Questions tagged [banach-tarski]
The banach-tarski tag has no usage guidance.
10
questions
3
votes
0
answers
178
views
Reverse-mathematical strength of Banach-Tarski
What is the reverse mathematical strength of the Banach-Tarski paradox?
The usual proof of Banach-Tarski should carry out in $\mathrm{ZF}+\mathrm{AC}_\kappa$, where $\kappa$ is the supremum of the ...
66
votes
9
answers
14k
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 ...
3
votes
0
answers
137
views
Can the Banach-Tarski paradox or Tarski's circle-squaring problem be done with hinges?
It is known for both the Banach-Tarski paradox and Tarski's circle-squaring problem that you can finitely partition the starting configuration, then continuously move these pieces (without ...
2
votes
1
answer
226
views
Paradoxical decomposition modulo finite sets
Suppose a group $G$ acts on an infinite set $X$ and $X$ has no non-empty $G$-paradoxical subsets. Is it possible for $X$ to have non-trivial $G$-paradoxical subsets modulo finite sets? I.e., can there ...
2
votes
0
answers
144
views
Are there sets in the unit cube that cannot be in the domain of any finitely-additive, isometry-invariant probability measure?
The Vitali construction implies (given choice) the existence of a set such that for any translation-invariant, countably additive probability measure on $[0,1]$, that set is nonmeasurable and has ...
38
votes
1
answer
3k
views
Is there any version of the Banach-Tarski paradox in ZF?
The Banach-Tarski paradox states that for a solid ball in 3‑dimensional space, there exists a decomposition into a finite number of disjoint subsets, which can then be put back together in a different ...
2
votes
0
answers
131
views
About Tarski number 7
Recall that a group $G$ admits a $(m+n)$-paradoxical decomposition if there exist positive integers $m$ and $n$, a partition $\{P_1,\dotsc,P_m,Q_1,\dotsc,Q_n\}$ of $G$ and elements $x_i, y_j$ of $G$ ...
8
votes
1
answer
937
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 ...
4
votes
1
answer
238
views
Latest progress on Tarski numbers
Two questions, the first: What is the smallest non negative integer that we do not know yet is the Tarski number of a group?
The second question is the same as in the title: What is the latest ...
8
votes
1
answer
235
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 ...