Implications

We have the following indirect implication of form equivalence classes:

95-F \(\Rightarrow\) 80
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
95-F \(\Rightarrow\) 333 clear
333 \(\Rightarrow\) 88 clear
88 \(\Rightarrow\) 80 clear

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

Howard-Rubin Number Statement
95-F:

Existence of Complementary Subspaces over a Field \(F\): If \(F\) is a field, then every vector space \(V\) over \(F\) has the property that if \(S\subseteq V\) is a subspace of \(V\), then there is a subspace \(S'\subseteq V\) such that \(S\cap S'= \{0\}\) and \(S\cup S'\) generates \(V\). H. Rubin/J. Rubin [1985], pp 119ff, and Jech [1973b], p 148 prob 10.4.

333:

\(MC(\infty,\infty,\mathrm{odd})\): For every set \(X\) of  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 \(x\) and \(|f(x)|\) is odd.

88:

  \(C(\infty ,2)\):  Every family of pairs has a choice function.

80:

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

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