Cohen \(\cal M8\): Apter's Model | Back to this models page
Description: Suppose \(\cal M \models ZFC +\) "There arecardinals \(\kappa \le \delta \le \lambda\) such that \(\kappa\) is a supercompact limit of supercompact cardinals, \(\lambda\) is a measurable cardinal, and \(\delta\) is \(\lambda\) supercompact." (See, for example, Drake [1974], Kanamori/Magador [1978], or Solovay/Reinhardt/Kanamori [1978] for information about large cardinals.) \(\cal M8\) is constructed by first forcing over the ground model \(\cal M\), constructing an inner model \(\cal M'\), doing an additional forcing argument over \(\cal M'\), and then constructing the final inner model \(\cal M8\)
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 |
|---|---|
| 8 | \(C(\aleph_{0},\infty)\): |
| 25 | \(\aleph _{\beta +1}\) is regular for all ordinals \(\beta\). |
| 91 | \(PW\): The power set of a well ordered set can be well ordered. |
Historical background: In this model \(\aleph_n\), for \(3\le n\le \omega\) aresingular cardinals so 25 (\(\aleph_{\alpha+1}\) is regular for all ordinals\(\alpha\).) is false. Also, the Axiom of Choice for a denumerable number ofsets, 8, is false. (Supercompact and measurable cardinals are used toconstruct the model.)
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