Hypothesis: HR 0: \(0 = 0\).
Conclusion: HR 81-n:
(For \(n\in\omega\)) \(K(n)\): For every set \(S\) there is an ordinal \(\alpha\) and a one to one function \(f: S \rightarrow {\cal P}^{n}(\alpha)\). (\({\cal P}^{0}(X) = X\) and \({\cal P}^{n+1}(X) = {\cal P}({\cal P}^{n}(X))\). (\(K(0)\) is equivalent to [1 AC] and \(K(1)\) is equivalent to the selection principle (Form 15)).
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal M16(n)\) Monro's Model I | Monro has constructed an \(\omega\)-sequence of models of \(ZF\) such that, <ul type="none"> <li>(*) For each ordinal \(\alpha\), \(\cal P^n(\alpha)\) is the same in the \(n\)th model and the \(n+1\)st model.</li> </ul> (\(\cal P^0(\alpha)=\alpha,\) and \(\cal P^{n+1}(\alpha)=\cal P(\cal P^n(\alpha))\).) He starts with a countable transitive model \(\cal M\models ZF + V = L\) |
Code: 3
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