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> AC by itself implies only vapourware. However these zombie-like objects like the voting scheme are necessary for the smooth functioning of the mathematical un

> AC by itself implies only vapourware. However these zombie-like objects like the voting scheme are necessary for the smooth functioning of the mathematical universe. Without them the universe is chaotically irregular. So, considering we just covered a bunch of reductio ad absurdum proofs, AC is false and the universe is chaotically irregular. I'm just being rhetorical, as I suspect it's more technical than that but why do people not see things that way? Whenever people talk about AC, all I hear is a bunch of reasons why ZFC is inconsistent or at the very least incoherent in some weaker sense. How is the Banach–Tarski paradox not just a refutation of the ZFC axioms taken as a whole?

lupire 4y ago
Yes, the computable, countbale universe is incompatible with ZFC. But ZFC is math, not computation or physics. ZFC is consistent with itself, which is good enough for math. You don't even need AoC to run into this. Even the "tame" smooth functions of classical physics (like f(x)=x^2) are a non-physical approximation to discrete reality.
- red_trumpet 4y ago
  Bold to assume that reality is discrete.
  - lupire 4y ago
    Even if it's not discrete at the bottom of QFT or something, it is discrete at the level where physics is applied in classical mechanics.
    - posterboy 4y ago
      Bold to assume that "it" is reality. Conversely, classical mechanics is discrete at the level where physics is applied. That it is applied to realia goes without saying. But it is not applied to reality as a whole.
      - Tainnor 4y ago
        Well, it's bold to assume anything about reality, because we only ever experience individual aspects of it. We can only try to model what we can observe. But as a counter to the assertion that "AC must be wrong because we can't really experience Banach-Tarski in the real world", it's perfectly valid to comment that "well, we also can't really observe continuous functions R -> R either". Or have you ever seen a least upper bound in real life?
  - syzarian 4y ago
    Everything appears to be discrete except possibly space-time. Is there a reason to believe space-time isn’t discrete other than using manifolds makes things easier mathematically?
    - leni536 4y ago
      There is no reason to believe that spacetime is anything but an approximation, which breaks down at Planck-length scales. We have events, and causal relationship between events. Events may not have precise spacetime coordinates.
      - syzarian 4y ago
        Events are discrete, right? Doesn’t everything then appear to be discete?
        leni536 4y ago
        Events themselves are discrete, but they might have contiguous properties, like the associated point in spacetime, assuming a the contiguous spacetime is not just an approximation.
    - lanstin 4y ago
      I believe the numbers in QFT are continuous. The electrons and what not are quantized but quantized over complex number spaces. U less one be Unless the fine structure constant or speed of light, or other apparent constants of the theories, are computable some how? Disclaimer: I can only get a few chapters into a QFT book without it all becoming meaningless to me.)
      - syzarian 4y ago
        I believe the universe is finite and so there can only be a finite number of realizations of quantum numbers. I have no real intuition in physics or training in physics so I know I could be completely wrong. It seems to me though that our experiences are such that believing the universe is discrete in all things is not completely unreasonable. Though this belief of mine might itself be unreasonable.
  - andrewla 4y ago
    If by "discrete" you mean rejecting the continuum, then I don't think it's as scary as it sounds. The rationals do not form a continuum, but they are "effectively" continuous -- between any two of them there is another. There are no "leaps" between them, and no concept of consecutive numbers. Computable numbers are similarly countable but allow the expression of numbers that can only be approximated by rationals. It's the reals that require "bold" assumptions -- numbers exist which are not possible to compute even approximately?
- Tainnor 4y ago
  > ZFC is consistent with itself, which is good enough for math. That's what we believe, but we don't know it. Nor do we have any hope of proving it in some reasonable way (via Gödel's second incompleteness theorem).
  - TuringTest 4y ago
    Math is necessarily consistent because, when we find inconsistencies, we remove them, so they are no longer valid math. Math is but a language to express your thoughts in a very precise and rational way. When you find out that two of your carefully expressed thoughts contradict each other, you tweak one or both so that they are no longer contradictory. That's why "the barber who shaves those who don't shave themselves" and "the set of all sets that don't contain themselves" are not considered valid sets anymore.
    - Tainnor 4y ago
      > Math is necessarily consistent because, when we find inconsistencies, we remove them, so they are no longer valid math. The former doesn't follow from the latter, because we may simply not yet have found any inconsistency, even though it exists. edit: Not sure if your point is just "if someone finds a contradiction in ZFC, it doesn't break maths, we'll just find a different foundation". Because I agree. But GP's point was about ZFC, a concrete set of axioms, being consistent.
tom-thistime 4y ago
I think he's talking about the mathematical 'universe,' not the universe we live in. Apparently it's not so obvious what's absurd in the mathematical universe. Alternatively, maybe you and I can define 'absurd' very solidly, then go and overturn large portions of current mathematics. But I'm guessing that's more difficult than it sounds.

