Cohen \(\cal M26\): Kanovei's Model I | Back to this models page
Description: Starting with a model of \(ZF + V = L\) and using forcing techniques due to Jensen [1968], Kanovei constructs a model of \(ZF\) in which there is an infinite Dedekind finite set \(A\) of generic reals that is in the class \(\varPi^1_n\), but there are no infinite Dedekind finite subsets of \(\Bbb R\) in the class \(\varSigma^1_n\), where \(n\in\omega\), \(n\ge 2\)
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 |
|---|---|
| 199(\(n\)) | (For \(n\in\omega-\{0,1\}\)) If all \(\varSigma^{1}_{n}\), Dedekind finite subsets of \({}^{\omega }\omega\) are finite, then all \(\varPi^1_n\) Dedekind finite subsets of \({}^{\omega} \omega\) are finite. |
Historical background: Thus,Form 199(\(n\)) is false.
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