Axiom Of Choice

We have the following indirect implication of form equivalence classes:

335-n \(\Rightarrow\) 413 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 \(\Rightarrow\) 67	clear
67 \(\Rightarrow\) 144	Axioms of multiple choice, Levy, A. 1962, Fund. Math.
144 \(\Rightarrow\) 413	Constructive order theory, Ern'e, M. 2001, Math. Logic Quart.

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.
67:	\(MC(\infty,\infty)\) \((MC)\), The Axiom of Multiple Choice: For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite).
144:	Every set is almost well orderable.
413:	Every infinite set \(S\) is the union of a set, well-ordered by inclusion, of subsets which are non-equipollent to \(S\).

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