Cohen \(\cal M7\): Cohen's Second Model | Back to this models page
Description: There are two denumerable subsets\(U=\{U_i:i\in\omega\}\) and \(V=\{V_i:i\in\omega\}\) of \(\cal P({\Bbb R})\)(neither of which is in the model) such that for each \(i\in\omega\), \(U_i\)and \(V_i\) cannot be distinguished in the model
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 |
|---|---|
| 191 | \(SVC\): There is a set \(S\) such that for every set \(a\), there is an ordinal \(\alpha\) and a function from \(S\times\alpha\) onto \(a\). |
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 |
|---|---|
| 18 | \(PUT(\aleph_{0},2,\aleph_{0})\): The union of a denumerable family of pairwise disjoint pairs has a denumerable subset. |
| 91 | \(PW\): The power set of a well ordered set can be well ordered. |
| 340 | Every Lindelöf metric space is separable. |
| 341 | Every Lindelöf metric space is second countable. |
| 389 | \(C(\aleph_0,2,\cal P({\Bbb R}))\): Every denumerable family of two element subsets of \(\cal P({\Bbb R})\) has a choice function. \ac{Keremedis} \cite{1999b}. |
Historical background: Therefore, in this modelthere is a denumerable set of pairs of subsets of \(\Bbb R\) whose union hasno denumerable subset, soForm 18 is false, and \(C(\aleph_0,2,\cal P({\BbbR}))\) (389) is also false. Blass [1979] has shown that in everysymmetric model there is a set \(X\) such that for each set \(a\), there isan ordinal \(\alpha\) and a function \(f\) mapping \(X\times\alpha\) onto \(a\)(191 is true). (See Note 59.) It is shown in Good/Tree [1995]that in this model there is a Lindel\"of metric space that is neithersecond countable nor separable (340 and 341 are false).
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