Hypothesis: HR 3: \(2m = m\): For all infinite cardinals \(m\), \(2m = m\).
Conclusion: HR 1:
\(C(\infty,\infty)\): The Axiom of Choice: Every set of non-empty sets has a choice function.
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal M6\) Sageev's Model I | Using iterated forcing, Sageev constructs \(\cal M6\) by adding a denumerable number of generic tree-like structuresto the ground model, a model of \(ZF + V = L\) |
| \(\cal N9\) Halpern/Howard Model | \(A\) is a set of atoms with the structureof the set \( \{s : s:\omega\longrightarrow\omega \wedge (\exists n)(\forall j > n)(s_j = 0)\}\) |
Code: 3
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