variety, can be written with a finite number of symbols. That's what it means to give something a name.
Counter-intuitively, this is not true.
The vast, vast majority of real numbers cannot be named, not even in principle. Their definitions would have to be infinitely long. Or to put it another way, no matter how close two named numbers are, there is an infinite number of reals in between them. If you say, okay, sure, but some of those might be named, then pick the two closest and then there is still an infinite number of other reals in between those two!
Another way to look at it is that the amount of information (measured in bits) between any two real numbers is literally infinite. If the reals were represented with binary digits, then a sequential subset of them would have a common finite prefix, and then all possible infinite bit strings would be the suffixes!
You're using "being named" in a very unusual sense. Most people would consider you not to have named something if you're literally never allowed to stop speaking (or writing) the name. Most reals "have infinitely long names" under your definition, and there is no finite time at which you've distinguished between reals with the same initial segment of "name", so really what's the use of the naming scheme at all?
(For a more compelling argument, the equality of real numbers is uncomputable, so it must be impossible to name them. If we could name them usefully, then we could determine equality by asking which ones had different names.)
> You're using "being named" in a very unusual sense.
No, I'm not.
> Most people would consider you not to have named something if you're literally never allowed to stop speaking (or writing) the name.
What's the connection? Names do not completely describe their referent. They're names. I know someone named Ko. Given that name, what can you tell me about Ko?
Naming a number in mathematics has different rules to assigning names to objects in the physical world.
You can "name" a number such as: "a positive number, that when multiplied by itself, the result is 2".
This is a finite statement that requires only a handful of 'bits' to represent, but it exactly and uniquely identifies the square root of two. If written with decimal digits, then the square root of two would require an infinite number of digits, but it can be defined ('named') with a finite number of bits.
It turns out that this is a rare property for real numbers. The vast majority do not have a shorthand name like this, only the infinitely long description exists.
What does naming have to do with AC though? This seems completely tangential to the axiom. It’s about the nature of sets, not the semantics of how you make the selection and reference it.
AC says: if there is a set of non-empty sets y with length x, I can construct a new set z of length x by taking a member from each set in y.
How does being able to name the constituents z or any member of y matter to the logic of the axiom?
This is all in response to a comment which basically said "why do people think AC is obvious - if you can't even name all elements of the set, why would you expect to be able to choose them". (Which is handwavy, of course, and leads to the wrong intuition about uncountable-but-well-ordered sets, and depending on how you look at it it may even lead you to conclude that you can't ever choose even a single element from an uncountable set.)
Honestly, I really am trying, but as far as I can see you are using the word "named" in a sense that allows most numbers to be called "0". I really can't see how your words make sense given any other definition. Is that correct?
That is essentially correct, and we actually do call many different numbers "0" in different contexts, but it's not necessary for any two numbers to share a name this way. You can give them all unique finite names with an infinitely large alphabet if you prefer.
You might consider that in the existing Chinese writing system, 口 and 囗 are distinct symbols.
> All real numbers, and all numbers of any other variety, can be written with a finite number of symbols. That's what it means to give something a name.
This is untrue and not particularly hard to prove by contradiction.
Suppose every elements in R can be named by a finite number of symbols.
You can build a bijection between R and the names of its elements (by definition a name points to a unique element and if elements have multiple names, you can easily well order a finite number symbols by building a lexicographic order and only consider the smallest name).
Or, you can also easily build a bijection between a finite number of symbols and N. That's just an encoding like ASCII or UTF-8. Therefore, the set of names is countably infinite.
Yet R is uncountable (see Cantor's diagonal argument).
Therefore, by contradiction, there has to exist elements in R which can't be named by a finite number of symbols.
> Or, you can also easily build a bijection between a finite number of symbols and N. That's just an encoding like ASCII or UTF-8. Therefore, the set of names is countably infinite.
You've confused two different senses of "a finite number of symbols". The requirement is that it takes a finite number of symbols to write down any given number, not that it takes a finite number of symbols to write down every number at once. (Though in fact that also takes a finite number of symbols; it is the precise meaning of ℝ.)
