Axiom Of Choice

We have the following indirect implication of form equivalence classes:

91 \(\Rightarrow\) 307 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
91 \(\Rightarrow\) 305	Equivalents of the Axiom of Choice II, Rubin, 1985, theorem 5.7
305 \(\Rightarrow\) 307	clear

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

Howard-Rubin Number	Statement
91:	\(PW\): The power set of a well ordered set can be well ordered.
305:	There are \(2^{\aleph_0}\) Vitali equivalence classes. (Vitali equivalence classes are equivalence classes of the real numbers under the relation \(x\equiv y\leftrightarrow(\exists q\in{\Bbb Q})(x-y=q)\).). \ac{Kanovei} \cite{1991}.
307:	If \(m\) is the cardinality of the set of Vitali equivalence classes, then \(H(m) = H(2^{\aleph_0})\), where \(H\) is Hartogs aleph function and the {\it 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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