(Not an explanation of how it works, but just of how it could be true)
IIRC it follows a principle similar to the Hilbert hotel[0]. The idea is that you can split an infinity into an infinity[1] of infinite parts.
for a simplex example imagine all the points (n,m) where n,m are natural numbers, lets call this set P. we can split P into two sets:
- Q defined by all the (n,m) in P where n<m
- R defined by all the (n,m) in P where n>=m
Now you can "bend" Q by mapping (n,m) into (n, m-n-1) and R by mapping (n,m) into (n-m,m).
These bent version of Q and R are both identical to the initial P set.
This was a very informal and messy proof, but the core idea is the same: split the set (like a sphere surface) into many sets, manipulate (rotate) each one taking advantage of their infinity, recompose them as needed.
I do not think I am able to legibly comunicate the idea behind Banach-Tarski, but hopefully this gives some intuition
IIRC it follows a principle similar to the Hilbert hotel[0]. The idea is that you can split an infinity into an infinity[1] of infinite parts.
for a simplex example imagine all the points (n,m) where n,m are natural numbers, lets call this set P. we can split P into two sets:
- Q defined by all the (n,m) in P where n<m - R defined by all the (n,m) in P where n>=m
Now you can "bend" Q by mapping (n,m) into (n, m-n-1) and R by mapping (n,m) into (n-m,m).
These bent version of Q and R are both identical to the initial P set.
This was a very informal and messy proof, but the core idea is the same: split the set (like a sphere surface) into many sets, manipulate (rotate) each one taking advantage of their infinity, recompose them as needed.
I do not think I am able to legibly comunicate the idea behind Banach-Tarski, but hopefully this gives some intuition
[0] https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Gra...
[1] the idea is that if K and H are two infinities then size(H * K) = max(H, K) for reasonable definitions of multiplication and size https://en.wikipedia.org/wiki/Cardinal_number#Cardinal_multi...