Implications

We have the following indirect implication of form equivalence classes:

181 \(\Rightarrow\) 209
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
181 \(\Rightarrow\) 8 clear
8 \(\Rightarrow\) 24 clear
24 \(\Rightarrow\) 26 Zermelo's Axiom of Choice, Moore, 1982, 66
Le¸cons sur la th´eorie des fonctions, Borel, [1898]
26 \(\Rightarrow\) 209 note-72

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

Howard-Rubin Number Statement
181:

\(C(2^{\aleph_0},\infty)\): Every set \(X\) of non-empty sets such that \(|X|=2^{\aleph_0}\) has a choice function.

8:

\(C(\aleph_{0},\infty)\):

24:

\(C(\aleph_0,2^{(2^{\aleph_0})})\): Every denumerable collection of non-empty sets each with power \(2^{(2^{\aleph_{0}})}\) has a choice function.

26:

\(UT(\aleph_{0},2^{\aleph_{0}},2^{\aleph_{0}})\): The union of denumerably many sets each of power \(2^{\aleph _{0}}\) has power \(2^{\aleph_{0}}\).

209:

There is an ordinal \(\alpha\) such that for all \(X\), if \(X\) is a denumerable union of denumerable sets then \({\cal P}(X)\) cannot be partitioned into \(\aleph_{\alpha}\) non-empty sets.

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