# Why can't Banach–Tarski make me a rich bastard? **URL:** \ **Category:** Factual Questions\ **Created:** [June 18, 2006, 3:17pm UTC](https://boards.straightdope.com/t/why-cant-banach-tarski-make-me-a-rich-bastard/361179 "2006-06-18T15:17:55Z")\ **Posts on this page:** 17\ **Page:** 2
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [June 19, 2006, 1:25pm UTC](https://boards.straightdope.com/t/why-cant-banach-tarski-make-me-a-rich-bastard/361179/21 "2006-06-19T13:25:40Z")
> [@Mathochist](#): > > Of course. I also know what’s purple and commutes, what’s green and very far away, and what’s grey and ignored by Grothendieck. I give. What’s green and very far away, and what’s grey and ignored by Grothendieck? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [June 19, 2006, 1:37pm UTC](https://boards.straightdope.com/t/why-cant-banach-tarski-make-me-a-rich-bastard/361179/22 "2006-06-19T13:37:03Z")
> [@ianzin](#): > > Agreed. It’s sometimes right and it’s sometimes wrong. The point is whether it’s right or wrong _in this specific instance_. I say it’s right, and that this sphere dissection thing can’t be done. Would be perfectly happy for a demonstration to be arranged to show me I’m wrong. As has been pointed out, there’s no possibility of actually doing this. There are good reasons to accept the axiom of choice, though–you can see some discussion in [this very old thread](http://boards.straightdope.com/sdmb/showthread.php?t=97465). Since that thread, I’ve learned that one of my undergrad professors doesn’t like the axiom of choice (I believe because of his work in orthonormal polynomials, but it’s been long enough that I’m not certain about that), but he’s definitely in the minority. ---
**Author:** ![Wendell\_Wagner](https://avatars.discourse-cdn.com/v4/letter/w/8491ac/32.png) [@Wendell\_Wagner](https://boards.straightdope.com/u/Wendell_Wagner)\ **Post date:** [June 19, 2006, 3:09pm UTC](https://boards.straightdope.com/t/why-cant-banach-tarski-make-me-a-rich-bastard/361179/23 "2006-06-19T15:09:16Z")
By Googling, I discovered that it’s the lime at infinity that’s green and very far away. I haven’t been able to figure out what’s gray and ignored by Grothendieck. I haven’t been able to guess at it either. I’ve tried looking at some articles about Grothendieck, but none of them talk about something he ignored. I believe in jokes of this sort when you say that something is gray, you’re usually looking for a pun on “elephant”, just as when you say that something is purple, you’re usually looking for a pun on “grape”, but I can’t think of any mathematical term that puns on “elephant”. ---
**Author:** ![MikeS](https://avatars.discourse-cdn.com/v4/letter/m/919ad9/32.png) [@MikeS](https://boards.straightdope.com/u/MikeS)\ **Post date:** [June 19, 2006, 3:09pm UTC](https://boards.straightdope.com/t/why-cant-banach-tarski-make-me-a-rich-bastard/361179/24 "2006-06-19T15:09:53Z")
> [@ultrafilter](#): > > I give. What’s green and very far away, and what’s grey and ignored by Grothendieck? Google will get you the first one, but **Mathochist** is going to have to enlighten us on the second. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [June 19, 2006, 5:21pm UTC](https://boards.straightdope.com/t/why-cant-banach-tarski-make-me-a-rich-bastard/361179/25 "2006-06-19T17:21:04Z")
> [@MikeS](#): > > Google will get you the first one, but **Mathochist** is going to have to enlighten us on the second. The irrelephant ideal. Incidentally for those who don’t want to Google, the other answer is the lime at infinity. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [June 19, 2006, 5:27pm UTC](https://boards.straightdope.com/t/why-cant-banach-tarski-make-me-a-rich-bastard/361179/26 "2006-06-19T17:27:02Z")
