Cohen \(\cal M9\): Feferman/Levy Model | Back to this models page
Description: Assume the ground model, \(\cal M\), satisfies \(ZF + GCH\) (the Generalized Continuum Hypothesis)
When the book was first being written, only the following form classes were known to be true in this model:
| Form Howard-Rubin Number | Statement |
|---|---|
| 0 | \(0 = 0\). |
When the book was first being written, only the following form classes were known to be false in this model:
| Form Howard-Rubin Number | Statement |
|---|---|
| 38 | \({\Bbb R}\) is not the union of a countable family of countable sets. |
Historical background: Let \(P\), theset of forcing conditions, be the set of all finite sets \(p\) of triples\((n, i,\alpha)\) that satisfy: (i) \(n\in\omega,\ i\in\omega,\\alpha\in\omega_n\); and (ii) \((n,i,\alpha)\), \((n,i,\beta)\in p\) implies\(\alpha=\beta\). Let \(G\) be a generic subset of \(P\) and let \(\cal M[G]\) bethe corresponding generic extension of \(\cal M\). \(\cal M9\) is a symmetricsubmodel of \(\cal M[G]\). In this model, \(\Bbb R\subseteq\bigcup \{R_n :n\in\omega\}\), where \(|R_n| = \aleph_{n+1}\) for \(n\in\omega\), and\(|\aleph_n| = |\aleph_0|\) for each \(n\in\omega\). Therefore, the set ofreal numbers is a countable union of countable sets (38 is false).
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