Axiom Of Choice

We have the following indirect implication of form equivalence classes:

174-alpha \(\Rightarrow\) 19 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
174-alpha \(\Rightarrow\) 43	"Representing multi-algebras by algebras, the axiom of choice and the axiom of dependent choice", Howard, P. 1981, Algebra Universalis
43 \(\Rightarrow\) 8	clear
8 \(\Rightarrow\) 27	clear
27 \(\Rightarrow\) 31	clear
31 \(\Rightarrow\) 34	clear
34 \(\Rightarrow\) 19	Sur les fonctions representables analytiquement, Lebesgue, H. 1905, J. Math. Pures Appl.

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

Howard-Rubin Number	Statement
174-alpha:	\(RM1,\aleph_{\alpha }\): The representation theorem for multi-algebras with \(\aleph_{\alpha }\) unary operations: Assume \((A,F)\) is a multi-algebra with \(\aleph_{\alpha }\) unary operations (and no other operations). Then there is an algebra \((B,G)\) with \(\aleph_{\alpha }\) unary operations and an equivalence relation \(E\) on \(B\) such that \((B/E,G/E)\) and \((A,F)\) are isomorphic multi-algebras.
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.
34:	\(\aleph_{1}\) is regular.
19:	A real function is analytically representable if and only if it is in Baire's classification. G.Moore [1982], equation (2.3.1).

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