Axiom Of Choice

We have the following indirect implication of form equivalence classes:

192 \(\Rightarrow\) 32 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
192 \(\Rightarrow\) 43	Injectivity, projectivity and the axiom of choice, Blass, A. 1979, Trans. Amer. Math. Soc.
43 \(\Rightarrow\) 8	clear
8 \(\Rightarrow\) 27	clear
27 \(\Rightarrow\) 31	clear
31 \(\Rightarrow\) 32	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.

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

Howard-Rubin Number	Statement
192:	\(EP\) sets: For every set \(A\) there is a projective set \(X\) and a function from \(X\) onto \(A\).
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:	\(C(\aleph_{0},\infty)\):
27:	\((\forall \alpha)( UT(\aleph_{0},\aleph_{\alpha}, \aleph_{\alpha}))\): The union of denumerably many sets each of power \(\aleph_{\alpha }\) has power \(\aleph_{\alpha}\). Moore, G. [1982], p 36.
31:	\(UT(\aleph_{0},\aleph_{0},\aleph_{0})\): The countable union theorem: The union of a denumerable set of denumerable sets is denumerable.
32:	\(C(\aleph_0,\le\aleph_0)\): Every denumerable set of non-empty countable sets has a choice function.

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