Implications

Hypothesis: HR 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.

Conclusion: HR 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 \(|X-Y| <\aleph_{\alpha}\).

List of models where hypothesis is true and the conclusion is false:

Name Statement
\(\cal N27\) Hickman's Model II Let \(A\) be a set with cardinality\(\aleph_1\) such that \(A=\{(a_{\alpha},b_{\beta}) : \alpha < \omega, \beta< \omega_1\}\)

Code: 3

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