Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
7 \(\Rightarrow\) 9	On the existence of large sets of Dedekind cardinals, Tarski, A. 1965, Notices Amer. Math. Soc. The Axiom of Choice, Jech, 1973b, page 161 problem 11.6
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
7:	There is no infinite decreasing sequence of cardinals.
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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