We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 335-n \(\Rightarrow\) 333 |
Bases for vector spaces over the two element field and the axiom of choice, Keremedis, K. 1996a, Proc. Amer. Math. Soc. |
| 333 \(\Rightarrow\) 67 | clear |
| 67 \(\Rightarrow\) 89 |
On cardinals and their successors, Jech, T. 1966a, Bull. Acad. Polon. Sci. S'er. Sci. Math. Astronom. Phys. |
| 89 \(\Rightarrow\) 254 |
Variations of Zorn's lemma, principles of cofinality, and Hausdorff's maximal principle, Part I and II, Harper, J. 1976, Notre Dame J. Formal Logic |
Here are the links and statements of the form equivalence classes referenced above:
| Howard-Rubin Number | Statement |
|---|---|
| 335-n: | Every quotient group of an Abelian group each of whose elements has order \(\le n\) has a set of representatives. |
| 333: | \(MC(\infty,\infty,\mathrm{odd})\): For every set \(X\) of sets such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\) and \(|f(x)|\) is odd. |
| 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. |
| 254: | \(Z(D,R,l)\): Every directed relation \((P,R)\) in which ramified subsets have least upper bounds, has a maximal element. |
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