Symbols are, in their simplest interpretation, bounded two-dimensional curves; there is no danger of running out of them before we run out of numbers.
> All real numbers, and all numbers of any other variety, can be written with a finite number of symbols.
This is false. In some sense, there exist numbers that can't be referred to. We can refer to the set of real numbers as a whole, but not some of the elements. https://en.wikipedia.org/wiki/Definable_real_number
Apparently by “finite number of symbols” he means, for example, {0 1 2 3 4 5 6 7 8 9 .} but he allows the representation of a number to contain infinitely many of them.
It's true that any real number can be written with a finite number of symbols. In fact, I can write any real number x with exactly one symbol, namely x, if I define x to mean that particular real number.
Now if you fix a particular formal language for defining real numbers, with a finite alphabet, then the language only has countably many words, hence there are reals not definable in the language. But the notion of "definability" here is not independent of the choice of formal language.
So "choosing an undefinable real number" amounts to "choosing a real number not definable in L", where L is some fixed formal language---and this isn't particularly hard to imagine; given a specific L, you can probably quite concretely construct a real number not definable in L.
I'm glad to have found this post. I discovered the Banach-Tarski theorem via Vsauce[0]. It was interesting but I couldn't get the significance of it. It either didn't seem like an unexpected result or too esoteric to appreciate.
There's phrasing in the post that could be misunderstood (later clarified) but can leave unclarity from assumed understanding of the earlier description.
> In R3, given a solid ball B of radius R, it is possible to partition B into finitely many pieces such that those pieces can be reassembled to form two solid balls B1 and B2 each of radius R.
"finitely many pieces" could be confusing because though we may be talking about 6 'pieces' those pieces have an uncountable infinity of radial line slices. It's also missing the rigid motion part which is key.
The part that didn't seem surprising (and is considered trivial) was in handling uncountable infinities of things. The 'number' of points on a line [0, 1) is the same as the number on a line [0, 2) so I wouldn't be surprised to map points from [0, 1) to [0, 2) filling the latter without 'gaps'. Similarly for areas. But what the theorem is saying is that this kind of mapping doesn't work in R1 nor R2 but does work in R3 (with rigid motions).
The part that makes B-T surprising is that the extra volume/ball can be constructed with rigid motions of those uncountably infinite sets. This is where it seems beyond me to appreciate: that one or two balls have the same uncountably number of radial line slices is considered trivial, and mapping using rigid motions is surprising.
For example, if you do the same rearrangement but instead of sets of radial lines, consider the set of points at the surface of those radial lines, we're basically working in R2. The reason why it can't be done is because if we instead of being on the surface of a sphere we're on a plane, then those same motions aren't distance preserving. Thinking geometrically doesn't seem weird: an extra dimension let's you do something you can't in lower ones.
I think the algebraic description that the post makes may be clearer to appreciate the difference, and I'll be giving it another read and more thought.
It is a remarkable theorem but you can't take a snooker ball and turn it into two snooker balls without being Paul Daniels (UK magician).
This is an excellent example of language going astray and confusing mathematic rigour with some sort of "reality". You can loosely model a "sphere in R3" with a snooker ball in errr the universe thingie which is probably R3ish or perhaps R3T1 or whatevs. Besides someone has spilt a whole pint of Guinness on the table and the balls are chipped.
Even if we get our match balls out that are not chipped, and are jolly shiney, they are still subject to things like Mr Planck's constant and the fine structure of matter.
The point is quite literally in the title - don't balls it up!
Rumor has it that Sir Isaac was investigating a mathematical praxis along such lines. For whatever reason the college isn’t willing to release those papers.
The difference between [0, 1] and B-T's shapes is that B-T's shapes have uncountably-infinite complexity of details.
This is what constructivists and finitists and "constructivism + excluded middle"-ists do not accept.
I always feel part of the confusion with Banach-Tarski is that lots of words don't use their "natural definitions", which makes the proof more surprising. People (not this article) often talking about "cutting" a sphere, which is really misleading.
