Implications

We have the following indirect implication of form equivalence classes:

218 \(\Rightarrow\) 397
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
218 \(\Rightarrow\) 67 clear
67 \(\Rightarrow\) 112 clear
112 \(\Rightarrow\) 395 clear
395 \(\Rightarrow\) 397 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 \(n\).

67:

\(MC(\infty,\infty)\) \((MC)\), The Axiom of Multiple Choice: For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite).

112:

\(MC(\infty,LO)\): For every family \(X\) of non-empty sets each of which can be linearly ordered there is a function \(f\) such that for all \(y\in X\), \(f(y)\) is a non-empty finite subset of \(y\).

395:

\(MC(LO,LO)\): For each linearly ordered family of non-empty linearly orderable sets \(X\), there is a function \(f\) such that for all \(x\in X\) \(f(x)\) is a non-empty, finite subset of \(x\).

397:

\(MC(WO,LO)\): For each well ordered family of non-empty linearly orderable sets \(X\), there is a function \(f\) such that for all \(x\in X\) \(f(x)\) is a non-empty, finite subset of \(x\).

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