We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 218 \(\Rightarrow\) 61 | clear |
| 61 \(\Rightarrow\) 45-n | clear |
| 45-n \(\Rightarrow\) 33-n | clear |
| 33-n \(\Rightarrow\) 47-n | clear |
| 47-n \(\Rightarrow\) 423 | clear |
| 423 \(\Rightarrow\) 374-n | 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. |
| 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. |
| 374-n: | \(UT(\aleph_0,n,\aleph_0)\) for \(n\in\omega -\{0,1\}\): The union of a denumerable set of \(n\)-element sets is denumerable. |
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