Hypothesis: HR 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 (\(\forall\beta < \aleph_{\alpha}\)) \(\{f(\gamma): \gamma < \beta\}\mathrel R f(\beta)\).
Conclusion: HR 51:
Cofinality Principle: Every linear ordering has a cofinal sub well ordering. Sierpi\'nski [1918], p 117.
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal M40(\kappa)\) Pincus' Model IV | The ground model \(\cal M\), is a model of \(ZF +\) the class form of \(AC\) |
Code: 3
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