> [@psychonaut](#): > > I’d settle for an approximation using a finite number of cuts. 😉 Ah, but any finite number of cuts can only give you a measurable set, and as has been pointed out the pieces are generally not measurable. In fact, at least one of the pieces (two, if I recall) is a Cantor dust. That is, it’s topologically equivalent to [the Cantor set](http://en.wikipedia.org/wiki/Cantor_set). ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [June 19, 2006, 5:31pm UTC](https://boards.straightdope.com/t/why-cant-banach-tarski-make-me-a-rich-bastard/361179/27 "2006-06-19T17:31:15Z")
> [@ultrafilter](#): > > As has been pointed out, there’s no possibility of actually doing this. There are good reasons to accept the axiom of choice, though–you can see some discussion in [this very old thread](http://boards.straightdope.com/sdmb/showthread.php?t=97465). Since that thread, I’ve learned that one of my undergrad professors doesn’t like the axiom of choice (I believe because of his work in orthonormal polynomials, but it’s been long enough that I’m not certain about that), but he’s definitely in the minority. Well, again the rule of thumb is “AC (or equivalent) is _really_ good in algebra and _really_ weird in analysis”. It’s also far from clear why it should hold. Still, I like my solution of having different topoi obeying different logics, and letting the mathematician in question choose which to use and develop the mathematics of. It’s analogous to the situation in geometry, where the parallel postulate can be accepted or rejected and different valid geometries spring from each choice. Then we just leave it to the physicists and philosophers which option obtains in the “real world”, and get back to doing some math. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [June 19, 2006, 6:39pm UTC](https://boards.straightdope.com/t/why-cant-banach-tarski-make-me-a-rich-bastard/361179/28 "2006-06-19T18:39:11Z")
> [@](#): > > and every sequence of increasing elements is bounded above. Of course… I had forgotten that sequences were allowed to be infinite. Is the bound required to be absolute? Because if so, I’m having a hard time thinking of any candidate spaces at all. And in case anyone is wondering, Abelian grapes are purple and commute, and Zorn’s Lemmon is yellow and equivalent to the A. of C. I knew those two, had to Google for the lime at infinity (which is incidentally very rare, even on Google), and had no clue about the irrelephant ideal. > [@](#): > > Then we just leave it to the physicists and philosophers which option obtains in the “real world”, and get back to doing some math. Sounds like the punchline to the old joke about having both a wife and a mistress ;). ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [June 19, 2006, 6:46pm UTC](https://boards.straightdope.com/t/why-cant-banach-tarski-make-me-a-rich-bastard/361179/29 "2006-06-19T18:46:17Z")
From the thread I linked to: > [@december](#): > > What’s the equivalent of the A of C and requests water for his school? Anyone know this one? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [June 19, 2006, 6:49pm UTC](https://boards.straightdope.com/t/why-cant-banach-tarski-make-me-a-rich-bastard/361179/30 "2006-06-19T18:49:10Z")
The well-ordering principal. Can’t believe it took me five years to get that. :smack: ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [June 19, 2006, 7:12pm UTC](https://boards.straightdope.com/t/why-cant-banach-tarski-make-me-a-rich-bastard/361179/31 "2006-06-19T19:12:28Z")
> [@Chronos](#): > > Of course… I had forgotten that sequences were allowed to be infinite. Is the bound required to be absolute? Because if so, I’m having a hard time thinking of any candidate spaces at all. “Spaces”. Are you thinking topological spaces? That’s far too much structure. One place it comes up a lot is in algebraic geometry. Consider the set of ideals of a Noetherian commutative unital ring R containing a fixed ideal I, partially-ordered by inclusion. The Noetherian condition is equivalent to stating that any increasing set of ideals terminates, and obviously the top element is an upper bound. Thus (by Zorn’s Lemma) there exists at least one maximal ideal P of R containing I. There are all sorts of similar applications in that field. > [@](#): > > Sounds like the punchline to the old joke about having both a wife and a mistress ;). Of course. We’ve got certain obligations to physics, and we screw around with philosophy for fun. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [June 19, 2006, 7:14pm UTC](https://boards.straightdope.com/t/why-cant-banach-tarski-make-me-a-rich-bastard/361179/32 "2006-06-19T19:14:44Z")