This result is, in many ways, quite similar to the idea I can "cut" the integers into the odd integers and even integers (but with many more fine details).
This is still a nice article, which explains the actual result well.
"So for those of you who don't know the result, here it is in simple, non-technical terms:"
proceeds to immediately use a character that can't even be copy/pasted due to needing MathJax to render it, refuses to elaborate further
Thankfully, it ain't hard to find the Banach-Tarski theorem in actual simple, non-technical terms, so for those wondering what in tarnation that character is: it just means three-dimensional space (and is usually rendered as ℝ³ instead of 𝕽³, at least so Wikipedia claims).
Had a laugh about that too. It’s sometimes hard to tell what’s »non-technical« when you’ve spent a lot of time on something, and even then what it means to be »technical« depends a lot on the audience. Non-technical here means knows basic math but is not familiar with measure theory; it does not mean _anyone_.
I like that they added my favorite math joke in their (linked at the very top) limited audience jokes section [1] (bottom left, starts with »Two mathematicians are in a bar«), which is also a about what basic math knowledge is.
> Non-technical here means knows basic math but is not familiar with measure theory; it does not mean _anyone_.
I guess "knows basic math" is relative; I tend to draw the "basic" line at "it's required for a high school diploma or GED", but I suppose I can blame my country's education system for that bar being much lower than what mathematicians seem to assume :)
Unrelated to the article in question, but using ℝ over 𝕽 for the reals is more of a modern development. If you read older articles and textbooks, many will use 𝕽 rather than the sleeker ℝ (most in my experience, but results will be heavily affected by your cut-off for 'old').
I don't have direct evidence for my speculations, but I presume the reason [fraktur](https://en.wikipedia.org/wiki/Fraktur) was more common in mathematics back in the days is largely down to articles having to be type-set using movable type. If you insisted on using ℝ over 𝕽, you were likely to make life considerably harder for your printer (which in turn meant higher printing costs), since they would be considerably more likely to have to cast new types. As printing was modernized and movable type was replaced by more flexible printing-technologies, this pragmatic reason for preferring one glyph over the other went away. Another explanation/contributing factor is that the switch seems to have occurred in tandem with the barycenter of mathematics switching away from continental Europe and towards the US in the post-WWII period (at least if we disregard Soviet mathematics which also flourished in this period, but which was largely published in Russian). The average American would probably be less familiar with 𝕽 and other fraktur glyphs than the average German.
I did get a chuckle at that, like the running joke about how a monad is “just a monoid in the category of endofunctors”. To be fair, you could delete the “R^3” part and the explanation can still be understood.
Hell, deleting the 𝕽³ part probably would've made it clearer: "ball" already implies 3D space for me, whereas seeing that qualified with some janky-ass medieval R had me second-guessing about it being one of those math quirks involving non-euclidean space or imaginary-number dimensions or whatever else mathematicians come up with in their efforts to compute the passcode on the Gates of Hell.
Yeah I feel you - I think these things come so naturally to mathematicians that it's hard for them to conceive of the idea that people like you and I exist :)
Here is a potentially daft question that I nonetheless would appreciate if someone could answer. Is it possible to deny the axiom of choice for the purposes of measures while accepting it for vector spaces? I am wondering if you could say, "there are two kinds of sets, ones equipped with a choice function and ones without it, and measurable sets are of the latter kind."
You probably could, though you would probably have to give up or weaken other axioms as well. You would need to prevent there being any bijections between the good sets and the bad sets, because you then you could use the bijection to define a choice function on the bad set. What set theorists usually do is just distinguish between when you use the axiom of choice and when you don't.
There's one super-subtle point, though: even if you ban using the axiom of choice, you can still construct sets where it is independent of the axioms of set theory whether they are measurable. You have to add "these sets are measurable" as an axiom.
