Description:
Form 207\((\alpha)\) implies Form 209
Content:
We prove that
Form 207(\(\beta\)) implies Form 209 for any ordinal \(\beta\). (Form 207(\(\beta\)) is \(UT(\aleph_{\beta},\aleph_{\beta },\lt 2^{\aleph_{\beta}})\) and Form 209 is \((\exists\) an ordinal\(\alpha)(\forall X)\) (If \(X\) is a denumerable union of denumerable sets then \({\cal P}(X)\) cannot be partitioned into
\(\aleph_{\alpha}\)non-empty sets.))
Assume Form 207\((\beta )\) is true. Then for any set \(X\), if \(X\) is a denumerable union of denumerable sets, then \(|X|<2^{\aleph_{\beta }}\). Choose an ordinal \(\alpha\) so that \(\neg(\aleph_{\alpha }\le| {\cal P}({\cal P}(2^{\aleph_{\beta }}))| )\). For any \(X\) which is the denumerable union of denumerable sets we have \(|{\cal P}({\cal P}(X))|\le| {\cal P}({\cal P}(2^{\aleph_{\beta }}))|\). A partition of \({\cal P}(X)\) into \(\aleph _{\alpha}\) non-empty sets would give \(\aleph_{\alpha }\le |{\cal P}({\cal P}(X))|\).
Similarly, Form 26 implies Form 209 and Form 31 implies Form 209.
Howard-Rubin number:
72
Type:
Theorem
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