Cohen \(\cal M35(\epsilon)\): David's Model | Back to this models page
Description: In Cohen's model \(\cal M1\), define sets \(B_n=\{x\subset\omega: |x\ \Delta\ a_n| <\omega\vee |x\ \Delta\ \omega-a_n| \le\omega\}\) (where \(\Delta\) is the symmetric difference)
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. |
| 179-epsilon | Suppose \(\epsilon > 0\) is an ordinal. \(\forall x\), \(x\in W(\epsilon\)). |
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. |
| 179-epsilon | Suppose \(\epsilon > 0\) is an ordinal. \(\forall x\), \(x\in W(\epsilon\)). |
Historical background: Let\(C\) be the function on \(\omega\) such that for each \(n\in\omega\),\(C(n)=B_n\). The symmetric submodel, \(N\), of \(\cal M1\) which is the set ofall sets that are hereditarily ordinal definable from \(\{C\}\cup \{a_n:n\in\omega\}\) has the property that \((\forall x\in N)(x\in W(1))\), but itis not the case that all sets are in \(W(0)\), the set of all well orderablesets. (See Note 25.) David proves that this procedure can be generalizedso that for any ordinal \(\epsilon\) there is a model, \(\cal M35(\epsilon)\),such that for each set \(x\) in the model, \(x\in W(\epsilon)\), and\(\epsilon\) is the smallest ordinal with this property. Thus, form179(\(\epsilon\)) is true, so every set is almost well orderable (144) istrue, butForm 179(\(\beta\)), \(\beta < \epsilon\), is false.
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