Implications

We have the following indirect implication of form equivalence classes:

109 \(\Rightarrow\) 80
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 \(\Rightarrow\) 111 Disasters in metric topology without choice, Keremedis, K. 2002, Comment. Math. Univ. Carolinae
111 \(\Rightarrow\) 80 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.

110:

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

111:

\(UT(WO,2,WO)\): The union of an infinite well ordered set of 2-element sets is an infinite well ordered set.

80:

\(C(\aleph_{0},2)\):  Every denumerable set of  pairs has  a  choice function.

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