Implications

We have the following indirect implication of form equivalence classes:

20 \(\Rightarrow\) 324
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
20 \(\Rightarrow\) 101 Partition principles and infinite sums of cardinal numbers, Higasikawa, M. 1995, Notre Dame J. Formal Logic
101 \(\Rightarrow\) 40 On some weak forms of the axiom of choice in set theory, Pelc, A. 1978, Bull. Acad. Polon. Sci. S'er. Sci. Math. Astronom. Phys.
40 \(\Rightarrow\) 165 clear
165 \(\Rightarrow\) 324 clear

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

Howard-Rubin Number Statement
20:

If \(\{A_{x}: x \in S \}\) 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 [1982] (1.4.12 and 1.7.8).

101:

Partition Principle:  If \(S\) is a partition of \(M\), then \(S \precsim M\).

40:

\(C(WO,\infty)\):  Every well orderable set of non-empty sets has a choice function. Moore, G. [1982], p 325.

165:

\(C(WO,WO)\):  Every well ordered family of non-empty, well orderable sets has a choice function.

324:

\(KW(WO,WO)\), The Kinna-Wagner Selection Principle for a well ordered family of well orderable sets: For every well ordered set \(M\) of well orderable sets, there is a function \(f\) such that for all \(A\in M\), if \(|A| > 1\) then \(\emptyset\neq f(A)\subsetneq A\). (See Form 15.)

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