Cohen \(\cal M31\): Szczepaniak's Model | Back to this models page
Description: Two models of \(ZF\), \(M_1\) and \(M_2\),\(M_1\subseteq M_2\), are constructed with the same ordinals (so \(L^1\) in\(M_1\) is the same as \(L^1\) in \(M_2\)), and a generic real \(a\in M_1\) so that \(a\not\in HOD\) in \(M_1\), but \(a\in HOD\) in \(M_2\)
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. |
| 230 | \(L^{1} = HOD\). |
Historical background: Thus, in at leastone of these models \(L^1\not=HOD\) and that is the model we call \(\calM31\). (See Note 82 for definitions.)
Back