Axiom Of Choice

We have the following indirect implication of form equivalence classes:

100 \(\Rightarrow\) 29 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
100 \(\Rightarrow\) 347	Partition principles and infinite sums of cardinal numbers, Higasikawa, M. 1995, Notre Dame J. Formal Logic
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 \(\Rightarrow\) 29	Zermelo's Axiom of Choice, Moore, 1982, page 324

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 \(y\prec x\).
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:	\(C(\aleph_{0},\infty)\):
29:	If \(\|S\| = \aleph_{0}\) and \(\{A_{x}: x\in S\}\) and \(\{B_{x}: x\in S\}\) are families of pairwise disjoint sets and \(\|A_{x}\| = \|B_{x}\|\) for all \(x\in S\), then \(\|\bigcup^{}_{x\in S} A_{x}\| = \|\bigcup^{}_{x\in S} B_{x}\|\). Moore, G. [1982], p 324.

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