Cohen \(\cal M19(\aleph)\): Monro's Model II | Back to this models page
Description: Let \(\cal M\) be a countable transitive model of \(ZF + V = L\) and let \(\aleph\) be a regular cardinal in \(\cal M\)
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 |
|---|---|
| 91 | \(PW\): The power set of a well ordered set can be well ordered. |
| 160 | No Dedekind finite set can be mapped onto an aleph. (See [9 E].) |
Historical background: A forcing condition is a partial function, \(p\), of cardinalityless than \(\aleph\) such that \(p: \aleph\times\aleph \to 2\). Let \(G\) be ageneric subset of the set of forcing conditions and let \(\cal M[G]\) be thecorresponding generic extension. Let \(p(\mu) = \{(\nu,i): (\mu,\nu,i)\inp\}\), for each \(\mu < \aleph\), let \(G_\mu=\{p(\mu): p\in G\}\), and \(G^*=\{G_\mu: \mu < \aleph\}\). Then \(\cal M19(\aleph)\) is the set of all setsin \(\cal M[G]\) that are constructible from \(G^*\). Monro proves that \(\calM19(\aleph)\) has the same ordinals and alephs as \(\cal M\) and that \(G^*\)is an infinite Dedekind finite set in \(\cal M19(\aleph)\). He also provesthat if \(x\) is any set that cannot be well ordered, then x can be mappedonto \(\aleph\). Thus, \(G^*\) can be mapped onto \(\aleph\) soForm 160 isfalse.
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