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 190:
There is a non-trivial injective Abelian group.
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal N28\) Blass' Permutation Model | The set \(A=\{a^i_{\xi}: i\in \Bbb Z, \xi\in\aleph_1\}\) |
Code: 5
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