Implications

We have the following indirect implication of form equivalence classes:

284 \(\Rightarrow\) 423
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
284 \(\Rightarrow\) 61 note-36
61 \(\Rightarrow\) 45-n clear
45-n \(\Rightarrow\) 33-n clear
33-n \(\Rightarrow\) 47-n clear
47-n \(\Rightarrow\) 423 clear

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.

45-n:

If \(n\in\omega-\{0,1\}\), \(C(\infty,n)\): Every set of \(n\)-element sets has a choice function.

33-n:

If \(n\in\omega-\{0,1\}\), \(C(LO,n)\):  Every linearly ordered set of \(n\) element sets has  a choice function.

47-n:

If \(n\in\omega-\{0,1\}\), \(C(WO,n)\): Every well ordered collection of \(n\)-element sets has a choice function.

423:

\(\forall n\in \omega-\{o,1\}\), \(C(\aleph_0, n)\) : For every \(n\in  \omega - \{0,1\}\), every denumerable set of \(n\) element sets has a choice function.

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