Implications

We have the following indirect implication of form equivalence classes:

309 \(\Rightarrow\) 93
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
309 \(\Rightarrow\) 93 clear

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

Howard-Rubin Number Statement
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 \(1\le i\le r\).

93:

There is a non-measurable subset of \({\Bbb R}\).

Comment:

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