For some reason, lately I’ve been seeing a bunch of mentions of Banach Tarski. B-T is a fascinating case of both how counter-intuitive math can be, and also how profoundly people can misunderstand things.

For those who aren’t familiar, Banach-Tarski refers to a topological/measure theory paradox. There are several variations on it, all of which are equivalent.

The simplest one is this: Suppose you have a sphere. You can take that sphere, and slice it into a *finite* number of pieces. Then you can take those pieces, and re-assemble them so that, without any gaps, you now have *two* spheres of the exact same size as the original.

Alternatively, it can be formulated so that you can take a sphere, slice it into a finite number of pieces, and then re-assemble those pieces into a *bigger* sphere.

This sure as heck *seems* wrong. It’s been cited as a reason to reject the axiom of choice, because the proof that you can do this relies on choice. It’s been cited by crackpots like EE Escultura as a reason for rejecting the theory of real numbers. And there are lots of attempts to explain why it works. For example, there’s one here that tries to explain it in terms of density. There’s a very cool visualization of it here, which tries to make sense of it by showing it in the hyperbolic plane. Personally, most of the attempts to explain it intuitively drive me crazy. One one level, intuitively, it doesn’t, and *can’t* make sense. But on another level, it’s actually pretty simple. No matter how hard you try, you’re never going to make the idea of turning a finite-sized object into a larger finite-sized object make sense. But on another level, once you think about infinite sets – well, it’s no problem.

The thing is, when you think about it carefully, it’s not really all that odd. It’s counterintuitive, but it’s not nearly as crazy as it sounds. What you need to remember is that we’re talking about a mathematical sphere – that is, an infinite collection of points in a space with a particular set of topological and measure relations.

Here’s an equivalent thing, which is a bit simpler to think about:

Take a line segment. How many points are in it? It’s infinite. So, from that infinite set, remove an infinite set of points. How many points are left? It’s still infinite. Now you’ve got two infinite sets of the same size. So, now you can use one of the sets to create the original line segment, and you can use the second one to create a second, identical line segment.

Still counterintuitive, but slightly easier.

How about this? Take the set of all natural numbers. Divide it into two sets: the set of even naturals, and the set of odd naturals. Now you have two infinite sets,

the set {0, 2, 4, 6, 8, …}, and the set {1, 3, 5, 7, 9, …}. The size of both of those sets is the ω – which is also the size of the original set you started with.

Now take the set of even numbers, and map it so that for any given value i, f(i) = i/2. Now you’ve got a copy of the set of natural numbers. Take the set of odd naturals, and map them with g(i) = (i-1)/2. Now you’ve got a *second* copy of the set of natural numbers. So you’ve created two identical copies of the set of natural numbers out of the original set of natural numbers.

The problem with Banach-Tarski is that we tend to think of it less in mathematical terms, and more in concrete terms. It’s often described as something like “You can slice up an orange, and then re-assemble it into two identical oranges“. Or “you can cut a baseball into pieces, and re-assemble it into a basketball.” Those are both obviously ridiculous. But they’re ridiculous because they violate one of our instinct that derives from the conservation of mass. You can’t turn one apple into two apples: there’s only a specific, finite amount of *stuff* in an apple, and you can’t turn it into two apples that are identical to the original.

But math doesn’t have to follow conservation of mass in that way. A sphere doesn’t *have* a mass. It’s just an uncountably infinite set of points with a particular collection of topological relationship and geometric relationships.

Going further down that path: Banach-Tarski relies deeply of the axiom of choice. The “pieces” that you cut have *non-measurable* volume. You’re “cutting” from the collection of points in the sphere in a way that requires you to make an *uncountably infinite* number of distinct “cuts” to produce each piece. It’s effectively a geometric version of “take every other real number, and put them into separate sets”. On that level, because you can’t actually *do* anything like that, it’s impossible and ridiculous. But you need to remember: we aren’t talking about apples or baseballs. We’re talking about sets. The “slices” in B-T aren’t something you can cut with a knife – they’re infinitely subdivided, not-contiguous pieces. Nothing in the real world has that property, and no real-world process has the ability to cut like that. But we’re not talking about the real world; we’re talking about abstractions. And on the level of abstractions, it’s no stranger than creating two copies of the set of real numbers.

Well, conservation of mass isn’t actually a real conservation law. It’s sort of approximate and somewhat valid for low energies.

