Hypothesis: HR 71-alpha:
\(W_{\aleph_{\alpha}}\): \((\forall x)(|x|\le\aleph_{\alpha }\) or \(|x|\ge \aleph_{\alpha})\). Jech [1973b], page 119.
Conclusion: HR 388:
Every infinite branching poset (a partially ordered set in which each element has at least two lower bounds) has either an infinite chain or an infinite antichain.
List of models where hypothesis is true and the conclusion is false:
| Name | Statement |
|---|---|
| \(\cal N16(\aleph_{\gamma})\) Levy's Model II | This is an extension of\(\cal N16\) in which \(A\) has cardinality \(\aleph_{\gamma}\) wherecf\((\aleph_{\gamma}) = \aleph_0\); \(\cal G\) is the group of allpermutations on \(A\); and \(S\) is the set of all subsets of \(A\) ofcardinality less that \(\aleph_{\gamma}\) |
| \(\cal N21(\aleph_{\alpha+1})\) Jensen's Model | We assume \(\aleph_{\alpha+1}\) is a regular cardinal |
Code: 3
Comments: