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\[(*)\; (\forall x)((x \hbox{ infinite and } x \hbox{ Dedekind finite) implies }{\cal P}(x) \hbox{ is Dedekind finite} )\]
and that Form 9 is "Dedekind finite = finite". Assume (\(*\)) and suppose that \(x\) is a set which
is infinite and not Dedekind infinite. Then by (\(*\)), \(Y = {\cal P}(x)\) is infinite and Dedekind finite. Applying (\(*\)) again,
\(z = {\cal P}({\cal P}(x))\) is infinite and Dedekind finite. But \(\{A_n:n\in\omega\}\)is a denumerable subset of \(z\) where
\(A_{n}=\{t\subseteq x: |t| = n\}\), a contradiction. Conversely assuming Form 9, the
hypotheses of (\(*\)) are never true.
Howard-Rubin number: 8
Type: Result
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