Cohen \(\cal M24\): Blass' Model | Back to this models page
Description: Let \(\cal M\) be a countable transitive model of \(ZFC + V = L\)
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. |
| 190 | There is a non-trivial injective Abelian group. |
Historical background: For each regular cardinal \(\xi\) adjoin generic sets\(b^i_{\xi}\subseteq\xi\) for each \(i\in{\Bbb Z}\) using Easton forcing. (SeeEaston [1970]. There is a proper class of generic sets.) callthe resulting forcing extension \(\cal M'\). In \(\cal M'\), let \(a^i_\xi\) bethe set of all subsets of \(\xi\) whose symmetric difference with\(b^i_{\xi}\) is a constructible set of cardinality less than \(\xi\). Foreach regular \(\xi\) and each \(i\in\Bbb Z\), define \(S(a^i_\xi) =a^{i+1}_\xi\). Let \(\cal M24\) be the submodel of \(\cal M'\) consisting ofall sets that are hereditarily ordinal definable from \(S\) and the\(b^i_{\xi}\)'s. Blass [1979] has shown that in this model thereis no non-trivial injective Abelian group. (See Note 60 for definitions.)
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