Implications

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We have the following indirect implication of form equivalence classes:

3 \(\Rightarrow\) 4 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
3 \(\Rightarrow\) 4	Russell's alternative to the axiom of choice, Howard, P. 1992, Z. Math. Logik Grundlagen Math. Cardinal addition and the axiom of choice, Howard, P. 1974, Bull. Amer. Math. Soc. note-27

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

Howard-Rubin Number	Statement
3:	\(2m = m\): For all infinite cardinals \(m\), \(2m = m\).
4:	Every infinite set is the union of some disjoint family of denumerable subsets. (Denumerable means \(\cong \aleph_0\).)

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