The simplest way to do it is to consider a class of sets where you know you didn't use the axiom of choice, and then declare that these sets are measurable. One common choice is the class of "projective sets". Usually set theorists impose a stronger property called determinacy, so they add to set theory the Axiom of Projective Determinacy. Then you can say "If I stick to the good sets -- projective sets -- nothing bad happens. Arbitrary choice functions take me out of the projective sets, where bad things can happen."
Not in a meaningful way I think. I mean you could weaken it to 'all finite vector spaces have a basis', but I think regular induction is enough to prove that, you don't need the axiom of choice.
After a semester of a senior math seminar with one student always bringing up "but you assumed the axiom of choice!" (mostly non-seriously), the gradstudent teaching it decided for his, and last, presentation to show this theorem. It was pretty wild as I had never heard of it before (being a physics-math person). Very cool.
(The seminar was on knot theory but I had more fun with Banach-Tarski).
This has always been my struggle with all of this "infinity" maths.
Infinity/Infinity is anything you like and many things you don't and these things always seem to boil down to dividing by infinity.
Visit each room in the hotel in turn, each one for half the total time spent going to and visiting the last. Without explanation of why we can divide infinity by infinity when we want to and not get total garbage.
Banach tarski, infinity/infinity = 2 because the numerator is exactly equaly to (infinity 2) then you cancel the infinity from top and bottom. Also (infinity 3) so clearly 2=3 and this is useful. infinity also equals infinity * infinity so 1 = infinity. As you say odd things happen when you divide by infinity and mostly they don't seem to be helpful things. They seem to be wholly invalid things. But not here?
I'm sure that's not it and I'm totally missing the point, but that point is being skipped over and handwaved away with a sneer and a muttering of "mathematical maturity" rather often.
The breakthrough for me was when someone described what it means for two sets of things to be the same size.
This is super basic, because innumerate shepherds can use it to count sheep. As your sheep go out in the morning, for each sheep, take a small rock from a pile and put it in a bag. When the sheep come back in, take a small rock from your bag and put it back on the pile. Any rocks left in your bag at the end of this? If yes, you've lost a sheep and have to go find it. If a sheep comes back and you don't have any rocks left, oops, you've got someone else's sheep, so you should probably see if any other local shepherds are missing any.
Or, how can you tell if you've got as many students in your auditorium as there are seats? You look for students without seats, or seats without students. If there are no spare students or spare seats, you have the same number of both.
In both cases, you don't actually need to know how many sheep, rocks, students or seats there are. You just know that the sets of "sheep that went out" and "rocks in a bag" and "sheep that came back" are the same size, and "students" and "seats" are the same size, because you can establish a 1-to-1 mapping between every element of each set.
This is the important result - if you can establish a 1-to-1 mapping between every element of two sets, those sets are the same size
Therefore, if you can show a 1-to-1 mapping between every element of "all integers" and every element of "all even integers", then those sets are the same size. And you can. For each x in the set of integers, you map it to 2x in the set of even integers. Every member of either set has a single equivalent member in the other.
And that's how you can make an infinite amount of space in your infinite-but-full hotel, by moving every guest from room x to room 2x.
> "And that's how you can make an infinite amount of space in your infinite-but-full hotel."
If you re-frame that as an inexhaustible resource than you can make any amount of space in it because by definition it is inexhaustible.
> "infinite-but-full..."
"If it were already full then..." It can't be full. By definition it is inexhaustible.
"but we have an infinite supply of hotel guests." So by definition we can't house them all because the supply cannot be exhausted.
"We fill this inexhaustible space with an inexhaustible quantity of stack-able objects..." It seems as nuts as saying "We apply this irresistible force to this immovable object.." The existence of one implies the other _cannot_ exist.
Pick one, reason with it. Include a second in your reasoning and... No.
A one to one mapping between two resources that we define to be inexhaustible boils down to take infinity and divide it by infinity. How can it be anything else? So yes you can get 2, or 10, or pi, or literally anything at all. Infinity - infinity is no better, division is repeated subtraction so you still get "Gerald" as the answer.
