We have the following indirect implication of form equivalence classes:
Implication | Reference |
---|---|
95-F \(\Rightarrow\) 333 | clear |
333 \(\Rightarrow\) 88 | clear |
88 \(\Rightarrow\) 276 |
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. |
276: | \(E(V'',III)\): For every set \(A\), \({\cal P}(A)\) is Dedekind finite if and only if \(A = \emptyset\) or \(2|{\cal P}(A)| > |{\cal P}(A)|\). \ac{Howard/Spi\u siak} \cite{1994}. |
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