Axiom Of Choice

We have the following indirect implication of form equivalence classes:

44 \(\Rightarrow\) 38 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
44 \(\Rightarrow\) 39	The Axiom of Choice, Jech, 1973b, page 120 theorem 8.1
39 \(\Rightarrow\) 8	clear
8 \(\Rightarrow\) 16	clear
16 \(\Rightarrow\) 6	L’axiome de M. Zermelo et son rˆole dans la th´eorie des ensembles et l’analyse, Sierpi'nski, W. 1918, Bull. Int. Acad. Sci. Cracovie Cl. Math. Nat.
6 \(\Rightarrow\) 5	L’axiome de M. Zermelo et son rˆole dans la th´eorie des ensembles et l’analyse, Sierpi'nski, W. 1918, Bull. Int. Acad. Sci. Cracovie Cl. Math. Nat.
5 \(\Rightarrow\) 38	Non-constructive properties of the real numbers, Howard, P. 2001, Math. Logic Quart.

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

Howard-Rubin Number	Statement
44:	\(DC(\aleph _{1})\): Given a relation \(R\) such that for every subset \(Y\) of a set \(X\) with \(\|Y\| < \aleph_{1}\) there is an \(x \in X\) with \(Y \mathrel R x\), then there is a function \(f: \aleph_{1} \rightarrow X\) such that \((\forall\beta < \aleph_{1}) (\{f(\gamma ): \gamma < b \} \mathrel R f(\beta))\).
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:	\(C(\aleph_{0},\infty)\):
16:	\(C(\aleph_{0},\le 2^{\aleph_{0}})\): Every denumerable collection of non-empty sets each with power \(\le 2^{\aleph_{0}}\) has a choice function.
6:	\(UT(\aleph_0,\aleph_0,\aleph_0,\Bbb R)\): The union of a denumerable family of denumerable subsets of \({\Bbb R}\) is denumerable.
5:	\(C(\aleph_0,\aleph_0,\Bbb R)\): Every denumerable set of non-empty denumerable subsets of \({\Bbb R}\) has a choice function.
38:	\({\Bbb R}\) is not the union of a countable family of countable sets.

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