Hypothesis: HR 0: \(0 = 0\).
Conclusion: HR 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.
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal M26\) Kanovei's Model I | Starting with a model of \(ZF + V = L\) and using forcing techniques due to <a href="/excerpts/Jensen-1968">Jensen [1968]</a>, 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 \(n\in\omega\), \(n\ge 2\) |
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