Hypothesis: HR 191:
\(SVC\): There is a set \(S\) such that for every set \(a\), there is an ordinal \(\alpha\) and a function from \(S\times\alpha\) onto \(a\).
Conclusion: HR 18:
\(PUT(\aleph_{0},2,\aleph_{0})\): The union of a denumerable family of pairwise disjoint pairs has a denumerable subset.
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal M7\) Cohen's Second Model | There are two denumerable subsets\(U=\{U_i:i\in\omega\}\) and \(V=\{V_i:i\in\omega\}\) of \(\cal P({\Bbb R})\)(neither of which is in the model) such that for each \(i\in\omega\), \(U_i\)and \(V_i\) cannot be distinguished in the model |
| \(\cal N2\) 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 \( A \). |
Code: 3
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