Sure two infinite things are, by some view of it, "The same size" where that size makes them not comparable. The set of points contained in a sphere in my hand has the "same size" as the set of points in the sphere we call the earth which is the same as the set of points in the universe. But so what? How does this tell us anything? We divided by infinity to define what a point is and there you have to stop or you get nonsense, even if it is convenient nonsense I would hesitate to build my house there. Maybe it doesn't make any sense to talk about size when it isn't finite. Maybe we can only really reason about it using limits for the same reason we note dividing by zero makes no sense in any way that is really useful?
If it helps, don't think about operations on infinite sets as "things that are true (or false)". Rather, think about it as "an abstract model with certain rules, that can produce interesting results if applied in careful ways."
Like √-1. Does that exist? Well, not really. But if we make up a pretend number i that we say is equal to √-1, we can perform interesting mathematical operations on it that can still produce useful results in the real world. But only if we follow a given set of rules carefully and consistently.
"All models are wrong, but some models are useful."
Infinity isn't real - at least, not in our universe. It certainly isn't a number. But it's a concept we can use to produce useful results sometimes, if we handle it carefully. But it's also a concept that can produce some weird results, even if we follow all the rules.
What I got from this post is that the weirdness here comes in, not so much from the Hilbert's Hotel phenomenon about cardinalities and sets that can be put into correspondence with their own proper subsets, but from looking more deeply at something relatively familiar and something that we ordinarily use to tame infinities: volume.
Even though there are infinitely many real numbers in [0,1], we have the idea that the unit cube [0,1]³ should have a finite volume of exactly 1, or the unit sphere { (x,y,z) | √(x²+y²+z²)≤1 } should have a finite volume of exactly 4π/3. Or indeed the unit line segment [0,1] should have a finite length of exactly 1, even though it contains infinitely many points.
This stuff has felt totally normal and appropriate in mathematics ever since Euclid: Euclid would probably agree that you can't count how many points are in a line segment, but still endorses talking about lengths of line segments (or at least ratios of lengths of different line segments).
While it feels like we know how to work with volumes, and that they're comfortably finite and well-behaved, things like Banach-Tarski suggest that if we want to have every set of points have a well-defined volume in any given number of dimensions, we're actually going to run into bad trouble. But the article suggests that there are several ways to avoid this trouble, including just saying that some weird geometric objects don't have a defined n-dimensional volume. Instead, maybe only some "nice" sets should have one?
>...we have the idea that the unit cube [0,1]³ should have a finite volume of exactly 1...
>...even though it contains infinitely many points.
There is the division by infinity.
Cut the volume in half. Still has infinite points. infinity/2 = infinity.
1 = 1/2, or 1=2 if you can cancel those.
Any set of points is some number of points 1,2, ..infinity. Points are infinitely small (division by infinity) so you'd have to infinitely many of them to have a volume other than zero. And you're back to infinity points * (1/infinity) vol of a point.
So yeah, no way I can see to have any set of points have a well defined volume because the volume of a point is a division by infinity, unless the set is finite in which case it is zero (or whatever you define a finite number divided infinitely to be - define it to be gerald if that helps? - Mathematical immaturity on display right there).
Yes, I see your point (no pun intended), but I would still maintain that this issue didn't necessarily bother mathematicians in the days before formalized set theory. The issue you mention is a reason that any definition of volume should not be based only on set cardinality, since there are those one-to-one correspondences with proper subsets (for infinite sets of points), yet volume should not be preserved by cutting something in half, or doubling something, in an ordinary way.
That is, if X is the unit line segment and Y is a pair of unit line segments joined end-to-end, we want the measure function µ to obey µ(Y) = 2µ(X), even though |X|=|Y| in set theory. And even though that's weird, formalized mathematics didn't get existentially ambitious enough to make anything truly bizarre out of it, I suppose, from Euclid all the way up until Vitali!
Yeah for me it simply doesn't. It isn't false, it just doesn't even make sense.
Define X and Y as you did but make one of the segments joined to make Y /be/ X.
