Axiom Of Choice

We have the following indirect implication of form equivalence classes:

174-alpha \(\Rightarrow\) 154 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 \(\Rightarrow\) 154	Kategoriesatze und multiples Auswahlaxiom, Brunner, N. 1983c, Z. Math. Logik Grundlagen Math.

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.
154:	Tychonoff's Compactness Theorem for Countably Many \(T_2\) Spaces: The product of countably many \(T_2\) compact spaces is compact.

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