This non-implication,
Form 412 \( \not \Rightarrow \)
Form 2,
whose code is 4, is constructed around a proven non-implication as follows:
| Hypothesis | Statement |
|---|---|
| Form 87-alpha | <p> \(DC(\aleph_{\alpha})\): Given a relation \(R\) such that for every subset \(Y\) of a set \(X\) with \(|Y|<\aleph_{\alpha}\), there is an \(x\in X\) with \(Y\mathrel R x\) then there is a function \(f:\aleph_{\alpha}\to X\) such that (\(\forall\beta < \aleph_{\alpha}\)) \(\{f(\gamma): \gamma < \beta\}\mathrel R f(\beta)\). </p> |
| Conclusion | Statement |
|---|---|
| Form 3 | \(2m = m\): For all infinite cardinals \(m\), \(2m = m\). |
The conclusion Form 412 \( \not \Rightarrow \) Form 2 then follows.
Finally, the
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal N2(\aleph_{\alpha})\) Jech's Model | This is an extension of \(\cal N2\) in which \(A=\{a_{\gamma} : \gamma\in\omega_{\alpha}\}\); \(B\) is the corresponding set of \(\aleph_{\alpha}\) pairs of elements of \(A\); \(\cal G\)is the group of all permutations on \(A\) that leave \(B\) point-wise fixed;and \(S\) is the set of all subsets of \(A\) of cardinality less than\(\aleph_{\alpha}\) |
| \(\cal N12(\aleph_{\alpha})\) A generalization of \(\cal N12(\aleph_1)\).Replace ``\(\aleph_1\)'' by ``\(\aleph_{\alpha}\)'' where \(\aleph_{\alpha}\) isa singular cardinal | Thus, \(|A|=\aleph_{\alpha}\); \(\cal G\) is the groupof all permutations on \(A\); and \(S\) is the set of all subsets of \(A\) withcardinality less than \(\aleph_{\alpha}\) |