Now if it is ever meaningful to have a one to one correspondence when infinity is lurking about surely the points on X correspond exactly, one to one, each with itself. Once all the points of X are accounted for, clearly half of Y remains, there is no other way. There is simply no point you can select in X that is free to correspond to the second half of Y if you select corresponding to itself in preference. You can never, ever select a point corresponding to the second half of Y if you admit preference to corresponding to itself in selection.
But X has an inexhaustible number of points to chose from by definition and so does Y. So if you start picking points at random from Y and matching them up to a previously unused point from X you can do that forever and never exhaust either of the sets of points. Thus X and Y correspond one to one and no points remain of either X or Y. And there are the same number of integers as there are integers that are even.
1=2 QED The walls are different measured heights but that doesn't matter because we've proved it. Yeah ok, but maybe it does to the poor sap who needs a useful roof?
In that case, I would say you're a finitist (which, as the name seems to suggest, is fine).
The set theory approach gives seemingly clear answers to all of these questions by talking about domains and ranges of functions, rather than talking about what is or isn't an "inexhaustible number". There are potentially different correspondences available, represented by different functions; some possible correspondences may follow the "if you select corresponding to itself in preference" pattern, while others don't. = and ≤ for cardinalities are defined using existence of certain functions between sets.
However, you don't have to believe that any infinite sets exist or that we should be allowed to quantify over them, or that we should attempt to define cardinality for infinite sets at all. Still, as Dana Scott said in a related context, "if you want more, you have to assume more".
If I asked you to divide an apple by a stone, what would your answer be? Infinity is simply not a thing that your carefully-honed real-world intuitions about division apply to. Hilbert's hotel is precisely a thought experiment to prove this: divide an infinitely large set among infinitely many people, and you can get many different results. Any true statement about infinities you construct with division is true only by coincidence, not for the usual maths reason that things are true (namely that they are the result of following some broadly-applicable rules).
There is no number called "infinity", so there's no reason to expect the rules of the arithmetic of numbers to apply to it; you have to carefully go and prove that the rules you want to apply have some meaning when you extend the language in this way, and moreover that these rules are true. For example, there are multiple meaningful ways to add infinite quantities, e.g. depending on whether you are considering the cardinals or the ordinals. Those concepts are the same in finite-land, but are different in infinite-land. Cardinals do not admit the notion of "division"; ordinals at least have the division algorithm.
Banach-Tarski from a constructivist/intuitionistic point of view:
"So it really seems that when we switch to constructive theory, it actually does something even better, it eliminates this pathology altogether, making balls behave more sensibly?"
I never understood this example used to explain it, vsauce made a video on the banach tarski theorem.
You make an infinite list of numbers between 0 and 1 chosen at random. Apparently you can make a new number that was never seen in the list before if you pick a digit from each number in the list and add one to it.
Say the list has numbers
0.36285728..
0.95825597..
0.47264112..
..
I can make a new number by taking the 3 from the first number the 5 from the second and the 2 from the third num and so on.
0.463..
I never understood
You don't take "3" from the first, you consider three, but choose something that's not "3", so the number you are constructing differs from that first number.
Then in the second place you don't take "5", you choose something that's not "5", so the number you are constructing differs from that second number.
And so on. So every time you have a list of real numbers, it cannot contain all real numbers ... you can always construct many, many, many numbers that are not in your list.
But I'm not sure how this is related to what we've been talking about. If you understand what I've said here and are still confused, maybe you can be a bit more specific. If you have not understood what I have said here, perhaps you can ask more specific questions.
Moreover, there cannot be a constructive proof using this construction, because the Banach-Tarski construction will fail if all sets are measurable (which can happen in the absence of Choice). A constructive proof would continue to succeed in such settings.
https://youtu.be/s86-Z-CbaHA?t=673 nicely visualizes it. It's an infinite number of infinitely thin filaments from the center of the sphere going outward in every direction.
* The proof uses the Axiom of Choice, so no, it's not constructive;
* The boundary of the pieces has to be unmeasurable, in that sense it's similar to Vitali's set[0];
* No, the pieces are more-or-less a "fog" of points.
