Axiom Of Choice

We have the following indirect implication of form equivalence classes:

264 \(\Rightarrow\) 252 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
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 \(\Rightarrow\) 79	clear
79 \(\Rightarrow\) 252	consequence of the axiom of choice, Ash, C. J. 1975, J. Austral. Math. Soc. Ser. A.

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number	Statement
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.
79:	\({\Bbb R}\) can be well ordered. Hilbert [1900], p 263.
252:	The additive groups of \(\Bbb Q\oplus\Bbb R\) and \(\Bbb R\) are isomorphic.

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