Axiom Of Choice

We have the following indirect implication of form equivalence classes:

430-p \(\Rightarrow\) 173 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
430-p \(\Rightarrow\) 67	clear
67 \(\Rightarrow\) 76	clear
76 \(\Rightarrow\) 173	Paracompactness of metric spaces and the axiom of choice, Howard, P. 2000a, Math. Logic Quart.

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

Howard-Rubin Number	Statement
430-p:	(Where \(p\) is a prime) \(AL21\)\((p)\): Every vector space over \(\mathbb Z_p\) has the property that for every subspace \(S\) of \(V\), there is a subspace \(S'\) of \(V\) such that \(S \cap S' = \{ 0 \}\) and \(S \cup S'\) generates \(V\) in other words such that \(V = S \oplus S'\). Rubin, H./Rubin, J [1985], p.119, AL21.
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).
76:	\(MC_\omega(\infty,\infty)\) (\(\omega\)-MC): For every family \(X\) of pairwise disjoint non-empty sets, there is a function \(f\) such that for each \(x\in X\), f(x) is a non-empty countable subset of \(x\).
173:	\(MPL\): Metric spaces are para-Lindelöf.

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