However, an orange or a baseball is made of atoms, and you can’t divide atoms without completely changing their behavior. So you can’t divide an orange or a baseball into the infinitely number of distinct cuts that are required. I think that’s the more salient point.

Atoms, quantum mechanics and relativity are not the only physical abstractions, and you certainly doesn’t require them for Banach-Tarski not to be a problem. For example, it’s not a problem in good old continuum classical mechanics, where conservation of mass is indeed a law.

The fact is that B-T is more counter-intuitive than the set-theoretic examples (essentially Hilbert’s hotel paradox), because no notion of isometry is involved there.

Thank you, Mark. That statement is so very clarifying on a number of fronts.

“a reason for rejecting the theory of real numbers”

Really? Rejecting the axiom of choice is one thing, but rejecting

the real numbersseems a bit inconvenient, at least if you want to use math to do something. Beyond appreciating its beauty, that is. The reals may be ugly, but they sure as heck are useful.You judge too quickly. One is perfectly justified in rejecting the real numbers as we know it. For example read about non-standard numbers and non-standard analysis, they take different real numbers than what you might know.

I’m confused. In your summary you say that you can cut it up into a finite number of slices, but in the rest of the description – particularly the real integer set theory stuff – you’re actually pulling out an infinite number of things and relying on that infinite property to make the duplicate.

Likewise in the end you say that the slices are uncountable infinite.

Did you mean to say finite initially or is that a typo?

If you did mean it, does anyone know what I’m missing? I just don’t see how you could do it with a finite set.

The trick is that the pieces of the sphere have an infinitely complex structure.

When I described the trick with the line segment, I divided the set of points in that segment into two sets. So I divided the set into two “slices”.

The trick that makes B-T seem so interesting is that with three dimensions, you can divide the set of points that make up the sphere into an infinitely complex – but

still fully connectedstructure. So you divide the set of points in the sphere into six slices, each of which contains a subset of the points in the original structure. Then you can put three slices back together into one sphere, and three into a second.The weird part is that you don’t need to stretch them. But that’s an artifact. You started with a measurable volume. But then, in the process of slicing out these infinitely complex pieces, you created a set of shapes that

don’thave measurable volume. That’s the real trick: you took something with volume, and by cutting it in an infinitely complex way, you turned it into something thatdoesn’thave measurable volume.Oh! I was just confused! I was shifting where the finite applied from the number of sets to the size of those sets. But that’s not what you said at all.

The interesting thing, for me, is that while I knew about the set-theory stuff, I never applied it to other areas like geometry – even though I know you can define a sphere as an infinite set. Mathematically anyway. Just never put it together before. It’s all quite interesting! Thanks for posting it!

Thank you for this post.

I recall this result from the measure theory course I took a long time ago. I always felt like it was a ‘cop-out’ to say that non-measurable sets existed given the Axiom of Choice.

I wonder if the Axiom of Choice can be restricted in some way so that some of the more counter intuitive results are avoided and we still get things that make math work the way we expect it to. And, I think, some progress has been made along these lines.

Your comments about this very counter intuitive result is a good way to think about this.

Well, there is an axiom of dependent choice, axiom of countable choice.

One of the weird results of AC (too me, at least) is the well-ordering of reals.

This is a nice post, but I kind of disagree with the title. A paradox is not something that just can’t be true; that’s a contradiction. A paradox is a situation in which the theory predicts something puzzling and bizarre. Contemplation of paradoxes is a great way to really understand a theory — sometimes they reveal the assumptions of the theory, sometimes they show the domain of applicability, and sometimes they reveal bizarre but real phenomena. In this case, I’d say this: the Banach-Tarski paradox really shows why non-measurable sets are a problem. They’re not just awkward sets with no volume; instead processes using them as intermediates can produce profoundly non-physical transformations.

“But math doesn’t have to follow conservation of mass in that way. A sphere doesn’t have a mass. It’s just an uncountably infinite set of points with a particular collection of topological relationship and geometric relationships.”

A sphere does have mass! That’s the whole point of measure theory: to give things a size. But with the axiom of choice, you can produce sets which can not be measured and it’s out of this that the B-T paradox works.

I understand you may have been trying to simplify things so that those who don’t know measure theory can understand, but the reason why the B-T paradox is interesting is that it respects measure theory, not cardinality.

