Implications

We have the following indirect implication of form equivalence classes:

218 \(\Rightarrow\) 120-K
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
218 \(\Rightarrow\) 61 clear
61 \(\Rightarrow\) 46-K clear
46-K \(\Rightarrow\) 120-K clear

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

Howard-Rubin Number Statement
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.

46-K:

If \(K\) is a finite subset of \(\omega-\{0,1\}\), \(C(\infty,K)\): For every \(n\in K\), every set of \(n\)-element sets has a choice function.

120-K:

If \(K\subseteq\omega-\{0,1\}\), \(C(LO,K)\): Every linearly ordered set of non-empty sets each of whose cardinality is in \(K\) has a choice function.

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