Hypothesis: HR 361:
In \(\Bbb R\), the union of a denumerable number of analytic sets is analytic. G. Moore [1982], pp 181 and 325.
Conclusion: HR 217:
Every infinite partially ordered set has either an infinite chain or an infinite antichain.
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal N2\) The Second Fraenkel Model | The set of atoms \(A=\{a_i : i\in\omega\}\) is partitioned into two element sets \(B =\{\{a_{2i},a_{2i+1}\} : i\in\omega\}\). \(\mathcal G \) is the group of all permutations of \( A \) that leave \( B \) pointwise fixed and \( S \) is the set of all finite subsets of \( A \). |
| \(\cal N34\) Pincus' Model X | For each \(q\in\Bbb Q\), let \(C_q=\{a_q,b_q\}\), a pair of atoms and let \(A=\bigcup_{q\in\Bbb Q}C_q\) |
Code: 3
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