Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
114 \(\Rightarrow\) 90	Products of compact spaces in the least permutation model, Brunner, N. 1985a, Z. Math. Logik Grundlagen Math.
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
114:	Every A-bounded \(T_2\) topological space is weakly Loeb. (\(A\)-bounded means amorphous subsets are relatively compact. Weakly Loeb means the set of non-empty closed subsets has a multiple choice function.)
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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