Axiom Of Choice

We have the following indirect implication of form equivalence classes:

7 \(\Rightarrow\) 249 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 \(\Rightarrow\) 10	Zermelo's Axiom of Choice, Moore, 1982, 322
10 \(\Rightarrow\) 249

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.
10:	\(C(\aleph_{0},< \aleph_{0})\): Every denumerable family of non-empty finite sets has a choice function.
249:	If \(T\) is an infinite tree in which every element has exactly 2 immediate successors then \(T\) has an infinite branch.

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