* I don't know why you have "proofs" in quotation marks, and I don't know what you mean by the second half of the sentence (even if I replace "contraction" with "contradiction").
I do feel like a careful reading, or perhaps re-reading, would let you ask more specific questions that we can help you with.
> I don't know why you have "proofs" in quotation marks
Because proof by contradiction is viewed by an entire category of mathematicians as "wrong" (please make sure to note the use of quotes in case you missed them).
"Proof" by contradiction, tertium non datur, proof by the absurd, whatever you want to call it, more often than not entirely fails to produce working examples, which makes the "proof" fare less useful than the one that explains how to build an actual exemplar.
The article states that all known versions of these 'paradoxes' arise from the axiom of choice. But I believe it isn't known yet whether denying the axiom of choice categorically prevents these paradoxes and allows 'perfect' measures to exist.
I asked a mathematician about what a wacky conclusion it is. He said that whenever you allow infinity, you get results like that. It relies on uncountably-infinite division of an object, which corresponds to no real-world experience anywhere in the universe. Real objects have, you know, atoms.
We use real numbers a lot, but we are careful never to rely on their more extreme properties anywhere it would matter. In practice, in fact, we use floating-point numbers, not reals, when doing actual calculations, and use numerical analysis to stay well clear of nonsensical results. If you tried to rely on BT in a real calculation, you would find a lot of NaNs and Infs.
In the real world we have atoms, but atoms are disturbances in the wave function of the universe, which may not be discrete.
We know atoms pop in and out of existence in matter anti-matter pairs. Maybe the universe is performing Banach Tarski under the hood, to make something from nothing?
> we use floating-point numbers, not reals, when doing actual calculations
I don't believe only calculations made by computers to be "actual calculations". Humans were calculating thousands of years before computers were invented, using integer or real numbers (e.g pi).
Banach Tarski, in some sense, really can't make sense in the universe because the Universe is... discreteish, as you can't get smaller than the Planck Lenth (1.6 x 10^(-35)), and while that's very small, as far as maths is concerned that means the number of "identifiable points" in a sphere of any finite size is itself finite, not even a countable infinity.
I realise that's all very dodgy, but of course we are also bringing in quantum mechanics -- but I consider it enough to say Banach Tarski doesn't apply :)
My understanding is that Planck Length is not a limit to the smallest possible thing; it's a limit to the most precise possible measurement. So there could in fact be lots of things happening at sub-Planck Length scales, but we could never observe them.
All programs manipulate symbols. All bits are is symbols.
It's just sometimes we use those symbols to represent a subset of the integers (e..g. with the popular binary notation)... or a subset of the rationals (e.g. with the popular floating point notation)... or a subset of the reals that happens to include a transcendental number because we decided that some symbol (or combination of symbols) represents some particular transedental number.
That is a symbolic manipulation. Wherever it comes down to actual numbers, you use adequate approximations to infinite summations for x and iy. Even nominally exact rational values are often idealizations of measurements: your house has no actual right angles, but eh, close enough.
e and pi arise in axiomatic systems we use to approximate our world. They never appear in nature. They do appear in formulas we find to approximate details of our world.
I say "actual numbers" to mean "numbers that refer to actual quantities or measures that can be taken". You might calculate that a stick must be exactly 1/pi meters long, but you will make the stick no better than 113/355 meters long.
I probably confused matters with my undefined expression "actual numbers".
e and pi are as actual as i or 0.
I think Turing introduced "computable numbers" which are a lot like the reals, but countable. You can write a program that produces each. So it includes integers, rationals, polynomic irrationals, and lots of transcendentals. But the set any particular person (or computer) operates on over the course of their existence is not just countable, but finite and not very large. You have to have expressed a representation of each in it at least once. We can only express a small number of numbers over the course of a life. Computers can do more of them. You might claim all those that programs you wrote have expressed.
