[thaumasiotes]

> But maybe this result that seems somewhat… odd, is an indication that those axioms have an odd corner somewhere.

The only way you're going to avoid getting results like this is with axioms like "there is no such thing as an infinite number". At that point, the real line doesn't exist (too many points) and it becomes impossible to duplicate spheres by dividing them at a level of fineness that also doesn't exist.

But that's not a productive approach to anything.

[mjw1007]

I was taught that dropping the axiom of choice was enough to make Banach-Tarski go away. That seems considerably short of "there is no such thing as an infinite number".

But the Twitter link at the top of this thread seems to have a rather more interesting way of doing so.

[thaumasiotes]

> I was taught that dropping the axiom of choice was enough to make Banach-Tarski go away. That seems considerably short of "there is no such thing as an infinite number".

The Banach-Tarski theorem is not the only theorem out there that bothers some people. Anything to do with infinities gets a large number of outraged rejections.