Implications

We have the following indirect implication of form equivalence classes:

427 \(\Rightarrow\) 414
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
427 \(\Rightarrow\) 67 clear
67 \(\Rightarrow\) 144 Axioms of multiple choice, Levy, A. 1962, Fund. Math.
144 \(\Rightarrow\) 414 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
427: \(\exists F\) AL20(\(F\)): There is a field \(F\) such that every vector space \(V\) over \(F\) has the property that every independent subset of \(V\) can be extended to a basis.  \ac{Bleicher} \cite{1964}, \ac{Rubin, H.\/Rubin, J \cite{1985, p.119, AL20}.
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.

414:

Every \(\cal W\)-frame is a \(\cal D\)-frame.  \ac{Ern\'e} \cite{2000}

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