We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 109 \(\Rightarrow\) 66 | clear |
| 66 \(\Rightarrow\) 67 |
Existence of a basis implies the axiom of choice, Blass, A. 1984a, Contemporary Mathematics |
| 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 \(\Rightarrow\) 51 |
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 |
| 51 \(\Rightarrow\) 208 |
Choice and cofinal well-ordered subsets, Morris, D.B. 1969, Notices Amer. Math. Soc. |
| 208 \(\Rightarrow\) 58 | 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. |
| 66: | Every vector space over a field has a basis. |
| 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. |
| 51: | Cofinality Principle: Every linear ordering has a cofinal sub well ordering. Sierpi\'nski [1918], p 117. |
| 208: | For all ordinals \(\alpha\), \(\aleph_{\alpha+1}\le 2^{\aleph_\alpha}\). |
| 58: |
There is an ordinal \(\alpha\) such that \(\aleph(2^{\aleph_{\alpha }})\neq\aleph_{\alpha +1}\). (\(\aleph(2^{\aleph_{\alpha}})\) is Hartogs' aleph, the least \(\aleph\) not \(\le 2^{\aleph _{\alpha}}\).) |
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