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What I mean is: if you imagine someone drawing up a requirements document for the team assigned to the task of axiomatising geometry, and somebody asked "Do we

What I mean is: if you imagine someone drawing up a requirements document for the team assigned to the task of axiomatising geometry, and somebody asked "Do we want our model of geometry to support cutting up a ball into five pieces, moving the pieces rigidly, and reassembling them into two copies?", I think their first idea would be to answer "no". So it isn't parallel to the intermediate value theorem, but opposite to it.

thaumasiotes 5y ago
The whole idea of a proof system is that there are some things you can't have without also having other things. The Banach-Tarski theorem is a consequence of things we want. You don't get to pick and choose everything at once.
- ncallaway 5y ago
  > The Banach-Tarski theorem is a consequence of things we want Is it? I think the parent comment is saying: “maybe we shouldn’t want things that result in Banach-Tarski” Maybe it’s a hint that the underlying axioms we’ve selected aren’t exactly what we want. You’re right that we can’t pick and choose the results of our axioms, but we do explicitly get to pick and choose the axioms we start with. If we choose bad axioms, we get nonsensical results. In general, it seems like we’ve picked _pretty good_ axioms that mostly give us sensible and useful results. But maybe this result that seems somewhat… odd, is an indication that those axioms have an odd corner somewhere.
  - thaumasiotes 5y ago
    > But maybe this result that seems somewhat… odd, is an indication that those axioms have an odd corner somewhere. The only way you're going to avoid getting results like this is with axioms like "there is no such thing as an infinite number". At that point, the real line doesn't exist (too many points) and it becomes impossible to duplicate spheres by dividing them at a level of fineness that also doesn't exist. But that's not a productive approach to anything.
    - mjw1007 5y ago
      I was taught that dropping the axiom of choice was enough to make Banach-Tarski go away. That seems considerably short of "there is no such thing as an infinite number". But the Twitter link at the top of this thread seems to have a rather more interesting way of doing so.
      - thaumasiotes 5y ago
        > I was taught that dropping the axiom of choice was enough to make Banach-Tarski go away. That seems considerably short of "there is no such thing as an infinite number". The Banach-Tarski theorem is not the only theorem out there that bothers some people. Anything to do with infinities gets a large number of outraged rejections.
    - deleted 5y ago
      [deleted]
  - bigbillheck 5y ago
    > Is it? I think the parent comment is saying: “maybe we shouldn’t want things that result in Banach-Tarski” OK, which axiom do you want to replace?
    - Smaug123 5y ago
      It's not about the axioms, which are perfectly fine as set theory. It's about the encoding of geometry in set theory, for which there are many options, some with nicer properties than others. Cf the original link in the thread, https://twitter.com/andrejbauer/status/1428471658088738818 https://twitter.com/andrejbauer/status/1428471658088738818, which argues that you should model locales rather than points (and I guess if your foundation is set theory then you should use set theory to model those locales, though it sounds rather painful to me, a layman).
    - ncallaway 5y ago
      Me? I have no idea. I was just trying to catch and diffuse what seemed like a miscommunication between two people. Actually selecting and proposing an axiom set is way outside my knowledge-base. My limited understanding is that the Axiom of Choice, in ZFC leads to Banach-Tarski, and if it’s removed Tarski doesn’t hold, but I don’t have nearly enough information to say if that’s worth exploring removing it.
      - thaumasiotes 5y ago
        The Axiom of Choice is one of the most controversial things in mathematics. There are many people who choose to work without it. I'll give a statement of it here: The Cartesian product of non-empty sets is itself non-empty. The Cartesian product of sets S_1, S_2, S_3, ... is of course the set of tuples (s_1, s_2, s_3, ...) such that s_1 ∈ S_1, s_2 ∈ S_2, s_3 ∈ S_3, ... . An element of the Cartesian product is a tuple with one element drawn from each of the sets being, um, Cartesianly multiplied. Thus, the Cartesian product of the three sets {1, 4}, {a, b}, and {@, 2} is the set {(1,a,@), (1,a,2), (1,b,@), (1,b,2), (4,a,@), (4,a,2), (4,b,@), (4,b,2)}. The Cartesian product of the three sets {1, 4}, {}, and {@, 2} is {}, the empty set, because no tuples exist such that the second element of the tuple belongs to the set {} (the second Cartesian factor). So all the Axiom of Choice asserts is that, if all of the Cartesian factors are nonempty, then a tuple exists with one element drawn from each of the Cartesian factors. The only way for it to be impossible for such a tuple to exist is if one of the factors itself has no elements. It's a theorem for finite Cartesian products, so all the dispute is over infinite products. It's probably also worth mentioning that the C in ZFC stands for the Axiom of Choice, which is an indicator that people have explored not using it. ZFC without the Axiom of Choice is ZF.
- Ericson2314 5y ago
  Did you read the original link? We don't want to use topological spaces, we want to use locales! We can get rid of the paradox while sacrificing very little.
pfortuny 5y ago
The thingis exactly that in the B-T paradox you are NOT “cutting”, you are “choosing” non-measurable subsets and reorganizing them. Thus, the operation of “cutting” is not taking place (there is no continuous function whose zeros gives you any of the subsets).

thaumasiotes 5y ago

The whole idea of a proof system is that there are some things you can't have without also having other things. The Banach-Tarski theorem is a consequence of things we want. You don't get to pick and choose everything at once.

ncallaway 5y ago
> The Banach-Tarski theorem is a consequence of things we want Is it? I think the parent comment is saying: “maybe we shouldn’t want things that result in Banach-Tarski” Maybe it’s a hint that the underlying axioms we’ve selected aren’t exactly what we want. You’re right that we can’t pick and choose the results of our axioms, but we do explicitly get to pick and choose the axioms we start with. If we choose bad axioms, we get nonsensical results. In general, it seems like we’ve picked _pretty good_ axioms that mostly give us sensible and useful results. But maybe this result that seems somewhat… odd, is an indication that those axioms have an odd corner somewhere.
- thaumasiotes 5y ago
  > But maybe this result that seems somewhat… odd, is an indication that those axioms have an odd corner somewhere. The only way you're going to avoid getting results like this is with axioms like "there is no such thing as an infinite number". At that point, the real line doesn't exist (too many points) and it becomes impossible to duplicate spheres by dividing them at a level of fineness that also doesn't exist. But that's not a productive approach to anything.
  - mjw1007 5y ago
    I was taught that dropping the axiom of choice was enough to make Banach-Tarski go away. That seems considerably short of "there is no such thing as an infinite number". But the Twitter link at the top of this thread seems to have a rather more interesting way of doing so.
    - thaumasiotes 5y ago
      > I was taught that dropping the axiom of choice was enough to make Banach-Tarski go away. That seems considerably short of "there is no such thing as an infinite number". The Banach-Tarski theorem is not the only theorem out there that bothers some people. Anything to do with infinities gets a large number of outraged rejections.
  - deleted 5y ago
    [deleted]
- bigbillheck 5y ago
  > Is it? I think the parent comment is saying: “maybe we shouldn’t want things that result in Banach-Tarski” OK, which axiom do you want to replace?
  - Smaug123 5y ago
    It's not about the axioms, which are perfectly fine as set theory. It's about the encoding of geometry in set theory, for which there are many options, some with nicer properties than others. Cf the original link in the thread, https://twitter.com/andrejbauer/status/1428471658088738818 https://twitter.com/andrejbauer/status/1428471658088738818, which argues that you should model locales rather than points (and I guess if your foundation is set theory then you should use set theory to model those locales, though it sounds rather painful to me, a layman).
  - ncallaway 5y ago
    Me? I have no idea. I was just trying to catch and diffuse what seemed like a miscommunication between two people. Actually selecting and proposing an axiom set is way outside my knowledge-base. My limited understanding is that the Axiom of Choice, in ZFC leads to Banach-Tarski, and if it’s removed Tarski doesn’t hold, but I don’t have nearly enough information to say if that’s worth exploring removing it.
    - thaumasiotes 5y ago
      The Axiom of Choice is one of the most controversial things in mathematics. There are many people who choose to work without it. I'll give a statement of it here: The Cartesian product of non-empty sets is itself non-empty. The Cartesian product of sets S_1, S_2, S_3, ... is of course the set of tuples (s_1, s_2, s_3, ...) such that s_1 ∈ S_1, s_2 ∈ S_2, s_3 ∈ S_3, ... . An element of the Cartesian product is a tuple with one element drawn from each of the sets being, um, Cartesianly multiplied. Thus, the Cartesian product of the three sets {1, 4}, {a, b}, and {@, 2} is the set {(1,a,@), (1,a,2), (1,b,@), (1,b,2), (4,a,@), (4,a,2), (4,b,@), (4,b,2)}. The Cartesian product of the three sets {1, 4}, {}, and {@, 2} is {}, the empty set, because no tuples exist such that the second element of the tuple belongs to the set {} (the second Cartesian factor). So all the Axiom of Choice asserts is that, if all of the Cartesian factors are nonempty, then a tuple exists with one element drawn from each of the Cartesian factors. The only way for it to be impossible for such a tuple to exist is if one of the factors itself has no elements. It's a theorem for finite Cartesian products, so all the dispute is over infinite products. It's probably also worth mentioning that the C in ZFC stands for the Axiom of Choice, which is an indicator that people have explored not using it. ZFC without the Axiom of Choice is ZF.
Ericson2314 5y ago
Did you read the original link? We don't want to use topological spaces, we want to use locales! We can get rid of the paradox while sacrificing very little.

pfortuny 5y ago

The thingis exactly that in the B-T paradox you are NOT “cutting”, you are “choosing” non-measurable subsets and reorganizing them. Thus, the operation of “cutting” is not taking place (there is no continuous function whose zeros gives you any of the subsets).