A sphere doesn’t have mass. It has volume.

But volume isn’t like mass. Volume isn’t conserved. You can take a sphere, and slice it in ways that

don’tpreserve volume. In fact, you cal slice in ways that make the whole concept of volume inapplicable. That’s the heart of B-T. Volume isn’t mass, and it doesn’t need to be conserved.Richard: a sphere doesn’t have a unique concept of mass (or volume), though we can define one on it.

MarkCC: volume is very much conserved by isometries.

Some people are confused mixing mathematics and physics. Not every mathematical concept is realizable physically.

When you teach a person to multiply by two, nobody thinks it leads to a physical contradiction because if you multiply by two the mass of a body, conservation of mass falls. They understand that multiplying by two do not represent an operation you can directly apply to a body.

But when you show people Banach-Tarski, they get blinded by complexity and forget that fact. They don’t understand that maybe the kind of cuts it requires are not physically realizable. No physical cut would leave a nonmeasurable quantity of mass you can not weight.

It’s the same situation. Multiplying by two is not an operation you can do on a physical object’s mass directly just because you can define it mathematically. It does not lead to a fall of conservation of mass. In the same way, the fact that you can define mathematically the kind of cuts required by Banach-Tarski does not mean they represent a physical operation, and they don’t lead to a paradox.

This is a common problem we already saw. Some people just think everything definable is real, and then they go crazy because you can define contradicting terms.

That didn’t really make B-T any less confusing. The strange thing isn’t that you can make a function that maps one sphere to two spheres (like you made two sets of integers out of one). It’s that it does that just by cutting it up (in a finite number of pieces!) and then moving and rotating them around.

This “paradox” was also mentioned in Feynman’s book. It’s not really a paradox at all. Just more confusion over infinity.

If space isn’t quantized then you could still maybe to a banach-tarski on space.

Provided you could manipulate peace like that.

One thing that’s confused me about B-T is that it’s often stated as being about spheres. But strictly speaking, a sphere is a surface, and B-T is about balls: Solid spheres. So I had the wrong mental model. I think that’s important, because the confusing thing is that it’s a finite number of connected pieces, (unlike the integers example). In 2d you’re very limited in the number of ways you can keep a set connected, but in a 3d ball a scattering of points on the surface can be connected in all sorts of weird ways by tendrils in bulk.

ive seen a proof that proved it for spheres first and then did balls as a corollary.

(talking about rays instead of points, and doing the center point separately)

it works just as well though.

‘Twas brillig, and the slithy toves. Did gyre and gimble in the wabe; All mimsy were the borogoves, And the mome raths outgrabe.

I think I get Alice in Wonderland now. Don’t get me wrong, I understand what you’re saying and this makes a sort of sense to me. It’s just kinda funny how deeply we’re into math territory that doesn’t have any semblance of connection with reality. Mind you, unlike Lewis Carroll I’m in the age of computers where bizarre seemingly nonsensical math can actually BE a part of reality by way of a computer program that uses those concepts.

I think this explanation of B-T leaves a little to be desired (although it’s clearly better than the ones you link– the authors of both of those are completely missing the point). What makes B-T so counterintuitive is the fact that there *is* a good notion of surface area on the sphere (or volume on the ball), and it *is* invariant under rigid motions. But you can still cut a sphere (or ball) into finitely many pieces, move those pieces around with rigid motions only, and put them back together into two spheres (or balls), each of which can be made equal to the original by a rigid motion. The important point is that the pieces into which you cut the sphere (or ball) do *not*, and in fact *cannot*, have a well-defined area (or volume) which is invariant under rigid motions. This is precisely what the paradox proves.

In the intervening time, the mathematical community has decided that this behavior is less distressing than the things that happen when one rejects the axiom of choice, which is what you would have to do in order to avoid the paradox.

Banach-Tarski is a perfectly good reason to reject the axiom of choice.

Ultimately we accept choice simply because we feel that ZFC is a more useful theory to work in than ZF or ZF+AD . However, Banach-Tarski shows that these benefits come at a cost: a less direct correspondence between R^3 and our concept of space. This isn’t a decisive reason to reject choice but it is a reason.

It’s no different than our practice of looking to properties of (non-curved) physical space in choosing the axioms of euclidean geometry. One relevant consideration in choosing our mathematical axioms is to create a close correspondence between non-mathematical concepts of interest and mathematical ones. This is as true for set theory as geometry.

