Axiom Of Choice

We have the following indirect implication of form equivalence classes:

109 \(\Rightarrow\) 396 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
109 \(\Rightarrow\) 66	clear
66 \(\Rightarrow\) 67	Existence of a basis implies the axiom of choice, Blass, A. 1984a, Contemporary Mathematics
67 \(\Rightarrow\) 112	clear
112 \(\Rightarrow\) 395	clear
395 \(\Rightarrow\) 396	clear

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

Howard-Rubin Number	Statement
109:	Every field \(F\) and every vector space \(V\) over \(F\) has the property that each linearly independent set \(A\subseteq V\) can be extended to a basis. H.Rubin/J.~Rubin [1985], pp 119ff.
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).
112:	\(MC(\infty,LO)\): For every family \(X\) of non-empty sets each of which can be linearly ordered there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\).
395:	\(MC(LO,LO)\): For each linearly ordered family of non-empty linearly orderable sets \(X\), there is a function \(f\) such that for all \(x\in X\) \(f(x)\) is a non-empty, finite subset of \(x\).
396:	\(MC(LO,WO)\): For each linearly ordered family of non-empty well orderable sets \(X\), there is a function \(f\) such that for all \(x\in X\) \(f(x)\) is a non-empty, finite subset of \(x\).

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