And, of course, the number has to have finite expression. Representing pi as a summation is finite. But presenting a numerical computation involving it has to be an approximation if it ever to finish.
How is "The circumferance of an idealized circle divided by its diameter" not a finite expression of π? Saying something cannot be expressed finitely in an integer-based numeral system, and saying that it admits no finite representation are two radically different statements.
Despite it being a non-starter from a pragmatic standpoint, we could for instance easily imagine a novel numeral type that encodes the set S = {a + b·π where a and b are integers} (we can encode integers quite easily and all we need to reposesent such a number in silico is to encode a and b). Using such a numeral type, we are able to do exact arithmetic if our operations are restricted to addition and subtraction (and if we are content with fractional representation of numbers as being considered "exact", we can also do division and multiplication although we would have to work within the larger set S' = { (a + b·π) / (c + d·π) where, a, b, c, and d are integers and c·d ≠ 0} rather than within S).
I don't think you have to be a Platonist to allow that numbers that do not exactly correspond to humanly measurable quantities can yet be part of a calculation.
Another example would be Binet's formula for Fibonacci n[1], which though it involves Phi, sqrt(5), etc., always evaluates to an integer, given integer n.
The existence of these irrationals in Platonic heaven doesn't follow from the formula, which afaik depends only on the usual rules of algebra and logic, though I admit I'm not sure what constructivists make of this type of calculation.
It feels to me like you're redefining what a number is to be very different to what anyone with a maths background would say a number is. Essentially you're saying that neither e nor pi are numbers?
The only numbers that may exist in nature are the non-negative rationals. Negative numbers are widely useful, but they don't exist in any meaningful way: they're an abstraction over reality. Irrational numbers definitely don't exist, and don't get me started on so-called "imaginary" numbers -- they're no less present in nature than the negative numbers are.
But the concepts are useful, and the symbols have meaning. The article mentions normal words having different meanings in mathematical language, and that impacts even every-day use of maths.
. . . and then I start thinking about that damn Bailey–Borwein–Plouffe formula.
Although ( handwave ) all the transcendentalness is magically manipulated away leaving a simple computation for the immediate reveal of any arbitrary n-th (hexadecimal) digit of π
You have an abstract definition of a sequence of digits, and an algorithm that reveals an actual digit from among them. It is surprising that you don't need to approximate an infinite series to get it, but everybody agrees that digit would show up there if you did one. You would still need to do an infinite amount of work to get the rest of them.
It is magic of a kind by wizards of a kind, or anyway indistinguishable from it.
Just to be clear, for any third party spectators, the surprise isn't that more work is required to get more digits .. the WTF moment for some is that no matter how large N is there's no need to do an increasing amount of work as N grows (there's no cost to "skipping to (not quite) the end").
It's another of those intuition challenging moments in math.
Atoms move according to path integrals which would be at least countably infinite in number even if time and space were discrete because they are allowed to loop back on themselves.
The main reason that we use real numbers is because we need real numbers to do calculus, at least in the normal way. And to do physics, we need to do calculus. You can keep approximating integrals as sums of very large number of terms, but it gets unwieldy to handle symbolically.
Real numbers almost always give the same result as more cumbersome methods, and we are mostly only interested in such results, so it is surprising when something wacky shows up to remind us of the tightrope we walk.
> It relies on uncountably-infinite division of an object
But the theorem claims finite division and not infinite?
In R³, given a solid ball B of radius R
it is possible to partition B into finitely many
pieces such that those pieces can be reassembled
to form two solid balls B1 and B2 each of
radius R
Finitely many pieces, but on infinitely variable boundaries. It was clever to make the proof allow a finite number in that place. Without, it would have attracted no attention.
- The 'problem' with B-T may be related to our notion of "space" (i.e. point-set topology) rather than any issues with the axiom of choice
- Most practical uses of the axiom of choice could get by with the axiom of countable choice, under which B-T doesn't hold
- Mathematics need not directly model the 'real' world or match our physical intuition
[0] https://mathoverflow.net/questions/260057/axiom-of-choice-ba...