I get the overall gist of the explanation mathematically, but I wonder, what does this situation say about the status of continuous space and maybe time, in particular, the idea that space is made up of points and instants? Does this mean that two objects can ultimately be composed of an equal infinity of points at a fundamental level, even though they have distinguished sizes on an approximate level?

If so, would a continuum (at least as defined here) thus be impossible at the risk of falling into such an odd result? Or, if this conception of the continuum is preserved, should we try to look at space and time in a different way (perhaps we can say that on an approximate scale, that our normal intuitions still apply, even though it does not apply at the fundamental level of points, if that makes any sense)?

I don’t have much of a background on the maths behind the paradox, but I do have some concerns about its implications for the concept of continuity.

I don’t understand your objection, William. Of course, any objects with extension in a continuum space fills infinite space points. I don’t see anything odd with this. I worry about your “equal” in “objects composed of an equal infinity of points even when they have distinguished sizes”. I believe you are comparing infinite sets in size with your common day intuition about countable objects. The “number” of “space points” is not a property of an object, and its always infinite. Equally infinite, if you want. But I may not understand your example.

Can you elaborate a bit more? If you are not sure, think about this: take an object with mass m in a continuum space, stretch it until you double its volume (obviously decreasing its density). In both states, the object occupies infinite space points. In the last state, double the volume. That’s why we invented the density property (which is not a property of space).

Do you see anything odd in this example? I’m trying to understand what do you believe it’s odd, and I think your answer may clarify it.

There is nothing odd in physical objects changing its volume, if mass is conserved. To know mass, you have to integrate density over volume. But Banach-Tarski involves (anyone correct me) nonmeassurable pieces, so mass is not defined in the middle of the process. It’s not a physical process. You don’t need to discard space, but operations that produce nonmeassurable volumes. That’s all. Banach-Tarsky is a mathematical result (not a paradox), and it’s not a problem for (even classical) physics, if you admit you can not cut a mass into pieces which you can not weight. A reasonable postulate. But thinking about the points of an object as separate entities is intuitive, but wrong.

Note I am not trying to object to the paradox. It makes sense to me mathematically. But I was wondering how this reasoning applies when we talk about physical space and time as being continuous. Of course this is beyond mathematics, but what does it mean then when we say that space and time are infinitely divisible and not discrete?

To continue, I am not so sure why you are bringing mass into this. Of course we can say that we can conserve the amount of one property in one object by making the desnity proportional to its size, but I think that is not relevant here. I am just talking about the amount of space in which an object occupies, which you agree is an infinite amount of points. If we deny atomism, then must we hold on to this idea that matter can occupy the same amount of space points (an infinite amount) at a fundamental level even if we double its volume (or as I am implying, that the amount of space points in which, for example, an orange occupies is equal to the sun’s)?

I brought density (and with it, mass) because it’s all the connection I see between space and objects. If I don’t speak about density, I see no connection between Banach-Tarski, space and objects.

Banach-Tarski speaks about space. How do you connect that with objects. You divide space into pieces, then you translate each piece to… But you can not cut and move space! All you can do physically is cut objects, and to relate it to Banach-Tarski you have to cut it into pieces you can not weight, and then don’t know what happens to the mass of the final double volume pieces. Even accepting you could cut objects into non-measurable pieces (which, of course, nobody would agree you can), all you have in the end is an object double the volume, which is not surprising. If you are not speaking of mass, there is nothing surprising, from my point of view. And if you speak about mass, you can not say how such a cut would affect density (because it’s non-measurable), so there is still no apparent physical problem.

To sum up, I brought density because it’s the only connection I see between space (and Banach-Tarski) and objects. If you are not speaking about density, how do you interpret all the cutting and moving of Banach-Tarski? What exactly do you find surprising in doubling the volume of an object, them?

And answering your question, of course, all that “amount of points” is nonsense. You know, you don’t measure space by an integer number of points. You measure it by an integral. That’s a problem well solved a long of time ago. “Number of points” is a very poor measurement for volumes.

But, again, I can’t explicitly address your doubts because I still have no idea what you find odd in all this. I’m perfectly confortable with the idea of both a classical apple and a classical Sun “filling infinite space points”, measuring their volume by an integral instead, which assumes a continuum space-time. Nothing odd to me. I see all this perfectly sensible. Where is the problem?

