We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 28-p \(\Rightarrow\) 427 | clear |
| 427 \(\Rightarrow\) 67 | clear |
| 67 \(\Rightarrow\) 112 | clear |
| 112 \(\Rightarrow\) 395 | clear |
| 395 \(\Rightarrow\) 396 | clear |
Here are the links and statements of the form equivalence classes referenced above:
| Howard-Rubin Number | Statement |
|---|---|
| 28-p: | (Where \(p\) is a prime) AL20(\(\mathbb Z_p\)): Every vector space \(V\) over \(\mathbb Z_p\) has the property that every linearly independent subset can be extended to a basis. (\(\mathbb Z_p\) is the \(p\) element field.) Rubin, H./Rubin, J. [1985], p. 119, Statement AL20 |
| 427: | \(\exists F\) AL20(\(F\)): There is a field \(F\) such that every vector space \(V\) over \(F\) has the property that every independent subset of \(V\) can be extended to a basis. \ac{Bleicher} \cite{1964}, \ac{Rubin, H.\/Rubin, J \cite{1985, p.119, AL20}. |
| 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 \(x\). |
| 396: | \(MC(LO,WO)\): For each linearly ordered family of non-empty well 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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