Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
87-alpha \(\Rightarrow\) 86-alpha	The interdependence of certain consequences of the axiom of choice, Levy, A. 1964, Fund. Math. The Axiom of Choice, Jech, 1973b, page 120 theorem 8.1 note-9
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
87-alpha:	\(DC(\aleph_{\alpha})\): Given a relation \(R\) such that for every subset \(Y\) of a set \(X\) with \(\|Y\|<\aleph_{\alpha}\), there is an \(x\in X\) with \(Y\mathrel R x\) then there is a function \(f:\aleph_{\alpha}\to X\) such that (\(\forall\beta < \aleph_{\alpha}\)) \(\{f(\gamma): \gamma < \beta\}\mathrel R f(\beta)\).
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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