Cohen \(\cal M21\): Felgner's Model II | Back to this models page
Description: Suppose \(\cal M \models ZF + V = L\). Define \(B=\{f: (\exists\alpha <\omega_1)f:\alpha\to\omega\}\)
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 |
|---|---|
| 8 | \(C(\aleph_{0},\infty)\): |
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. |
| 163 | Every non-well-orderable set has an infinite, Dedekind finite subset. |
| 231 | \(UT(WO,WO,WO)\): The union of a well ordered collection of well orderable sets is well orderable. |
Historical background: \(\cal M21\)is constructed using forcing conditions which are finite partial functionsfrom \(\omega \times B\) to \(\{0,1\}\). In this model, there is a wellordered set of well orderable sets whose union cannot be well ordered (231is false), but the Axiom of Choice for a denumerable set (8) is true. Itfollows from Brunner [1982a] that in this model there is a setthat cannot be well ordered and does not have an infinite Dedekind finitesubset, (163 is false). (Form 8 plusForm 163 iff AC.)
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