Implications

We have the following indirect implication of form equivalence classes:

87-alpha \(\Rightarrow\) 183-alpha
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
87-alpha \(\Rightarrow\) 71-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
71-alpha \(\Rightarrow\) 183-alpha clear

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 (\(\forall\beta < \aleph_{\alpha}\)) \(\{f(\gamma): \gamma < \beta\}\mathrel R f(\beta)\).

71-alpha:  

\(W_{\aleph_{\alpha}}\): \((\forall x)(|x|\le\aleph_{\alpha }\) or \(|x|\ge \aleph_{\alpha})\). Jech [1973b], page 119.

183-alpha:

There are no \(\aleph_{\alpha}\) minimal  sets.  That is, there are no sets \(X\) such that

  1. \(|X|\) is incomparable with \(\aleph_{\alpha}\)
  2. \(\aleph_{\beta}<|X|\) for every \(\beta < \alpha \) and
  3. \(\forall Y\subseteq X, |Y|<\aleph_{\alpha}\) or \(|X-Y| <\aleph_{\alpha}\).

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