Implications

We have the following indirect implication of form equivalence classes:

109 \(\Rightarrow\) 373-n
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
109 \(\Rightarrow\) 66 clear
66 \(\Rightarrow\) 110 clear
110 \(\Rightarrow\) 373-n The vector space Kinna-Wagner Principle is equivalent to the axiom of choice, Keremedis, K. 2001a, Math. Logic Quart.

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.

110:

Every vector space over \(\Bbb Q\) has a basis.

373-n:

(For \(n\in\omega\), \(n\ge 2\).) \(PC(\aleph_0,n,\infty)\): Every denumerable set of \(n\)-element sets has an infinite subset with a choice function.

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