Cohen \(\cal M28\): Morris' Model II | Back to this models page
Description: Morris constructs a generic extension of acountable standard model of ZFC in which there is a proper class ofgeneric sets
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. |
| 209 | There is an ordinal \(\alpha\) such that for all \(X\), if \(X\) is a denumerable union of denumerable sets then \({\cal P}(X)\) cannot be partitioned into \(\aleph_{\alpha}\) non-empty sets. |
Historical background: ( The generic sets are added at each level \(R_{\alpha} =\{x: \hbox{ rank of \)x\( is } < \alpha\}\).) In this model, for each ordinal\(\alpha\), there is a set \(X\) such that \(X\) is a denumerable union ofdenumerable sets and \(\cal P(X)\) can be partioned into \(\aleph_{\alpha}\)nonempty sets.
Back