Implications

We have the following indirect implication of form equivalence classes:

335-n \(\Rightarrow\) 330
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 \(\Rightarrow\) 328 clear
328 \(\Rightarrow\) 330 clear

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).

328:

\(MC(WO,\infty)\): For  every well ordered set \(X\) such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that and for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\). (See Form 67.)

330:

\(MC(WO,WO)\): For every well ordered set \(X\) of well orderable 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 \(x\).  (See Form 67.)

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