Axiom Of Choice

We have the following indirect implication of form equivalence classes:

43 \(\Rightarrow\) 26 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
43 \(\Rightarrow\) 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]

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

Howard-Rubin Number	Statement
43:	\(DC(\omega)\) (DC), Principle of Dependent Choices: If \(S\) is a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\) then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\). See Tarski [1948], p 96, Levy [1964], p. 136.
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 \(2^{\aleph_{0}}\).

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