Cohen \(\cal M41\): Kanovei's Model III | Back to this models page
Description: Let \(\Bbb P\) be the set of conditions from the model in Jensen [1968]
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 |
|---|---|
| 194 | \(C(\varPi^1_2)\) or \(AC(\varPi^1_2)\): If \(P\in \omega\times{}^{\omega}\omega\), \(P\) has domain \(\omega\), and \(P\) is in \(\varPi^1_2\), then there is a sequence of elements \(\langle x_{k}: k\in\omega\rangle\) of \({}^{\omega}\omega\) with \(\langle k,x_{k}\rangle \in P\) for all \(k\in\omega\). Kanovei [1979]. |
Historical background: \(\Bbb P\) is a subset of theset of all constructible perfect trees. \(\cal M41\) is the symmetric partof \(L[\boldsymbol\alpha]\) where \(\boldsymbol\alpha = \langle \alpha_n :n\in\omega \rangle\) is \(\Bbb P^\omega\) generic. In Kanovei [1976] it is shown that there is a \(\varPi^1_2\) subset \(P\) of\(\omega\times{}^\omega\omega\) with \(\hbox{dom }P = \omega\) and for whichno sequence \(\langle x_n : n\in\omega \rangle\) with \(\langle k,x_k \rangle\in P\) for all \(k \in\omega\) exists (form 194 is false).
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