Hypothesis: HR 350:
\(MC(\aleph_0,\aleph_0)\): For every denumerable set \(X\) of non-empty denumerable sets there is a function \(f\) such that for all \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\).
Conclusion: HR 171:
If \((P,\le)\) is a partial order such that \(P\) is the denumerable union of finite sets and all antichains in \(P\) are finite then for each denumerable family \({\cal D}\) of dense sets there is a \({\cal D}\) generic filter.
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal N6\) Levy's Model I | \(A=\{a_n : n\in\omega\}\) and \(A = \bigcup \{P_n: n\in\omega\}\), where \(P_0 = \{a_0\}\), \(P_1 = \{a_1,a_2\}\), \(P_2 =\{a_3,a_4,a_5\}\), \(P_3 = \{a_6,a_7,a_8,a_9,a_{10}\}\), \(\cdots\); in generalfor \(n>0\), \(|P_n| = p_n\), where \(p_n\) is the \(n\)th prime |
Code: 3
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