Description: For all \(n\in\omega\), \(\neg (\hbox{ZF} \vdash K(n+1) \rightarrow K(n) )\)
Content:
Monro [1972] shows that for all \(n\in\omega\), \(\neg (\hbox{ZF} \vdash K(n+1) \rightarrow K(n) )\). \(K(n)\) is Form 81(\(n\)): 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 [1 E] and \(K(1)\) is [15 A]. Monro [1972] also says that in an unpublished model due to H. Stewart \(K(2)\) holds, Form 85 (\(C(\infty,\aleph_{0})\)) fails and Form 64 (there is no amorphous set) fails. (\(A\) is amorphous if \(A\) is infinite and is not the union of two non-empty, disjoint infinite sets.)
Howard-Rubin number: 90
Type: Summary
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