Implications

We have the following indirect implication of form equivalence classes:

284 \(\Rightarrow\) 93
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
284 \(\Rightarrow\) 61 note-36
61 \(\Rightarrow\) 88 clear
88 \(\Rightarrow\) 93 The Axiom of Choice, Jech, 1973b, page 7 problem 10

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

Howard-Rubin Number Statement
284:

A system of linear equations over a field \(F\) has a solution in \(F\) if and only if every finite sub-system has a solution in \(F\).

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.

93:

There is a non-measurable subset of \({\Bbb R}\).

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