We have the following indirect implication of form equivalence classes:
Implication | Reference |
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264 \(\Rightarrow\) 202 |
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 |
202 \(\Rightarrow\) 91 | note-75 |
91 \(\Rightarrow\) 313 | Equivalents of the Axiom of Choice II, Rubin, 1985, theorem 5.7 |
Here are the links and statements of the form equivalence classes referenced above:
Howard-Rubin Number | Statement |
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264: | \(H(C,P)\): Every connected relation \((X,R)\) contains a \(\subseteq\)-maximal partially ordered set. |
202: | \(C(LO,\infty)\): Every linearly ordered family of non-empty sets has a choice function. |
91: | \(PW\): The power set of a well ordered set can be well ordered. |
313: | \(\Bbb Z\) (the set of integers under addition) is amenable. (\(G\) is {\it amenable} if there is a finitely additive measure \(\mu\) on \(\cal P(G)\) such that \(\mu(G) = 1\) and \(\forall A\subseteq G, \forall g\in G\), \(\mu(gA)=\mu(A)\).) |
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