= 27d567bc2d48d9f40e9ccc20ea370b15)
JGWeissman comments on Explained: Gödel's theorem and the Banach-Tarski Paradox - Less Wrong Discussion
You are viewing a comment permalink. View the original post to see all comments and the full post content.
= 783df68a0f980790206b9ea87794c5b6)
Loading…= b715d02e85f7b3b4ae65ca3d3e450868)
Subscribe to RSS Feed
Powered by Reddit
= f037147d6e6c911a85753b9abdedda8d)
You are viewing a comment permalink. View the original post to see all comments and the full post content.
Comments (40)
The explanation of Banach-Tarski misses the point, which is that by using only volume-preserving transformations (no stretching) on subsets of the sphere, you can rearrange it into two spheres of the same size, provided you use the axiom of choice to define the subsets (which turn out to be immeasurable and escape the volume-preserving property of the transformations.) Here is a good explanation.
How can both of these be true? Either the transformations always preserve volume or they don't.
Roughly speaking the problem is that mathematicians cannot come up with a meaningful definition of volume that applies to all sets of points (when I say cannot, I mean literally impossible, not just that they tried really hard then gave up). Instead, we have a definition that applies to a very large collection of sets of points, but not all of them.
Sets from that collection have a well defined volume, and any transformation which always leaves this unchanged is called volume preserving.
Sets from outside it, which the sets in the Banach Tarski paradox are, don't have a defined volume at all, and thus can interact with volume-preserving transformations in all sorts of weird ways.
I like this explanation put together by students and faculty at the University of Copenhagen.