Cohen \(\cal M2(\kappa)\): Feferman/Pincus Model | Back to this models page
Description: This is an extension of \(\cal M2\) in which there are \(\kappa\) generic sets, where \(\kappa\) is a regular cardinal
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 |
|---|---|
| 222 | There is a non-principal measure on \(\cal P(\omega)\). |
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 |
|---|---|
| 52 | Hahn-Banach Theorem: If \(V\) is a real vector space and \(p: V \rightarrow {\Bbb R}\) satisfies \(p(x+y) \le p(x) + p(y)\) and \((\forall t > 0)( p(tx) = tp(x) )\) and \(S\) is a subspace of \(V\) and \(f:S \rightarrow {\Bbb R}\) is linear and satisfies \((\forall x \in S)( f(x) \le p(x) )\) then \(f\) can be extended to \(f^{*} : V \rightarrow {\Bbb R}\) such that \(f^{*}\) is linear and \((\forall x \in V)(f^{*}(x) \le p(x))\). |
| 91 | \(PW\): The power set of a well ordered set can be well ordered. |
Historical background: (\(\cal M2\) is \(\cal M2(\aleph_0)\).) Pincus proves that theHahn-Banach Theorem (52) is false in this model. If \(\kappa\) issufficiently large, then there is a non-trivial measure on \(\calP(\omega)\) which is 0 on finite sets (222 is true). \(\kappa\) has to belarge enough so thatForm 222 is boundable in sets of rank \(\mu\), where\(\mu\) is less than \(\kappa\).
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