Axiom Of Choice

We have the following indirect implication of form equivalence classes:

66 \(\Rightarrow\) 329 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 \(\Rightarrow\) 329	clear

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).
329:	\(MC(\infty,WO)\): For every set \(M\) of well orderable sets such that for all \(x\in X\), \(\|x\|\ge 1\), there is a function \(f\) such that for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\). (See Form 67.)

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