lupire 4y ago

Yes, the computable, countbale universe is incompatible with ZFC. But ZFC is math, not computation or physics. ZFC is consistent with itself, which is good enough for math. You don't even need AoC to run into this. Even the "tame" smooth functions of classical physics (like f(x)=x^2) are a non-physical approximation to discrete reality.

red_trumpet 4y ago
Bold to assume that reality is discrete.
- lupire 4y ago
  Even if it's not discrete at the bottom of QFT or something, it is discrete at the level where physics is applied in classical mechanics.
  - posterboy 4y ago
    Bold to assume that "it" is reality. Conversely, classical mechanics is discrete at the level where physics is applied. That it is applied to realia goes without saying. But it is not applied to reality as a whole.
    - Tainnor 4y ago
      Well, it's bold to assume anything about reality, because we only ever experience individual aspects of it. We can only try to model what we can observe. But as a counter to the assertion that "AC must be wrong because we can't really experience Banach-Tarski in the real world", it's perfectly valid to comment that "well, we also can't really observe continuous functions R -> R either". Or have you ever seen a least upper bound in real life?
- syzarian 4y ago
  Everything appears to be discrete except possibly space-time. Is there a reason to believe space-time isn’t discrete other than using manifolds makes things easier mathematically?
  - leni536 4y ago
    There is no reason to believe that spacetime is anything but an approximation, which breaks down at Planck-length scales. We have events, and causal relationship between events. Events may not have precise spacetime coordinates.
    - syzarian 4y ago
      Events are discrete, right? Doesn’t everything then appear to be discete?
      - leni536 4y ago
        Events themselves are discrete, but they might have contiguous properties, like the associated point in spacetime, assuming a the contiguous spacetime is not just an approximation.
  - lanstin 4y ago
    I believe the numbers in QFT are continuous. The electrons and what not are quantized but quantized over complex number spaces. U less one be Unless the fine structure constant or speed of light, or other apparent constants of the theories, are computable some how? Disclaimer: I can only get a few chapters into a QFT book without it all becoming meaningless to me.)
    - syzarian 4y ago
      I believe the universe is finite and so there can only be a finite number of realizations of quantum numbers. I have no real intuition in physics or training in physics so I know I could be completely wrong. It seems to me though that our experiences are such that believing the universe is discrete in all things is not completely unreasonable. Though this belief of mine might itself be unreasonable.
- andrewla 4y ago
  If by "discrete" you mean rejecting the continuum, then I don't think it's as scary as it sounds. The rationals do not form a continuum, but they are "effectively" continuous -- between any two of them there is another. There are no "leaps" between them, and no concept of consecutive numbers. Computable numbers are similarly countable but allow the expression of numbers that can only be approximated by rationals. It's the reals that require "bold" assumptions -- numbers exist which are not possible to compute even approximately?
Tainnor 4y ago
> ZFC is consistent with itself, which is good enough for math. That's what we believe, but we don't know it. Nor do we have any hope of proving it in some reasonable way (via Gödel's second incompleteness theorem).
- TuringTest 4y ago
  Math is necessarily consistent because, when we find inconsistencies, we remove them, so they are no longer valid math. Math is but a language to express your thoughts in a very precise and rational way. When you find out that two of your carefully expressed thoughts contradict each other, you tweak one or both so that they are no longer contradictory. That's why "the barber who shaves those who don't shave themselves" and "the set of all sets that don't contain themselves" are not considered valid sets anymore.
  - Tainnor 4y ago
    > Math is necessarily consistent because, when we find inconsistencies, we remove them, so they are no longer valid math. The former doesn't follow from the latter, because we may simply not yet have found any inconsistency, even though it exists. edit: Not sure if your point is just "if someone finds a contradiction in ZFC, it doesn't break maths, we'll just find a different foundation". Because I agree. But GP's point was about ZFC, a concrete set of axioms, being consistent.

tom-thistime 4y ago

I think he's talking about the mathematical 'universe,' not the universe we live in. Apparently it's not so obvious what's absurd in the mathematical universe. Alternatively, maybe you and I can define 'absurd' very solidly, then go and overturn large portions of current mathematics. But I'm guessing that's more difficult than it sounds.