Cohen \(\cal M15\): Feferman/Blass Model | Back to this models page
Description: Blass constructs a model similar to Feferman's model, \(\cal M2\)
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 |
|---|---|
| 144 | Every set is almost well orderable. |
| 253 | \L o\'s' Theorem: If \(M=\langle A,R_j\rangle_{j\in J}\) is a relational system, \(X\) any set and \({\cal F}\) an ultrafilter in \({\cal P}(X)\), then \(M\) and \(M^{X}/{\cal F}\) are elementarily equivalent. |
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 |
|---|---|
| 206 | The existence of a non-principal ultrafilter: There exists an infinite set \(X\) and a non-principal ultrafilter on \(X\). |
Historical background: Suppose \(\cal M\models\) ZF + V = L. Byforcing with finite partial functions from \(\omega\times\omega\) into 2, ageneric extension \(\cal M[G]\) of \(\cal M\) is obtained that contains an\(\omega\)-sequence of generic reals \(a_n\subseteq\omega\). For each\(x\subseteq\omega\), let \(\delta(x)\) be the set of reals whose symmetricdifference with \(x\) is finite. Let \(f\) be the function in \(\cal M[G]\) suchthat for all \(n\in\omega\), \(f(n) = \{\delta(a_n), \delta(\omega - a_n)\}\).Let \(S=\bigcup_{n\in\omega} (\delta(a_n)\cup\delta(\omega - a_n))\cup\{f\}\). \(\cal M15\) is the submodel of \(\cal M[G]\) that is hereditarilyordinal definable from \(S\) and \(\bigcup S\). In this model, every set isalmost well orderable (144 is true) and all ultrafilters are principal(206 is false). Howard has shown that ifForm 206 is false, then \L os'Theorem (253) is true.
Back