Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
183-alpha \(\Rightarrow\) 183-alpha

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

Howard-Rubin Number Statement
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 \(|X-Y| <\aleph_{\alpha}\).
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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