Axiom Of Choice

We have the following indirect implication of form equivalence classes:

152 \(\Rightarrow\) 390 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
152 \(\Rightarrow\) 4	Russell's alternative to the axiom of choice, Howard, P. 1992, Z. Math. Logik Grundlagen Math. note-27 note-27 note-27
4 \(\Rightarrow\) 9	clear
9 \(\Rightarrow\) 64	The independence of various definitions of finiteness, Levy, A. 1958, Fund. Math. clear
64 \(\Rightarrow\) 390	clear

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

Howard-Rubin Number	Statement
152:	\(D_{\aleph_{0}}\): Every non-well-orderable set is the union of a pairwise disjoint, well orderable family of denumerable sets. (See note 27 for \(D_{\kappa}\), \(\kappa\) a well ordered cardinal.)
4:	Every infinite set is the union of some disjoint family of denumerable subsets. (Denumerable means \(\cong \aleph_0\).)
9:	Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite.
64:	\(E(I,Ia)\) There are no amorphous sets. (Equivalently, every infinite set is the union of two disjoint infinite sets.)
390:	Every infinite set can be partitioned either into two infinite sets or infinitely many sets, each of which has at least two elements. Ash [1983].

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