Don't the Banach-Tarski Paradox Prove 1 = 2?

[talanum1]

The Banach-Tarski paradox proves we can decompose 1 sphere into 2. Then to me this is 1 = 2.

 

[Cassandro]

A paradox doesn't prove anything.

 

[Hafar1014]

If 10 plus 10 is 20 how come 11 plus 11 is twenty too

 

[scruffy]

The Cantor set is equally vexing.

 

[scruffy]

The Banach-Tarski paradox proves we can decompose 1 sphere into 2. Then to me this is 1 = 2.

Here is some input from the concept of "space filling curves". I'm sure you know what they are, Peano curves and such. There's one called Hilbert curve we could talk about.

For the simplest construction, consider a square of unit area. You're a bug starting at the lower right corner, and you'll walk up until you hit a boundary, at which point you'll turn in the direction that has continuity AND where you haven't been before. You'll move a distance dx in this direction, and then immediately turn again, in the opposite direction from which you started. When you can no longer satisfy the AND condition you're done, you have filled the space inside the square.

Now, consider that the bug lives in Flatland. It is one dimensional. So it can't do the dx operation because it's in a different dimension. But consider what happens if we have an uncertainty principle ("quantum fluctuations") in play in this dimension. At some point, the bug will be able to traverse dx - it just has to wait for the right conditions.

And, we can play games with the topology, we can glue two edges of the square together in such a way that they're offset by the distance dx, that way the bug can just keep going round and round without ever actually having to go sideways.