Hypothesis: HR 32:
\(C(\aleph_0,\le\aleph_0)\): Every denumerable set of non-empty countable sets has a choice function.
Conclusion: HR 31:
\(UT(\aleph_{0},\aleph_{0},\aleph_{0})\): The countable union theorem: The union of a denumerable set of denumerable sets is denumerable.
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal M12(\aleph)\) Truss' Model I | This is a variation of Solovay's model, <a href="/models/Solovay-1">\(\cal M5(\aleph)\)</a> in which \(\aleph\) is singular |
| \(\cal M20\) Felgner's Model I | Let \(\cal M\) be a model of \(ZF + V = L\). Felgner defines forcing conditions that force \(\aleph_{\omega}\) in \(\cal M\) to be \(\aleph_1\) |
| \(\cal N18\) Howard's Model I | Let \(B= {B_n: n\in\omega}\) where the \(B_n\)'sare pairwise disjoint and each is countably infinite and let \(A=\bigcup B\) |
Code: 3
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