Axiom Of Choice

We have the following indirect implication of form equivalence classes:

430-p \(\Rightarrow\) 413 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
430-p \(\Rightarrow\) 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
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).
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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