I am sorry, I am not very well informed about the maths behind the paradox (as I said initially), or with mathematics in general. So I hope you can bare with me.

I still don’t see why we must speak necessarily about the density of mass (or any other physical property) in an object when I am focusing more about the amount of space in which an object occupies (which I assuming is a separate issue). Doubling the volume shouldn’t necessarily be suprising, even though it is counter-intuitive but that is the paradox, but can we say the same for space and time (considering they are physically continuous?) and again, I am not trying to object to the paradox in any way, but just looking into its implications.

As for your point about the number of points being a “very poor measurement of volume” I agree (in part due to my perhaps misguided reasoning above), but I don’t know what you mean by measuring volume by “integrals”. What does it mean to measure by integrals? Is that another word for measuring by approximate intervals (such as metres), or is it something else?

Don’t worry, if there is something I say you don’t understand, it’s my fault. That’s just different backgrounds.

I spoke about density because I didn’t expected you find counter-intuitive that you can change the volume of an object. Take an ideal gas (a physical object), in which volume is inversely proportional to pressure. If you half the pressure, you double the volume. If you heat a metal, it expands. Physical objects change volume, which is not an invariant of an object. And it can be formalized very naturally.

I see very intuitive to think about objects changing volume in a continuum space. I know you see something counter-intuitive that makes you uncomfortable with the concept of a continuum space, but I still don’t have a clue about what it is.

About the measurement of volume by integrals, well, I would not go technical, I don’t believe you want me to speak about metric tensors XD But, more or less, it means what you would expect: that you measure volume by taking a cube, and seeing how many cubes you can pack inside the volume of your object. When you can no longer add a complete cube, divide the cube into (for example) eight identical subcubes, and fill the rest of the space, until you can no longer do this. Take that procedure to infinity. The volume would be n complete cubes + (m subcubes)/8 + (m’ subsubcubes)/64 + … = n’ cubes. This is the volume of your object, in your physical unit (your cube). This way, the sun takes a lot more cubes than your orange. It’s a good way to measure volumes. When you take this procedure (more or less) to math, it takes the form of an integral. But it’s not approximate, in a mathematical model of the space it can be made exact.

The only thing I find counter-intuitive is the entire situation itself (that all matter in space is composed of an “equally infinite” number of points, using your terminology, which is where the paradox arises), but that should already be expected when discussing something like the BT paradox. Again, I was just trying to see how this applies to our idea of a continuous space made of points (I still didn’t get a clear answer on whether or not the same rules apply). There isn’t anything more to it as far as I am concerned.

As for the description about integrals, I think I am starting to get the picture. So by comparing measurements by integrals, you can compare the number of cubes of a particular size (say a m^3) and subcubes that fit into an object, and differentiate them at certain levels. In this way, we can say that the sun has more cubes than an orange, is that correct? When looking at it in terms of a mathematical space I can see how it can be made exact.

I don’t believe your problem has anything to do with Banach-Tarski. Suppose Banach-Tarski were not true (for example, discard the axiom of choice). You would still say

“Does this mean that two objects can ultimately be composed of an equal infinity of points at a fundamental level”

(which, by the way, is your terminology XD). In my opinion, what is causing you trouble is your intuition that infinite is a number you can compare like you compare 3 and 7, and that this “number of points” (which is always infinite) has any meaning. The solution is accepting you can not measure volumes by number of points, and use volumes instead. This problem is completely unrelated to Banach-Tarski, and holds even if BT were false.

About the lack of a clear answer for your question, err… can you repeat the question, please?

Well, I just assumed that committing oneself to a continuum of points would also mean commiting oneself to something like the BT paradox, because my understanding of the paradox does not seem to be that different from the phrase you quoted. It is like the example above with dividing the set of natural numbers into even and odd numbers which are both identical in size. Like the BT paradox, I am assuming that this is because we are dealing with infinities (but this may be a wrong interpetation), which of course are not intuitive to think about. I have heard about the axiom of choice and its relation to BT, but have no idea what it is ( I believe I am treading further into the abstract realm of mathematics then I initially expected LOL).

So although in a continuum, every interval is ultimately made up of an infinite amount of points, but it is incorrect to compare them at this fundamental level. We can only measure them using other intervals (or volumes as you say). Does this sound right?

