Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
113 \(\Rightarrow\) 8	Tychonoff's theorem implies AC, Kelley, J.L. 1950, Fund. Math. Products of compact spaces in the least permutation model, Brunner, N. 1985a, Z. Math. Logik Grundlagen Math.
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
113:	Tychonoff's Compactness Theorem for Countably Many Spaces: The product of a countable set of compact spaces is compact.
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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