Axiom Of Choice

We have the following indirect implication of form equivalence classes:

66 \(\Rightarrow\) 415 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
66 \(\Rightarrow\) 67	Existence of a basis implies the axiom of choice, Blass, A. 1984a, Contemporary Mathematics
67 \(\Rightarrow\) 144	Axioms of multiple choice, Levy, A. 1962, Fund. Math.
144 \(\Rightarrow\) 415	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
66:	Every vector space over a field has a basis.
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.
415:	Every \(\cal W\)-compactly generated complete lattice is algebraic. \ac{Ern\'e} \cite{2000}.

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