We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 218 \(\Rightarrow\) 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 |
|---|---|
| 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 \(n\). |
| 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. |
Comment: