Axiom Of Choice

We have the following indirect implication of form equivalence classes:

133 \(\Rightarrow\) 306 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
133 \(\Rightarrow\) 90	Dedekind-Endlichkeit und Wohlordenbarkeit, Brunner, N. 1982a, Monatsh. Math.
90 \(\Rightarrow\) 91	The Axiom of Choice, Jech, 1973b, page 133
91 \(\Rightarrow\) 79	clear
79 \(\Rightarrow\) 203	clear
203 \(\Rightarrow\) 306	clear

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

Howard-Rubin Number	Statement
133:	Every set is either well orderable or has an infinite amorphous subset.
90:	\(LW\): Every linearly ordered set can be well ordered. Jech [1973b], p 133.
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.
203:	\(C\)(disjoint,\(\subseteq\Bbb R)\): Every partition of \({\cal P}(\omega)\) into non-empty subsets has a choice function.
306:	The set of Vitali equivalence classes is linearly orderable. (Vitali equivalence classes are equivalence classes of the real numbers under the relation \(x\equiv y\leftrightarrow (\exists q\in{\Bbb Q})(x-y = q)\).).

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