> [@ultrafilter](#): > > From the thread I linked to: > > Anyone know this one? The Well-Ordering Principal. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [June 19, 2006, 8:27pm UTC](https://boards.straightdope.com/t/why-cant-banach-tarski-make-me-a-rich-bastard/361179/33 "2006-06-19T20:27:31Z")
> [@](#): > > “Spaces”. Are you thinking topological spaces? That’s far too much structure. No, I’m thinking “Big ol’ set that you do a bunch of interesting stuff inside of”. Forgive me if I’m not more specific, but I’m a physicist, not a mathematician. And the punchline I was thinking of was “It’s better to have both a wife and a mistress, because that way, your wife will think you’re with your mistress, and your mistress will think you’re with your wife, and you can stay up at the office and do math.”. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [June 19, 2006, 9:04pm UTC](https://boards.straightdope.com/t/why-cant-banach-tarski-make-me-a-rich-bastard/361179/34 "2006-06-19T21:04:49Z")
> [@Chronos](#): > > And the punchline I was thinking of was “It’s better to have both a wife and a mistress, because that way, your wife will think you’re with your mistress, and your mistress will think you’re with your wife, and you can stay up at the office and do math.”. I know. I was drawing an analogy and running with it. When you distract the physicists with philosophy and the philosophers with physics you can be left alone to do some math. ---
**Author:** ![Wendell\_Wagner](https://avatars.discourse-cdn.com/v4/letter/w/8491ac/32.png) [@Wendell\_Wagner](https://boards.straightdope.com/u/Wendell_Wagner)\ **Post date:** [June 20, 2006, 3:11am UTC](https://boards.straightdope.com/t/why-cant-banach-tarski-make-me-a-rich-bastard/361179/35 "2006-06-20T03:11:47Z")
At work today, I asked another mathematician who knows a little more about the sort of thing that Grothendieck does what the answer to the riddle could be. He said that Grothendieck did geometry in a very abstract fashion. It’s almost as though you could forget about everything you learned by reading Euclid. So I came up with the following: Q.: What’s gray and ignored by Grothendieck? A.: _Euclid’s Elephants_. I also came up with the following: Q.: What wades in the river and vanishes only on a single line? A.: Riemann’s hippotamus. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [June 20, 2006, 7:22am UTC](https://boards.straightdope.com/t/why-cant-banach-tarski-make-me-a-rich-bastard/361179/36 "2006-06-20T07:22:09Z")
> [@Wendell Wagner](#): > > Grothendieck did geometry in a very abstract fashion. It’s almost as though you could forget about everything you learned by reading Euclid. Well, Grothendieck worked in _algebraic_ geometry, which is (or at least was before Grothendieck) more about using geometric techniques to solve systems of polynomial equations. Euclid never really came into it that much. ---
**Author:** ![naita](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/naita/32/5862_2.png) [@naita](https://boards.straightdope.com/u/naita)\ **Post date:** [June 20, 2006, 10:31am UTC](https://boards.straightdope.com/t/why-cant-banach-tarski-make-me-a-rich-bastard/361179/37 "2006-06-20T10:31:36Z")
> [@ianzin](#): > > What next? A math theorem that proves you can lick your own elbows? Considering the number of people who _can_ lick their own elbows, as evidenced by the all the photos Brainiac got when they made the statement that no one can, that theorem wouldn’t have to be very complex. Or counter intuitive. [Previous page](https://boards.straightdope.com/t/why-cant-banach-tarski-make-me-a-rich-bastard/361179.md?page=1)