Hypothesis: HR 309:
The Banach-Tarski Paradox: There are three finite partitions \(\{P_1,\ldots\), \(P_n\}\), \(\{Q_1,\ldots,Q_r\}\) and \(\{S_1,\ldots,S_n, T_1,\ldots,T_r\}\) of \(B^3 = \{x\in {\Bbb R}^3 : |x| \le 1\}\) such that \(P_i\) is congruent to \(S_i\) for \(1\le i\le n\) and \(Q_i\) is congruent to \(T_i\) for \(1\le i\le r\).
Conclusion: HR 237:
The order of any group is divisible by the order of any of its subgroups, (i.e., if \(H\) is a subgroup of \(G\) then there is a set \(A\) such that \(|H\times A| = |G|\).)
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal N32\) Hickman's Model III | This is a variation of \(\cal N1\) |
Code: 3
Comments: