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This must be the most unintuitive result of all of mathematics. Its very interesting what a seemingly simple axiom like the axiom of choice can lead to -- simpl

This must be the most unintuitive result of all of mathematics. Its very interesting what a seemingly simple axiom like the axiom of choice can lead to -- simple as in 'even a 9-year old can understand it', the consequences are rather enormous and not simple at all.

tsimionescu 2y ago
Honestly, it's not that surprising if you learned the properties of infinity before, especially of uncountable infinity. If 2*Inf == Inf, and if a sphere has an infinity of points, it's not that surprising that you can make two spheres from those same points. The construction itself is of course much more impressive, I'm not downplaying it, but I don't think it's less intuitive than other properties of infinity. My personal reckoning with this was learning that there are as many numbers in the [0,1] interval of the real line as on the whole real line.
- bubblyworld 2y ago
  The BT paradox includes the requirement that the pieces are separated and put back together using isometries of R3, which is _way_ more restrictive than isomorphism of sets (what you're talking about). So it's quite surprising from that point of view!
- petters 2y ago
  Really? I think this is on a completely different level of intuition. There are five pieces here that are only rotated and translated.
stared 2y ago
"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" Well, the axiom of choice gives a lot of counterintuitive examples, with the Banach-Tarski paradox being the easiest to imagine by a non-mathematician. Yet, I know no consequences that would be measurable in physics. To my knowledge, AoC is more like glue, which (paradoxically) makes quite a few things smoother, e.g., all Hilbert spaces have a basis. Otherwise one runs in a lot of theorems, in all corners of maths, with "this is always true for finite, for infinite we know that there are no counterexamples, yet we cannot prove that for all cases".
- dist-epoch 2y ago
  My understanding is the general consensus is that no physical infinity can exist. So Axiom of Choice/Banach-Tarski doesn't really apply in physics since they are only interesting when talking about infinite sets.
  - nairboon 2y ago
    In what sense could it exsist then, if infinity is not physically realizable? Does infinity even exist?
    - dist-epoch 2y ago
      > David Hilbert famously argued that infinity cannot exist in physical reality. The consequence of this statement — still under debate today — has far-reaching implications. https://www.nature.com/articles/s41567-018-0238-1 https://www.nature.com/articles/s41567-018-0238-1
  - stared 2y ago
    Well, in principle, the Universe can be infinite. Sure, we cannot measure infinity, but to be fair, all mathematical concepts (when looked at closely enough) are not something we measure directly. Even if a kindergarten-level maths of "there are three apples," we do an abstraction. We need to decide that something is a separate object, an apple (how big or small should a fruit be an apple? if there is a bite, is it an apple? etc, etc) - usually with an assumption that all apples are the same (which we know is not true, but serves as an useful approximation). pretend that
  - thechao 2y ago
    As far as we can tell, GR implies, and we have measured, space-time is completely continuous. Draw a square on a piece of paper; or, better yet, outline a cube with some sticks: within that square (or cube) is an infinite set of points of either the integral or real cardinality — whichever you’d like. The “no physical infinity” thing sounds like a very Greek sort of axiom — like their “nature abhors a vacuum” thing, etc.

tsimionescu 2y ago

Honestly, it's not that surprising if you learned the properties of infinity before, especially of uncountable infinity. If 2*Inf == Inf, and if a sphere has an infinity of points, it's not that surprising that you can make two spheres from those same points. The construction itself is of course much more impressive, I'm not downplaying it, but I don't think it's less intuitive than other properties of infinity. My personal reckoning with this was learning that there are as many numbers in the [0,1] interval of the real line as on the whole real line.

bubblyworld 2y ago
The BT paradox includes the requirement that the pieces are separated and put back together using isometries of R3, which is _way_ more restrictive than isomorphism of sets (what you're talking about). So it's quite surprising from that point of view!
petters 2y ago
Really? I think this is on a completely different level of intuition. There are five pieces here that are only rotated and translated.

stared 2y ago

"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" Well, the axiom of choice gives a lot of counterintuitive examples, with the Banach-Tarski paradox being the easiest to imagine by a non-mathematician. Yet, I know no consequences that would be measurable in physics. To my knowledge, AoC is more like glue, which (paradoxically) makes quite a few things smoother, e.g., all Hilbert spaces have a basis. Otherwise one runs in a lot of theorems, in all corners of maths, with "this is always true for finite, for infinite we know that there are no counterexamples, yet we cannot prove that for all cases".

dist-epoch 2y ago
My understanding is the general consensus is that no physical infinity can exist. So Axiom of Choice/Banach-Tarski doesn't really apply in physics since they are only interesting when talking about infinite sets.
- nairboon 2y ago
  In what sense could it exsist then, if infinity is not physically realizable? Does infinity even exist?
  - dist-epoch 2y ago
    > David Hilbert famously argued that infinity cannot exist in physical reality. The consequence of this statement — still under debate today — has far-reaching implications. https://www.nature.com/articles/s41567-018-0238-1 https://www.nature.com/articles/s41567-018-0238-1
- stared 2y ago
  Well, in principle, the Universe can be infinite. Sure, we cannot measure infinity, but to be fair, all mathematical concepts (when looked at closely enough) are not something we measure directly. Even if a kindergarten-level maths of "there are three apples," we do an abstraction. We need to decide that something is a separate object, an apple (how big or small should a fruit be an apple? if there is a bite, is it an apple? etc, etc) - usually with an assumption that all apples are the same (which we know is not true, but serves as an useful approximation). pretend that
- thechao 2y ago
  As far as we can tell, GR implies, and we have measured, space-time is completely continuous. Draw a square on a piece of paper; or, better yet, outline a cube with some sticks: within that square (or cube) is an infinite set of points of either the integral or real cardinality — whichever you’d like. The “no physical infinity” thing sounds like a very Greek sort of axiom — like their “nature abhors a vacuum” thing, etc.