Axiom Of Choice

We have the following indirect implication of form equivalence classes:

174-alpha \(\Rightarrow\) 196-alpha given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
174-alpha \(\Rightarrow\) 86-alpha	"Representing multi-algebras by algebras, the axiom of choice and the axiom of dependent choice", Howard, P. 1981, Algebra Universalis
86-alpha \(\Rightarrow\) 196-alpha	Successive large cardinals, Bull Jr., E. L. 1978, Ann. Math. Logic

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.
86-alpha:	\(C(\aleph_{\alpha},\infty)\): If \(X\) is a set of non-empty sets such that \(\|X\| = \aleph_{\alpha }\), then \(X\) has a choice function.
196-alpha:	\(\aleph_{\alpha}\) and \(\aleph_{\alpha+1}\) are not both measurable.

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