Axiom Of Choice

We have the following indirect implication of form equivalence classes:

164 \(\Rightarrow\) 313 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
164 \(\Rightarrow\) 91	Dedekind-Endlichkeit und Wohlordenbarkeit, Brunner, N. 1982a, Monatsh. Math.
91 \(\Rightarrow\) 313	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
164:	Every non-well-orderable set has an infinite subset with a Dedekind finite power set.
91:	\(PW\): The power set of a well ordered set can be well ordered.
313:	\(\Bbb Z\) (the set of integers under addition) is amenable. (\(G\) is {\it amenable} if there is a finitely additive measure \(\mu\) on \(\cal P(G)\) such that \(\mu(G) = 1\) and \(\forall A\subseteq G, \forall g\in G\), \(\mu(gA)=\mu(A)\).)

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