Axiom Of Choice

We have the following indirect implication of form equivalence classes:

91 \(\Rightarrow\) 309 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
91 \(\Rightarrow\) 309	Equivalents of the Axiom of Choice II, Rubin, 1985, theorem 5.7

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number	Statement
91:	\(PW\): The power set of a well ordered set can be well ordered.
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 \(1\le i\le r\).

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