Axiom Of Choice

We have the following indirect implication of form equivalence classes:

100 \(\Rightarrow\) 64 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
100 \(\Rightarrow\) 9	On the existence of large sets of Dedekind cardinals, Tarski, A. 1965, Notices Amer. Math. Soc. The Axiom of Choice, Jech, 1973b, page 162 problem 11.8
9 \(\Rightarrow\) 64	The independence of various definitions of finiteness, Levy, A. 1958, Fund. Math. clear

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

Howard-Rubin Number	Statement
100:	Weak Partition Principle: For all sets \(x\) and \(y\), if \(x\precsim^* y\), then it is not the case that \(y\prec x\).
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.)

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