Description:
Form 144 is true in \(\cal N14\),\(\cal N15\), \(\cal N17\), \(\cal N18\), \(\cal N36(\beta)\), \(\cal N37\), and \(\cal N41\).
Content:
A proof that
Form 144 is true in \(\cal N14\),\(\cal N15\), \(\cal N17\), \(\cal N18\), \(\cal N36(\beta)\), \(\cal N37\), and \(\cal N41\).
We will prove that form [144 B], the set induction principle, is true in \(\cal N14\). The proof can be modified to show that Form 413 is true in the other models listed above. It suffices to show that in \(\cal N41\), every set \(S\) is the union of a family \(Y\) of well ordered sets such that \(Y\) is well ordered by \(\subseteq\). Assume \(S\) is a set in \(\cal N41\). For each \(m\in\omega\), let \(S_m = \{ z\in S : z\hbox{ has a support contained in }\bigcup_{n\le m} B_n\}\). (See the description of \(\cal N41\) for the definition of \(B_n\).) Each element of \(S_m\) has support\(\bigcup_{n\le m} B_n\) and therefore \(S_m\) is well ordered in \(\cal N41\). Further, \(S_m\) has empty support and therefore the ordering \(\le*\) on \(Y = \{ S_m : m\in\omega\}\) defined by\(S_m\le* S_k\) if and only if \(m\le k\) is in the model. But this ordering has order type \(\omega\). Since it is clear that \(S= \bigcup Y\) the proof is completed.
Howard-Rubin number:
155
Type:
Proof
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