As for the question, again I will try to rephrase it:

How should we look at the physical continuum of space and time? Does the BT paradox apply to this conception of space and time?

I may get back to rephrasing it again, but for now, I gotta get to bed.

Yes, that sounds right to me. I would not even say that comparing sets with infinite elements by their number of elements is incorrect. I would say it’s completely useless, as you have seen. So mathematicians had to come up with useful ways to measure space, like volumes. That’s right. Once you accept that infinite is not a common number you can add, subtract and compare in the usual way, everything else will sound better Almost everyone has trouble with infinites from time to time XD.

As for the question of how should we look at the physical continuum spacetime of classical physics… it’s a somewhat big question for a post. Basically, you can see it as a four-dimensional affine space with certain additional structure (a mapping from points to time, an euclidean metric, a galilean group, etc., and observers, which introduce a connection between non-simultaneous points). That’s probably too technical and boring for this discussion. What’s important is that, as any continuum space, it’s an uncountable set, which is the word mathematicians have to remember you can not count their elements, so your naive approach of saying “it has infinite elements” is useless. But, as it has a metric, you can measure things by integration. Exactly as we’ve seen before. I can not list every treatment you can do on this kind of space, but you can read anything about affine spaces or uncountable sets.

Banach-Tarski is not directly applicable to classical physical space because it’s defined on undefined operations. What does it mean to cut or rigidly translate space? You are just defining mappings from an initial volume A to a final volume B = 2A (there is nothing surprising in that), or you have to associate these operations to physical processes. The only natural way I can think of is working with objects instead of space, but of course then BT requires unphysical operations (like cutting into non-weightable pieces), and it’s ill defined. Even if you could realize these cuts, BT say nothing about what happens to density, for example, and if the final pieces do conserve mass, there would be nothing surprising in the operation.

So, to sum up, Banach-Tarski does not speak about physical space, and to apply it to physics, you would have to associate physical operations to the operations BT uses, and there is no sensible way to do this in a physical realizable way in which the surprise remains

Okay, thanks for clearing that up. I actually conceded before that looking at the spacetime continuum in terms of points is useless (as there is no distinction between intervals), but still valid as far as I am concerned. Good to know I wasn’t wrong about that. And I agree that infinity is quite a mysterious and frustrating concept to think about.

This is the best description I’ve found online of the paradox. You put it in terms that the educated lay person can understand but still retain a good sense of the theory without over-simplifying the ideas.

I’m writing the introduction for a book I’m editing, The Nebula Showcase 2013, which is a Year’s Best anthology for the Nebula Awards given by the Science Fiction and Fantasy Writers of America (of 2011, actually; the book will be published in 2013, but the awards were given this year for stories that appeared in 2011). One of the works on the ballot is called “The Axiom of Choice” by David W. Goldman, and offers a clever use of the axiom to frame his story of a musician.

I talk about the axiom in my introduction as part of my thoughts on the story, and I’ve been looking for a good source/quote to reference for readers interested in finding out more. This post is the one I’d like to use as a reference.

I can’t hyperlink the post in the hardcopy book, so I’ll have to give the entire address. Do you by any chance have a tinyurl or similar shortening of the link? If not, I can make one, but if you already have one, I’ll use that.

Here’s a link to Goldman’s story. It was published in the New Haven Review:

http://www.newhavenreview.com/wp-content/uploads/2012/01/NHR-9-Goldman.pdf

Best,

Catherine Asaro, Editor

Nebula Awards Showcase 2013

Thank you for the praise, and I would be absolutely honored. I don’t have a short url, so please feel free to go ahead and make one.

Mark

Thanks! Here it is:

http://tinyurl.com/AxiomChoiceBlog

I’ll let you know when the book comes out (probably around May 2013).

Best,

Cat

Hello. The publisher of the book I’m editing is having me proof and verify the quotes for my introduction. So I’d like to verify this footnote about your blog article. Let me know if I’ve spelled your name correctly and if you see anything that doesn’t look right. Thanks — Catherine

1. Editor’s note: After I wrote this introduction, some of my early readers had questions about the Banach-Tarski paradox and axiom of choice. I did a web search and found a number of sites that talk about the concepts. The one I liked best was an essay in the blog, Good Math, Bad Math, written by Mark Chu-Carroll. If you’d like a look, see tinyurl.com/AxiomChoiceBlog.