Axiom Of Choice

We have the following indirect implication of form equivalence classes:

430-p \(\Rightarrow\) 82 given by the following sequence of implications, with a reference to its direct proof:

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).
126:	\(MC(\aleph_0,\infty)\), Countable axiom of multiple choice: For every denumerable set \(X\) of non-empty sets there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\).
82:	\(E(I,III)\) (Howard/Yorke [1989]): If \(X\) is infinite then \(\cal P(X)\) is Dedekind infinite. (\(X\) is finite \(\Leftrightarrow {\cal P}(X)\) is Dedekind finite.)

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