Axiom Of Choice

We have the following indirect implication of form equivalence classes:

335-n \(\Rightarrow\) 276 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
335-n \(\Rightarrow\) 333	Bases for vector spaces over the two element field and the axiom of choice, Keremedis, K. 1996a, Proc. Amer. Math. Soc.
333 \(\Rightarrow\) 88	clear
88 \(\Rightarrow\) 276

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

Howard-Rubin Number	Statement
335-n:	Every quotient group of an Abelian group each of whose elements has order \(\le n\) has a set of representatives.
333:	\(MC(\infty,\infty,\mathrm{odd})\): For every set \(X\) of sets such that for all \(x\in X\), \(\|x\|\ge 1\), there is a function \(f\) such that for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\) and \(\|f(x)\|\) is odd.
88:	\(C(\infty ,2)\): Every family of pairs has a choice function.
276:	\(E(V'',III)\): For every set \(A\), \({\cal P}(A)\) is Dedekind finite if and only if \(A = \emptyset\) or \(2\|{\cal P}(A)\| > \|{\cal P}(A)\|\). \ac{Howard/Spi\u siak} \cite{1994}.

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