Implications

We have the following indirect implication of form equivalence classes:

284 \(\Rightarrow\) 12
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
284 \(\Rightarrow\) 61 note-36
61 \(\Rightarrow\) 11 clear
11 \(\Rightarrow\) 12 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.

11:

A Form of Restricted Choice for Families of Finite Sets: For every infinite set \(A\), \(A\) has an infinite subset \(B\) such that for every \(n\in\omega\), \(n>0\), the set of all \(n\) element subsets of \(B\) has a choice function. De la Cruz/Di Prisco [1998b]

12:

A Form of Restricted Choice for Families of Finite Sets: For every infinite set \(A\) and every \(n\in\omega\), there is an infinite subset \(B\) of \(A\) such the set of all \(n\) element subsets of \(B\) has a choice function. De la Cruz/Di Prisco} [1998b]

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