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Definition: \(|x|\le^*|y|\) (or \(x\precsim^* y\)) if \(y\) can be mapped onto \(x\).
Theorem: [40 B] implies Form 208.
Proof: [40 B] is "For all ordinals \(\alpha\) and all sets \(x\) if \(\aleph_{\alpha}\le^* |x|\), then \(\aleph_{\alpha}\le |x|\)." and Form 208 is "\(2^{\aleph_{\alpha}}\ge\aleph_{\alpha+1}\)" (Form 208 depends on the ordinal \(\alpha\)).) Assume [40 B]. As is shown in Kunen [1980], chapter 1, \(|\aleph_{\alpha}\times\aleph_{\alpha}| =\aleph_{\alpha }\) and therefore, \(|{\cal P}(\aleph_{\alpha}\times\aleph_{\alpha})| = |{\cal P}(\aleph_{\alpha})|\). The function \(f\) from \({\cal P} (\aleph_{\alpha}\times\aleph_{\alpha })\) defined by \(f(R)\) = the order type of \(R\), if \(R\) is a well ordering and \(f(R) = 0\) otherwise is onto \(\aleph_{\alpha +1}\). It follows that \(\aleph_{\alpha+1}\le^*| {\cal P}(\aleph_{\alpha})|\) and therefore by [40 B], \(\aleph_{\alpha+1} \le |{\cal P}(\aleph_{\alpha})|\).
Howard-Rubin number: 69
Type: Proof
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