Axiom Of Choice

We have the following indirect implication of form equivalence classes:

2 \(\Rightarrow\) 75 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
2 \(\Rightarrow\) 3	On successors in cardinal arithmetic, Truss, J. K. 1973c, Fund. Math.
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
4 \(\Rightarrow\) 405	clear
405 \(\Rightarrow\) 75	clear

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

Howard-Rubin Number	Statement
2:	Existence of successor cardinals: For every cardinal \(m\) there is a cardinal \(n\) such that \(m < n\) and \((\forall p < n)(p \le m)\).
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 \(\cong \aleph_0\).)
405:	Every infinite set can be partitioned into sets each of which is countable and has at least two elements.
75:	If a set has at least two elements, then it can be partitioned into well orderable subsets, each of which has at least two elements.

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