We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 28-p \(\Rightarrow\) 427 | clear |
| 427 \(\Rightarrow\) 67 | clear |
| 67 \(\Rightarrow\) 89 |
On cardinals and their successors, Jech, T. 1966a, Bull. Acad. Polon. Sci. S'er. Sci. Math. Astronom. Phys. |
| 89 \(\Rightarrow\) 90 | The Axiom of Choice, Jech, 1973b, page 133 |
| 90 \(\Rightarrow\) 91 | The Axiom of Choice, Jech, 1973b, page 133 |
| 91 \(\Rightarrow\) 79 | clear |
| 79 \(\Rightarrow\) 70 | clear |
| 70 \(\Rightarrow\) 206 | 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). |
| 89: | Antichain Principle: Every partially ordered set has a maximal antichain. Jech [1973b], p 133. |
| 90: | \(LW\): Every linearly ordered set can be well ordered. Jech [1973b], p 133. |
| 91: | \(PW\): The power set of a well ordered set can be well ordered. |
| 79: | \({\Bbb R}\) can be well ordered. Hilbert [1900], p 263. |
| 70: | There is a non-trivial ultrafilter on \(\omega\). Jech [1973b], prob 5.24. |
| 206: | The existence of a non-principal ultrafilter: There exists an infinite set \(X\) and a non-principal ultrafilter on \(X\). |
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