# Questions tagged [banach-tarski]

The banach-tarski tag has no usage guidance.

3
questions

**51**

votes

**10**answers

8k views

### Axiom of choice, Banach-Tarski and reality

The following is not a proper mathematical question but more of a metamathematical one. I hope it is nonetheless appropriate for this site.
One of the non-obvious consequences of the axiom of choice ...

**7**

votes

**1**answer

592 views

### Equivalence of the Banach-Tarski paradox

I am working on the Banach-Tarski paradox and the fact that the Hahn-Banach theorem implies that paradox. The proof involves the equivalence of the Hahn-Banach theorem and the fact that for every ...

**8**

votes

**1**answer

222 views

### Paradoxical spherical caps

All spherical caps (i.e. sets $C_L:=\{(x,y,z)|x^2+y^2+z^2=1, z\geq L\}$) admit a paradoxical decomposition in the sense of Banach-Tarski, meaning $C_L \tilde{} 2C_L$; here $\tilde{}$ stands for the ...