Implications

We have the following indirect implication of form equivalence classes:

109 \(\Rightarrow\) 276
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
109 \(\Rightarrow\) 218 Equivalents of the Axiom of Choice II, Rubin/Rubin, 1985, page 120
218 \(\Rightarrow\) 61 clear
61 \(\Rightarrow\) 88 clear
88 \(\Rightarrow\) 276

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.

218:

\((\forall n\in\omega - \{0\}) MC(\infty,\infty \), relatively prime to \(n\)): \(\forall n\in\omega -\{0\}\), if \(X\) is a set of non-empty sets, then  there  is  a function \(f\) such that for all \(x\in X\), \(f(x)\) is a non-empty, finite subset of \(x\) and \(|f(x)|\) is relatively prime to \(n\).

61:

\((\forall n\in\omega, n\ge 2\))\((C(\infty,n))\): For each \(n\in\omega\), \(n\ge 2\), every set of \(n\) element  sets has a choice function.

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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