We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 109 \(\Rightarrow\) 218 | Equivalents of the Axiom of Choice II, Rubin/Rubin, 1985, page 120 |
| 218 \(\Rightarrow\) 61 | clear |
| 61 \(\Rightarrow\) 45-n | clear |
| 45-n \(\Rightarrow\) 33-n | clear |
| 33-n \(\Rightarrow\) 47-n | clear |
| 47-n \(\Rightarrow\) 288-n | clear |
Here are the links and statements of the form equivalence classes referenced above:
| Howard-Rubin Number | Statement |
|---|---|
| 109: | Every field \(F\) and every vector space \(V\) over \(F\) has the property that each linearly independent set \(A\subseteq V\) can be extended to a basis. H.Rubin/J.~Rubin [1985], pp 119ff. |
| 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. |
| 288-n: | If \(n\in\omega-\{0,1\}\), \(C(\aleph_0,n)\): Every denumerable set of \(n\)-element sets has a choice function. |
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