Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
112 \(\Rightarrow\) 90	Equivalents of the Axiom of Choice II, Rubin/Rubin, 1985, page 79
90 \(\Rightarrow\) 91	The Axiom of Choice, Jech, 1973b, page 133
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
112:	\(MC(\infty,LO)\): For every family \(X\) of non-empty sets each of which can be linearly ordered there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\).
90:	\(LW\): Every linearly ordered set can be well ordered. Jech [1973b], p 133.
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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