Axiom Of Choice

We have the following indirect implication of form equivalence classes:

347 \(\Rightarrow\) 10 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
347 \(\Rightarrow\) 40	Partition principles and infinite sums of cardinal numbers, Higasikawa, M. 1995, Notre Dame J. Formal Logic
40 \(\Rightarrow\) 39	clear
39 \(\Rightarrow\) 8	clear
8 \(\Rightarrow\) 9	Was sind und was sollen die Zollen?, Dedekind, [1888]
9 \(\Rightarrow\) 10	Zermelo's Axiom of Choice, Moore, 1982, 322

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

Howard-Rubin Number	Statement
347:	Idemmultiple Partition Principle: If \(y\) is idemmultiple (\(2\times y\approx y\)) and \(x\precsim ^* y\), then \(x\precsim y\).
40:	\(C(WO,\infty)\): Every well orderable set of non-empty sets has a choice function. Moore, G. [1982], p 325.
39:	\(C(\aleph_{1},\infty)\): Every set \(A\) of non-empty sets such that \(\vert A\vert = \aleph_{1}\) has a choice function. Moore, G. [1982], p. 202.
8:	\(C(\aleph_{0},\infty)\):
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.

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